Treatise on Analysis I-VIII (8-Vols all in one) 0122155505, 0122155025, 0122155033, 0122155041, 012215505X, 0122155068, 0122155076, 0122155084

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F O U N D A T I O N S OF MODERN ANALYSIS

This is a volume in PURE AND APPLIED MATHEMATICS A series of Monographs and Textbooks Editors: PAULA. SMITHAND SAMUEL EILENBERG A list of recent titles in this series appears at the end of this volume.

Volume 10 TREATISE ON ANALYSIS 10-1. Chapters I-XI, Foundations of Modern Analysis, enlarged and corrected printing, 1969 10-11. Chapters XII-XV, enlarged and corrected printing, 1976 10-111. Chapters XVI-XVII, 1972 10-IV. Chapters XVIII-XX, 1974 10-V. Chapter XXI, 1977 10-VI. Chapters XXII, 1978

FOUNDATIONS OF

MODERN ANALYSIS Enlarged and Corrected Printing

J. DIEUDONNE Universitt de Nice Facultt des Sciences Parc Valrose, Nice, France

ACADEMIC PRESS

N e w York and London

A Subsidiary o f Harcourt Brace Jovanovlch, Publishers

1969

0

COPYRIGHT 1960, 1969, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED N O PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New York. New York 10003

United Kingdom Edition published by ACADEMIC P R E S S , INC. (LONDON) LTD. 24/28 Oval Road. London NWl

LIBRARY OF CONGRESS CATALOG CARD NUMBER: 60-8049 T H I R D(ENLARGEDA N D C O R R E C T E D )

PRINTING

AMS 1968 Subject Classification 0001

PRINTED IN THE UNITED STATES OF AMERICA

PREFACE TO THE ENLARGED A N D CORRECTED PRINTING

This book is the first volume of a treatise which will eventually consist of four volumes. It is also an enlarged and corrected printing, essentially without changes, of my Foundations of Modern Analysis,” published in 1960. Many readers, colleagues, and friends have urged me to write a sequel to that book, and in the end I became convinced that there was a place for a survey of modern analysis, somewhere between the minimum tool kit ” of an elementary nature which I had intended to write, and specialist monographs leading to the frontiers of research. My experience of teaching has also persuaded me that the mathematical apprentice, after taking the first step of “ Foundations,” needs further guidance and a kind of general bird’s eye-view of his subject before he is launched onto the ocean of mathematical literature or set on the narrow path of his own topic of research. Thus 1 have finally been led to attempt to write an equivalent, for the mathematicians of 1970, of what the “ Cours d’Analyse ” of Jordan, Picard, and Goursat were for mathematical students between 1880 and 1920. It is manifestly out of the question to attempt encyclopedic coverage, and certainly superfluous to rewrite the works of N. Bourbaki. I have therefore been obliged to cut ruthlessly in order to keep within limits comparable to those of the classical treatises. I have opted for breadth rather than depth, in the opinion that it is better to show the reader rudiments of many branches of modern analysis rather than to provide him with a complete and detailed exposition of a small number of topics. Experience seems to show that the student usually finds a new theory difficult to grasp at a first reading. He needs to return to it several times before he becomes really familiar with it and can distinguish for himself which are the essential ideas and which results are of minor importance, and only then will he be able to apply it intelligently. The chapters of this treatise are “

“

V

vi

PREFACE TO THE ENLARGED AND CORRECTED PRINTING

therefore samples rather than complete theories: indeed, I have systematically tried not to be exhaustive. The works quoted in the bibliography will always enable the reader to go deeper into any particular theory. However, I have refused to distort the main ideas of analysis by presenting them in too specialized a form, and thereby obscuring their power and generality. It gives a false impression, for example, if differential geometry is restricted to two or three dimensions, or if integration is restricted to Lebesgue measure, on the pretext of making these subjects more accessible or “ intuitive.” On the other hand I do not believe that the essential content of the ideas involved is lost, in a first study, by restricting attention to separable metrizable topological spaces. The mathematicians of my own generation were certainly right to banish hypotheses of countability wherever they were not needed: this was the only way to get a clear understanding. But now the situation is well understood: the most central parts of analysis (let us say those which turn on the notion of a finite-dimensional manifold) involve only separable metrizable spaces, in the great majority of important applications. Moreover, there exists a general technique, which is effective and usually easy to apply, for passing from a proof based on hypotheses of countability to a general proof. Broadly speaking, the recipe is to “replace sequences by filters.” Often, it should be said, the result is simply to make the original proof more elegant. At the risk of being reviled as a reactionary I have therefore taken as my motto “only the countable exists at infinity”: I believe that the beginner will do better to concentrate his attention on the real difficulties involved in concepts such as differential manifolds and integration, without having at the same time to worry about secondary topological problems which he will meet rather seldom in practice.? In this text, the whole structure of analysis is built up from the foundations. The only things assumed at the outset are the rules of logic and the usual properties of the natural numbers, and with these two exceptions all the proofs in the text rest on the axioms and theorems proved earlier.$ Nevertheless this treatise (including the first volume) is not suitable for students who have not yet covered the first two years of an undergraduate honours course in mathematics.

t In the same spirit I have abstained (sometimes at the cost of greater length) from the use of transfinite induction in separable metrizable spaces: not in the name of philosophical scruples which are no longer relevant, but because it seems to me to be unethical to ban the uncountable with one hand whilst letting it in surreptitiously with the other. This logical order is not followed so rigorously in the problems and in some of the examples, which contain definitions and results that have not up to that point appeared in the text, or will not appear at all.

PREFACE TO THE ENLARGED AND CORRECTED PRINTING

vii

A striking characteristic of the elementary parts of analysis is the small amount of algebra required. Effectively all that is needed is some elementary linear algebra (which is included in an appendix at the end of the first volume, for the reader’s convenience). However, the role played by algebra increases in the subsequent volumes, and we shall finally leave the reader at the point where this role becomes preponderant, notably with the appearance of advanced commutative algebra and homological algebra. As reference books in algebra we have taken R. Godement’s “Abstract Algebra,”§ and S. A. Lang’s “Algebra ”11 which we shall possibly augment in certain directions by means of appendices. As with the first volume, I have benefited greatly during the preparation of this work from access to numerous unpublished manuscripts of N. Bourbaki and his collaborators. To them alone is due any originality in the presentation of certain topics. Nice, France April, 1969

J. DIEUDONNB

9 Godement, R., “Abstract Algebra.” Houghton-Mifflin, New York, 1968. (Original French edition published by Hermann, Paris, 1963.) 7 Lang, S . A., “Algebra.” Addison-Wesley, Reading, Massachusetts, 1965.

This Page Intentionally Left Blank

PREFACE

This volume is an outgrowth of a course intended for first year graduate students or exceptionally advanced undergraduates in their junior or senior year. The purpose of the course (taught at Northwestern University in 19561957) was twofold: (a) to provide the necessary elementary background for all branches of modern mathematics involving “ analysis ” (which in fact means everywhere, with the possible exception of logic and pure algebra); (b) to train the student in the use of the most fundamental mathematical tool of our time-the axiomatic method (with which he will have had very little contact, if any at all, during his undergraduate years). It will be very apparent to the reader that we have everywhere emphasized the conceptual aspect of every notion, rather than its computational aspect, which was the main concern of classical analysis; this is true not only of the text, but also of most of the problems. We have included a rather large number of problems in order to supplement the text and to indicate further interesting developments. The problems will at the same time afford the student an opportunity of testing his grasp of the material presented. Although this volume includes considerable material generally treated in more elementary courses (including what is usually called “ advanced calculus”) the point of view from which this material is considered is completely different from the treatment it usually receives in these courses. The fundamental concepts of function theory and of calculus have been presented within the framework of a theory which is sufficiently general to reveal the scope, the power, and the true nature of these concepts far better than it is possible under the usual restrictions of “classical analysis.” It is not necessary to emphasize the well-known “ economy of thought ” which results from such a general treatment; but it may be pointed out that there is a corresponding “economy of notation,” which does away with hordes of indices, much in the same way as “ vector algebra ” simplifies classical analytical geometry. This has also as a consequence the necessity of a strict adherence to axiomatic methods, with no appeal whatsoever to geometric intuition,” at least in the formal proofs: a necessity which we have emphasized by deliberately abstaining from introducing any diagram in the book. My opinion is that the “

ix

x

PREFACE

graduate student of today must, as soon as possible, get a thorough training in this abstract and axiomatic way of thinking, if he is ever to understand what is currently going on in mathematical research. This volume aims to help the student to build up this “ intuition of the abstract” which is so essential in the mind of a modern mathematician. It is clear that students must have a good working knowledge of classical analysis before approaching this course. From the strictly logical point of view, however, the exposition is not based on any previous knowledge, with the exception of: 1. The first rules of mathematical logic, mathematical induction, and the fundamental properties of (positive and negative) integers. 2. Elementary linear algebra (over a field) for which the reader may consult Halmos [I I], Jacobson [13], or Bourbaki [4]; these books, however, contain much more material than we will actually need (for instance we shall not use the theory of duality and the reader will know enough if he is familiar with the notions of vector subspace, hyperplane, direct sum, linear mapping, linear form, dimension, and codimension). In the proof of each statement, we rely exclusively on the axioms and on theorems already proved in the text, with the two exceptions just mentioned. This rigorous sequence of logical steps is somewhat relaxed in the examples and problems, where we will often apply definitions or results which have not yet been (or ever will never be) proved in the text. There is certainly room for a wide divergence of opinion as to what parts of analysis a student should learn during his first graduate year. Since we wanted to keep the contents of this book within the limits of what can materially be taught during a single academic year, some topics had to be eliminated. Certain topics were not included because they are too specialized, others because they may require more mathematical maturity than can usually be expected of a first-year graduate student or because the material has undoubtedly been covered in advanced calculus courses. If we were to propose a general program of graduate study for mathematicians we would recommend that every graduate student should be expected to be familiar with the contents of this book, whatever his future field of specialization may be. I would like to express my gratitude to the mathematicians who have helped me in preparing these lectures, especially to H. Cartan and N. Bourbaki, who allowed me access to unpublished lecture notes and manuscripts, which greatly influenced the final form of this book. My best thanks also go to my colleagues in the Mathematics Department of Northwestern University, who made it possible for me to teach this course along the lines 1 had planned and greatly encouraged me with their constructive criticism. April, I960

J. DIEUDONN~

CONTENTS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Preface to the Enlarged and Corrected Printing Preface.

v ix xv

Chapter I

ELEMENTS OF THE THEORY OF SETS

. . . . . . . . . . . . . .

1

1. Elements and sets. 2. Boolean algebra. 3. Product of two sets. 4. Mappings.

5. Direct and inverse images. 6 . Surjective, injective, and bijective mappings. 7. Composition of mappings. 8. Families of elements. Union, intersection, and products of families of sets. Equivalence relations. 9. Denumerable sets. Chapter II

REAL NUMBERS

. . . . . . . . . . . . . . . . . . . . . . . .

16

. . . . . . . . . . . . . . . . . . . . . . . .

27

1. Axioms of the real numbers. 2. Order propertics of the real numbers. 3. Least upper bound and greatest lower bound.

Chapter I I I

METRIC SPACES

I . Distances and metric spaces. 2. Examples of distances. 3. Isometries. 4. Balls, spheres, diameter. 5. Open sets. 6. Neighborhoods. 7. Interior of a set. 8. Closed sets, cluster points, closure of a set. 9. Dense subsets; separable spaces. 10. Subspaces of a metric space. 1 1 . Continuous mappings. 12. Homeomorphisms. Equivalent distances. 13. Limits. 14. Cauchy sequences, complete spaces. 15. Elementary extension theorems. 16. Compact spaces. 17. Compact sets. 18. Locally compact spaces. 19. Connected spaces and connected sets. 20. Product of two metric spaces. Chapter I V

ADDITIONAL PROPERTIES OF THE REAL LINE

. . . . . . . . . .

I . Continuity of algebraic operations. 2. Monotone functions. 3. Logarithms and exponentials. 4. Complex numbers. 5. The Tietze-Urysohn extension theorem.

79

xi

xii

CONTENTS

Chapter V

. . . . . . . . . . . . . . . . . . . . . . .

91

. . . . . . . . . . . . . . . . . . . . . . . .

115

NORMED SPACES

3. Absolutely convergent series. 4. Subspaces and finite products of normed spaces. 5. Condition of continuity of a multilinear mapping. 6. Equivalent norms. 7 . Spaces of continuous multilinear mappings. 8. Closed hyperplanes and continuous linear forms. 9. Finite dimensional normed spaces. 10. Separable normed spaces. 1. Normed spaces and Banach spaces. 2. Series in a normed space.

Chapter V I

HILBERT SPACES

I . Hermitian forms. 2. Positive hermitian forms. 3. Orthogonal projection on a complete subspace. 4. Hilbert sum of Hilbert spaces. 5. Orthonormal systems. 6. Orthonormalization.

Chapter V I I

SPACES OF CONTINUOUS FUNCTIONS . . . . . . . . . . . . . 132 I. Spaces of bounded functions. 2. Spaces of bounded continuous functions. 3. The Stone-Weierstrass approximation theorem. continuous sets. 6. Regulated functions.

Chapter V l l l

DIFFERENTIAL CALCULUS

4. Applications.

5. Equi-

. . . . . . . . . . . . . . . . . . 147

1. Derivative of a continuous mapping. 2. Formal rules of derivation.

3. Derivatives in spaces of continuous linear functions. 4. Derivatives of functions of one variable. 5. The meanvalue theorem. 6. Applications of themeanvalue theorem. 7 . Primitives and integrals. 8. Application: the number e. 9. Partial derivatives. 10. Jacobians. 11. Derivative of an integraldependingonaparameter. 12. Higher derivatives, 13. Differential operators. 14. Taylor’s formula. Chapter I X

ANALYTIC FUNCTIONS

. . . . . . . . . . . . . . . . . . . .

I . Power series. 2. Substitution of power series in a power series. 3. Analytic functions. 4. The principle of analytic continuation. 5. Examples of analytic functions; the exponential function; the number T . 6. Integration along a road. 7. Primitive of an analytic function in a simply connected domain. 8. Index of a point with respect to a circuit. 9. The Cauchy formula. 10. Characterization of analytic functions of complex variables. 11. Liouville’s thoerem. 12. Convergent sequences of analytic functions. 13. Equicontinuous sets of analytic functions. 14. The Laurent series. 15. Isolated singular points; poles; zeros; residues. 16. The theorem of residues. 17. Meromorphic functions.

197

Appendix to Chapter I X

APPLICATION OF ANALYTIC FUNCTIONS TO PLANE TOPOLOGY 1. Index of a point with respect to a loop. 2. Essential mappings in the unit circle.

3. Cuts of the plane.

4. Simple arcs and simple closed curves.

251

CONTENTS Chapter X

EXISTENCE THEOREMS

....................

xiii

264

1. The method of successive approximations. 2. Implicit functions. 3. The rank theorem. 4. Differential equations. 5. Comparison of solutions of differential

equations. 6. Linear differential equations. 7. Dependence of the solution on parameters. 8. Dependence of the solution on initial conditions. 9. The theorem of Frobenius. Chapter XI

ELEMENTARY SPECTRAL THEORY.

. . . . . . . . . . . . . . .

312

1 . Spectrum of a continuous operator. 2. Compact operators. 3. The theory of F. Riesz. 4.Spectrum of a compact operator. 5. Compact operators in Hilbert spaces. 6. The Fredholm integral equation. 7. The Stiirm-Liouville problem. Appendix

ELEMENTS OF LINEAR ALGEBRA. . . . . . . . . . . . . . . . . 358 I. Vector spaces. 2. Linear mappings. 3. Direct sums of subspaces. 4. Bases. Dimension and codimension. 5 . Matrices. 6. Multilinear mappings. Determinants. 7. Minors of a determinant.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References Index.

380 381

This Page Intentionally Left Blank

N OTATl0NS

In the following definitions the first digit refers to the number of the chapter in which the notation occurs and the second to the section within the chapter.

equals: 1.1 is different from: 1.1 is an element of, belongs to: 1.1 is not an element of: 1.1 is a subset of, is contained in: 1.1 contains : 1.1 is not contained in: 1.1 the set of elements of X having property P: 1.1 the empty set: 1.1 the set having a as unique element: 1.1 the set of subsets of X: 1.1 complement of Y in X: 1.2 union: 1.2 intersection : 1.2 ordered pair: 1.3 first and second projection: 1.3 cross sections of G c X x Y : 1.3 product of two sets: 1.3 product of n sets: 1.3 ith projection: 1.3 partial projection: 1.3 product of n sets equal to X: 1.3 value of the mapping F at x: 1.4 xv

xvi

NOTATIONS

set of mappings of X into Y : 1.4 identity mapping of X: 1.4 mapping: 1.4 direct image: 1.5 inverse image : 1.5 inverse image of a one element set ( y } : 1.5 partial mappings of a mapping F of A c X x Y into Z: 1.5 natural injection: 1.6 inverse mapping of a bijective mapping: 1.6 composed mapping: 1.7 family: 1.8 set of natural integers: 1.8 set of elements of a finite sequence: 1.8 union of a family of sets: 1.8 intersection of a family of sets: 1.8 quotient set of a set X by an equivalence relation R : 1.8

product of a family of sets: 1.8

Z 1.u.b. X, sup X g.1.b. X, inf X SUP f ( x > , inf f W

projection on a partial product: 1.8 mapping into a product of sets: 1.8 set of real numbers: 2.1 sum of real numbers: 2.1 product of real numbers: 2.1 element of R: 2.1 opposite of a real number: 2.1 element of R: 2.1 inverse in R: 2.1 order relation in R: 2.1 relation in R: 2.1 intervals in R: 2.1 set of real numbers > O (resp. > O ) : 2.2 absolute value, positive and negative part of a real number: 2.2 set of rational numbers: 2.2 set of positive or negative integers: 2.2 least upper bound of a set: 2.3 greatest lower bound of a set: 2.3 supremum and infimum off in A : 2.3

R

extended real line: 3.3

Q

xsA

XEA

NOTATIONS

xvii

points at infinity in R : 3.3 order relation in E:3.3 X G Y ,y > x distance of two sets: 3.4 d(A,B) B(u; r), B'(a; r), S(a;r ) open ball, closed ball, sphere of center a and radius r : 3.4 diameter: 3.4 interior: 3.7 closure: 3.8 frontier: 3.8 limit of a function: 3.13

+a,--co

limit of a sequence: 3.13 oscillation of a function: 3.14 logarithm of a real number: 4.3 exponential of base a (x real) : 4.3 set of complex numbers: 4.4 sum, product of complex numbers: 4.4 elements of C: 4.4 real and imaginary parts: 4.4 conjugate of a complex number : 4.4 absolute value of a complex number: 4.4 sum and product by a scalar in a vector space: 5.1 element of a vector space: 5.1 norm: 5.1 sum of a series, series: 5.2 sum of an absolutely summable family: 5.3 space of sequences tending to 0: 5.3, prob. 5 space of linear continuous mappings: 5.7 norm of a linear continuous mapping: 5.7 space of multilinear continuous mappings: 5.7 space of absolutely convergent series: 5.7, prob. 1 space of bounded sequences: 5.7, prob. 1 scalar product: 6.2 orthogonal projection: 6.3 Hilbert spaces of sequences: 6.5 spaces of bounded mappings: 7.1 space of continuous mappings : 7.2 space of bounded continuous mappings: 7.2 limits to the right, to the left: 7.6 (total) derivative at x,: 8.1

xviii

NOTATIONS

f Df f’(C0, D+f(.> f,’(P>, D-f(P>

derivative (as a function): 8.1 derivative on the right: 8.4 derivative on the left: 8.4 integral: 8.7

’3

S,”(t)

d5

e, exp(x), log x (x real): 8.8 D, f ( a l ,az),D z f ( a l ,az) partial derivatives: 8.9

a

fji(4,,. ..,t,), -f(T1, . . . ,5,)

at;i

partial derivatives: 8.10

f ”(x,), Dzf(xo),f‘P’(x,), DPf(x,) higher derivatives: 8.12 f * P regularization: 8.12, prob. 2 space o f p times continuously differentiable mappings: &‘,“’(A) 8.13. (acomposite index) : 8.13 lal, M a Da7 D,. ezl exp(z) (z complex): 9.5 sin z, cos z sine and cosine: 9.5 71 9.5 9

log z, am(z),

(:),

(1

+

2)‘

(z, t complex numbers): 9.5, prob. 8

opposite path: 9.6 juxtaposition of paths: 9.6 integral along a road: 9.6 f(z) C-lZ index with respect to a circuit: 9.8 .i(a : Y) primary factor: 9.12, prob. 1 E(z, P) gamma function: 9.12, prob. 2 r(z) Euler’s constant: 9.12, prob. 2 Y integral along an endless road: 9.12, prob. 3 f ( 4c-iJz order of a function at a point: 9.15 w(a ;f 1, 4 4 WE) algebra of operators: 11.1 uu composed operator: 11.1 SP(4 spectrum: 11.1 E(L-1, u) eigenspace : 11.1 u“ continuous extension: 11.2 N(A), N(A; u), F(A), F(A: u) subspaces attached to an eigenvalue of a compact operator: 11.4 order of an eigenvalue: 11.4 k ( 4 , M A ; u) U* adjoint operator: 11.5 YO

Y1

s,

v YZ

CHAPTER I

ELEMENTS OF THE THEORY OF SETS

We do not try in this chapter to put set theory on an axiomatic basis; this can however be done, and we refer the interested reader to Kelley [I51 and Bourbaki [3] for a complete axiomatic description. Statements appearing in this chapter and which are not accompanied by a proof or a definition may be considered as axioms connecting undefined terms. The chapter starts with some elementary definitions and formulas about sets, subsets and product sets (Sections 1.Ito 1.3); the bulk of the chapter is devoted to the fundamental notion of mapping, which is the modern extension of the classical concept of a (numerica1)function of one or several numerical “ variables.” Two points related to this concept deserve some comment : 1. The all-important (and characteristic) property of a mapping is that it associates to any “value” of the variable a single element; in other words, there is no such thing as a “ multiple-valued ” function, despite many books to the contrary. It is of course perfectly legitimate to define a mapping whose values are subsets of a given set, which may have more than one element; but such definitions are in practice useless (at least in elementary analysis), because it is impossible to define in a sensible way algebraic operations on the “values” of such functions. We return to this question in Chapter IX.

2. The student should as soon as possible become familiar with the idea that a functionfis a single object, which may itself “vary” and is in general to be thought of as a “point” in a large “functional space”; indeed, it may be said that one of the main differences between the classical and the modern concepts of analysis is that, in classical mathematics, when one writes . f ( x ) , f is visualized as “fixed” and x as “variable,” whereas nowadays both f 1

2

I ELEMENTS OF THE THEORY OF SETS

and x are considered as “variables” (and sometimes it is x which is fixed, and ,f which becomes the “ varying” object). Section 1.9 gives the most elementary properties of denumerable sets; this is the beginning of the vast theory of “cardinal numbers” developed by Cantor and his followers, and for which the interested reader may consult Bourbaki ([3], Chapter 111) o r (for more details) Bachmann [2]. It turns out, however, that, with the exception of the negative result that the real numbers d o not form a denumerable set (see (2.2.17)), one very seldom needs more than these elementary properties in the applications of set theory to analysis.

1. E L E M E N T S AND SETS

We are dealing with objects, some of which are called sets. Objects are susceptible of having properties, or relations with one another. Objects are denoted by symbols (chiefly letters), properties o r relations by combinations of the symbols of the objects which are involved in them, and of some other symbols, characteristic of the property o r relation under consideration. The relation x = y means that the objects denoted by the symbols x and y are the same; its negation is written x # y . If X is a set, the relation x E X means that x is a n element of the set X, o r belongs to X; the negation of that relation is written x $ X. If X and Y are two sets, the relation X c Y means that every element of X is a n element of Y (in other words, it is equivalent to the relation (Vx)(x E X * x E Y)); we have X c X, and the relation (X c Y and Y c Z) implies X c Z. If X c Y and Y c X, then X = Y, in other words, two sets are equal if and only if they have the same elements. If X c Y, one says that X is contained in Y , o r that Y coritains X, or that X is a subset of Y ; one also writes Y 3 X. The negation of X c Y is written X Q Y. Given a set X, and a property P, there is a unique subset of X whose elements are all elements x E X for which P(x) is true; that subset is written {x E X I P(x)}. The relation {x E XI P(x)} c {x E X I Q(x)} is equivalent to (Vx E X)(P(x) * Q(x)); the relation {x E X I P(x)} = { x E X I Q(x)) is equivalent to (Vx E X)(P(x)oQ(x)). We have, for instance, X = {x E X I x = x} and X = {x E X I x E X}. The set = {x E X I x # x} is called the empty set of X ; it contains no element. If P is any property, the relation x E * P(x) is true for every x, since the negation of x E is true for every x (remember that Q * P means “ n o t Q o r P”). Therefore, if X and Y are sets, x E implies x E in other words c By,and similarly Byc @, hence ox= Dy, all empty sets are equal, hence noted @. If a is an object, the set having a as unique element is written { a } .

a,

ax

aV,

ax

a,

a,

2 BOOLEAN ALGEBRA

3

If X is a set, there is a (unique) set the elements of which are all subsets of X; it written v(X). We have @ E ‘$(X), X E v(X); the relations x E X , {x} E v(X) are equivalent; the relations Y c X, Y E v(X) are equivalent.

PROBLEM

Show that the set of all subsets of a finite set having having 2“ elements.

II

elements (n 3 0) is a finite set

2. B O O L E A N ALGEBRA

If X, Y are two sets such that Y c X, the set {x E X I x $ Y) is a subset of X called the diference of X and Y or the complenient of Y with respect to X, and written X - Y or Y (or Y when there is no possible confusion). Given two sets X, Y, there is a set whose elements are those which belong to both X and Y, namely {x E XI x E Y}; it is called the intersection o f X a n d Y and written X n Y. There is also a set whose elements are those which belong to one at least of the two sets X, Y ; it is called the union of X and Y and written X u Y.

tx

The following propositions follow at once from the definitions: (1.2.1)

x-X=@,

x-@=X.

(1.2.2)

x u x = x ,

XnX=X.

(1.2.3)

XuY=YuX,

XnY=YnX.

(1.2.4)

The relations X c Y, X u Y

XcXuY,

(1.2.5) (1.2.6)

(1.2.8)

X n Y = X are equivalent.

XnYcX.

The relation “ X c Z and Y c Z ” is equivalent to X u Y c Z; the relation

(1.2.7)

= Y,

“

Z c X and Z

c

Y ” is equivalent to Z

c

X n Y.

X u (Y u Z) = (X u Y) u Z,

written

X u Y u Z.

X n (Y n Z) = (X n Y) n Z,

written

X n Y n Z.

X u (Y n Z) = (X u Y) n (X u Z) X n (Y u Z) = (X n Y) u (X n Z) (distributivity).

4

I

(1.2.9)

ELEMENTS OF THE THEORY OF SETS

For subsets X, Y of a set E (with

c written for gE)

c cc X) = x;

c (X ” J’) = cc XI n (C Y), (Xn Y) = 0.

-

20

II REAL NUMBERS

For an interval of origin a and extremity 6 , the positive number b - a is called the length of the interval. For any real number x, we define 1x1 as equal to x if x > 0, to - x if x < 0, hence I - X I = 1x1; 1x1 is called the absolute value of x; 1x1 = 0 is equivalent to x = 0. We write x + = (x (x()/2(positive part of x), x - = (1x1 - x)/2 (negative part of x) so that x+ = x if x > 0, x + = 0 if x Q 0, x- = 0 ifx > 0, x- = -x if x Q 0, and x = x+ - x - , 1x1 = x + x - .

+

+

(2.2.9) Zfa > 0, the relation 1x1 < a is equivalent to -a 1x1 < a to -a < x < a.

< x < a, the relation

For if x > 0, x > -a is always satisfied and 1x1 < a (resp. 1x1 < a) is equivalent to x Q a (resp. x < a ) ; and if x < 0, x < a is always satisfied and 1x1 Q a (resp. 1x1 < a) is equivalent to - x Q a (resp. - x < a).

(2.2.10)

For any pair of real nunibers x, y , Ix 11x1 -

+ yl < 1x1 + lyl

and

IYI I Q lx - Yl.

The first relation is evident by definition and from (2.2.8) when x, y are both positive or both negative. If for instance x Q 0 Q y , then x + y < y ~ y + l x l = l u l + I x l , and x + ~ > ~ > x - I y I -lxl-Iyl. = From the first inequality follows 1x1 = Iy (x - y)l < Iyl lx - yl and lyl = lx ( y - x)l < 1x1 ly - XI whence - Ix - yl d 1x1 - lyl Q Ix - yl. By induction, it follows from (2.2.10) that

+

1x1 (2.2.11) I f z

+

+

+ + . * . + x,I Q x2

1x1

+

1x21

+

+

* * -

+ 1X"l.

> 0 , the relation x < y implies xz < yz.

For by (2.2.7), x Q y implies 0 Q y - x, hence 0 < z ( y - x) from (11.5).

= zy

- zx

(2.2.12) The relations x < 0 and y 2 0 iniply xy Q 0 ; the relations x < 0 and y < 0 imply xy > 0. Same results with < replaced by 0 unless x = 0.

The first statements follow from (11.5) and ( - x ) y = -(xy), (-x)( - y ) = xy; on the other hand, xy = 0 implies x = 0 or y = 0. (2.2.12) implies that lxyl = 1x1 lyl for any pair of real numbers x, y.

-

2 ORDER PROPERTIES OF T H E REAL NUMBERS

21

From (2.2.12) and (1.7) it follows that 1 = 1' > 0, hence, by (2.2.8), the real number n * I ( 1 added n times) is > O for n > 0 ; this shows that the mapping n n * 1 of the natural integers into R is injective, and preserves order relations, addition and multiplication; hence natural integers are identified to real numbers by means of that mapping. --f

(2.2.1 3) r f x > 0, x-' > 0. For z > 0, the relation x Q y (resp. x < y ) is equivalent to xz < yz (resp. xz < y z ) . The relation 0 < x < y is equivalent to 0 < y - l < x-l, and to 0 < X" < y" for every integer n > 0.

The first statement follows from the fact that xx-I = 1 > 0, hence x - l > 0 by (2.2.12); the second follows from the first and (2.2.11), since x = (xz)z-'. The third is an obvious consequence of the second. The last follows by induction on the integer n > 0 from the relations xn < x"-ly < y".

Remark. An open interval ]a,b[ of R (with a < b) is not empty, for the relation b - a > 0 implies, by (2.2.13), (b - a)/2 > 0 ; hence a < ( a b)/2 < 6 . From that remark one deduces:

+

(2.2.14) Let J,,

. . . ,J, be n open intervals, no two of which have common n

points, and let 1 be an interual containing (1

< k < n), 1 the length of

I, 1,

+ l2 +

U1 J k ; then, i f ! , + I,, < I.

is the length of Jk

k= * * -

Let I = ]a,b[, J, = ] c k ,dk[. For each k # 1, we have either ck < d, < c, or dl Q ck < d,, otherwise J, n J, would not be empty. For n = 1, the property is immediate as a d c1 < d, < b, hence -cl < - a, and d, - c1 < b - a. Use induction on n ; let J i l , . . . , J i p be the intervals contained in ]a, cl[, and J i l , ..., Jjn-,-n the intervals contained in Id,, b[; then

C lih < c1 - a, lj, < b - dl by induction, and I, + 1lih+ E l j , < d, - c1 + c1 - a + b - dl = b - a . P

n-1-p-

h= 1

k= 1

h

Il

+ l2 +

+ I,, =

k

Real numbers of the form +r/s, where r and s are natural integers, s # 0, are called rational numbers. Those for which s = 1 are called integers (positive or negative) and the set of all integers is written Z.

22

II

(2.2.1 5)

REAL NUMBERS

The set Q of rational numbers is denumerable.

As Q is the union of Q n R + and Q A ( - R + ) , it is enough to prove Q n R, denumerable. But there is a surjective mapping (m, n) -+ m/n of the subset of N x N consisting of the pairs such that n # 0, onto Q n R + , hence the result by (1.9.2), (1.9.3), and (1.9.4).

(2.2.16) Every open interval in

R contains an injinite set of rational numbers.

It is enough to prove that ]a,b[ contains one rational number c, for then ]a, c[ contains a rational number, and induction proves the final result. Let x = b - a > 0 ; by (111) there is an integer n > l/x, hence l / n < x by (2.2.13). We can suppose b > 0 (otherwise we consider the interval 1-b, -a[ with -a > 0). By (111) there is an integer k > 0 such that b < k / n ; let h be the smallest integer such that b < h/n. Then (h - I)/n < 6; let us show that (h - l ) / n > a ; if not, we would have b - a = x < I/n by (2.2.14), contradicting the definition of n.

(2.2.1 7) The set of real numbers is not denumerable.

We argue by contradiction. Suppose we had a bijection n + x , from N onto R. We define a subsequence n-+p(n) of integers by induction in the following way: p ( 0 ) = 0, p ( 1 ) is the smallest value of n such that x, > xo . Suppose that p(n) has been defined for n < 2m - 1, and that x ~ ( ~< x~ , -, ~~ ~) ~then - , ~the; set ]x,,(~,,,-~), x ~ ( ~ ~is- infinite ~ ) [ by (2.2.16), and we define p(2m) to be the smallest integer k > p(2m - 1) such that X p ( 2 m - 2 ) < xk < then we define p(2m + 1 ) as the smallest integer k >p(2m) such that x , , ( ~ ~ 0, there exists a 6 > 0 such that the relation d(x, y ) < 6 implies d’(f(x),f ( y ) ) < E . From this definition and (3.11.2), it follows that

(3.1 1.7) A uniformly continuous mapping is continuous.

The converse is not true in general: for instance, the function x -,xz is not uniformly continuous in R, since for given a > 0, the difference (x a)’ - x2 = a(2x a) can take arbitrarily large values (see however (3 .I6.5)). The examples given above (constant mapping, natural injection) are uniformly continuous.

+

(3.11.8)

+

For any nonempty subset A of E , x -,d(x, A) is uniformly con-

tinuous. This follows from the definition and (3.4.2).

12 HOMEOMORPHISMS, EQUIVALENT DISTANCES 47

Iff is a uniformly continuous mapping of E into E‘, g a uniformly continuous mapping of E’ into E“, then h = g o f is unijormly continuous. (3.11.9)

Indeed, given any E > 0, there is 6 > 0 such that d’(x’, y‘) < 6 implies d”(g(x’),g(y’)) < E ; then there is q > 0 such that d(x, y ) < r] implies d’( f ( x ) ,f ( y ) ) < 6; therefore d(x, y ) < q implies d”(h(x),h(y)) E.

-=

PROBLEMS

1. Let f b e a mapping of a metric space E into a metric space E’. Show that the following properties are equivalent : (a) f i s continuous; (b) for every subset A’ of E‘,f-l(A’) c (f-I(A’))’; (c) for every subset A‘ of E’,f-I(A’) ‘f-’(A’). Give an example of a continuous mappingfand a subset A‘ c E’ such that f-’(A’)is not the closure of f-’(A’). 2. For any metric space E, any number r > 0 and any subset A of E, the set V;(A) of points x E E such that d(x, A) < r is closed (use (3.11.8)). 3. In a metric space E, let A, B be two nonempty subsets such that A n = A n B = 0. Show that there exists an open set U 3 A and an open set V 2 B such that U n V = 0 (consider the function x d(x, A) - d(x, B)). 4. Let f be a continuous mapping of R into itself. (a) Show that iffis uniformly continuous in R, there exist two real numbers a > 0, 2 0 such that If(x)l < a 1x1 B for every x E R. (b) Show that iffis monotone and bounded in R,fis uniformly continuous in R.

s

--f

+

12. HOMEOMORPHISMS, EQUIVALENT DISTANCES

A mapping f of a metric space E into a metric space E’ is called a homeomorphism if: (1) it is a bijection; (2) both f and its inverse mappingf-’ are continuous. Such a mapping is also said to be bicontinuous. The inverse mapping f is then a homeomorphism of E’ onto E. Iff is a homeomorphism of E onto E’, g a homeomorphism of E’ onto E”, g of is a homeomorphism of E onto E” by (3.11.5). A homeomorphism may fail to be uniformly continuous (for instance, the homeomorphism x + x 3 of R onto itself). Two metric spaces E, E‘ are homeomorphic if there exists a homeomorphism of E onto E‘. Two spaces homeomorphic to a thrid one are homeomorphic. By abuse of language, a space homeomorphic to a discrete metric space (3.2.5)is called a discrete space, even if the distance in nos defined as in

-’

(3.2.5).

48

Ill METRIC SPACES

An isometry is always uniformly continuous by definition, hence a homeomorphism. For instance, the extended real line R is by definition homeomorphic t o the subspace [ - I , 11 of R. Let d,, d, be two distances on a set E; this defines two metric spaces on E, which have to be considered as distinct (although they have the same “underlying set”); let E,, E, be these spaces. If the identity mapping x -+ x of El onto E, is a homeomorphism, d l , d , are called equivaletit distances (or topologically equiidetit distances) on E ; from (3.1 1.4),we see that this means the families ofopeii sets are the saiiie in El and E, . The family of open sets of a metric space E is often called the topology of E (cf. Section 12.1); equivalent distances are thus those giving rise to the same topology. It may be observed here that the definitions of neighborhoods, closed sets, cluster point, closure, interior, exterior, dense sets, frontier, continuous fLiMctioii only depend on the topologies of the spaces under consideration; they are topological tiotiotis; on the other hand, the notions of balls, splwres, diameter, bounded set, uiiifortnly coiitiiiuous function are not topological notions. Topological properties of a metric space are itii~ariar?tunder honieoniorpliisms. With the preceding notations, it may happen that the identity mapping x + x of El into E, is continuous but not bicontinuous: for instance, take E = R, d2(x,y) = Ix -)>I and for dl(.x,y ) the distance defined in (3.2.5) taking only values 0 and 1. In such a case, the distance d, (resp. the topology of El) is said t o b e j u e r than the distance d2 (resp. the topology of E,).

PROBLEMS

Let u be an irrational number .: 0; for each rational number x .: 0, let f.(x) be the unique real number such that 0 0 into the interval 10, u[ of R, and that f b ( Q $ ) is dense in 10, u [ . Deduce from that result and from Problem 1 in Section 2.2 that there exists a bijective continuous mapping of Q onto itself which is not bicontinuous (compare to (4.2.2)). Let f b e a continuous mapping of a metric space E into a metric space F. (a) Let (V,) be a covering of F by open subsets; show that if, for each /I the , restriction o f f t o f - ‘(VJ is a homomorphism of the subspacef-l(VJ of E onto the subspace VA of F, f i s a homeoniorphism of E onto F. (b) Give an example of a niappingfwhich is not injective, and of a covering (U,) of E by open subsets, such that the restriction o f / t o each of the U, is a homeomorphism of the subspace U , of E onto the subspacef(Ui,) of F (one can take both E and F discrete). Let E, F, G be three metric spaces,fa continuous mapping of E into F,g a continuous f a homeomorphism of mapping of F into G. Show that if f is surjective and g t ~ is E onto G , thenf is a honieomorphism of E onto F and g is a homeomorphism of F onto G.

13

LIMITS 49

13. LIMITS

Let E be a metric space, A a subset of E, a a cluster point of A. Suppose first that a does not belong to A. Then, iff is a mapping of A into a metric space E’, we say that f (x) has a limit a’ E E’ bithen x E A tends to a (or also that a‘ is a limit off at the point a E A with respect to A), if the mapping g of A u { a } into E’ defined by taking g(x) =f(x) for x E A, g(a) = a’, is continuous at the point a ; we then write a‘ = lim f(x). If a E A , we use the x-ra. x E

A

same language and notation to mean that f is continuous at the point a, with a’ = f ( a ) .

(3 . I 3 . I ) In order that a’ E E‘ be limit off (x) when x E A tends to a, a necessary and suflcient condition is that, for eilery neighborhood V’ of a’ in E‘, there exist a neighborhood V of a in E such that f ( V n A) c V‘.

(3.13.2) In order that a’ E E‘ be limit off(x) lithen x E A tends to a, a necessary and suflcient condition is that, for ever)’ c > 0, there exist a 6 > 0 such that the relations x E A, d(x, a ) < 6 imply d’(a’,f(x)) < E .

These criteria are mere translations of the definitions.

(3.13.3) A mapping can only have one limit with respect to A at a given point a E A.

For if a’, b‘ were two limits off at the point a, it follows from (3.13.2) and the triangle inequality that, for any E > 0, we would have d’(a’, b’) < 24 which is absurd if a’ # b’.

(3.13.4) Let f be a mapping of E into E’. In order that f be continuous at a point x o E E such that xo is a cluster point of E - {xo} (which means xo is not isolated in E (3.10.10)), a necessary and suflcient condition is that .f(xo) = lim f’(x). X-XO.

x E E - 1x0)

Mere restatement of definitions.

50

I l l METRIC SPACES

(3.13.5) Suppose a' = lim f ( x ) . Then, for every subset B c A such that x-a, x E A

a E 8 , a' is also the limit in particular when B

of f a t the point

=V A

a, with respect to B. This applies

A, where V is a neighborhood of a.

Obvious consequence of the definition and (3.1 1.6).

(3.13.6) Suppose f has a limit a' at the point a E A with respect to A ; 5 f g is a mapping of E' into E", continuous at the point a', then g(a') = lim g(f ( x ) ) . x'a.

xEA

This follows at once from (3.11.5).

(3.13.7) If a'

-

=

lirn f ( x ) , then a' ef(A). x-a.

xcA

For by (3.13.1), for every neighborhood V' of a', V' nf ( A ) contains f ( V n A), which is not empty since a E A. An important case is that of limits of sequences: in the extended real line, we consider the point +a, which is a cluster point of the set N of natural integers. A mapping of N into a metric space E is a sequence n -,x , of points of E; if a E E is limit of that mapping at + co, with respect to N, we say that a is limit of the sequence (x,) (or that the sequence (x,) converges to a ) and write a = lim x,. The criteria (3.13.1) and (3.13.2) become here: n-m

(3.13.8) In order that a = lirn x, , a necessary and suficient condition is that, n-m

for every neighborhood V of a, there exist an integer no such that the relation n 2 no implies x, E V (in other words, V contains all x , with the exception of a finite number of indices).

(3.13.9) In order that a = lirn x,, a necessary and sufficient condition is n- m

that, for every E > 0, there exist an integer no such that the relation n 2 no implies d(a, x,) < E .

This last criterion can also be written lirn d(a, x,) = 0. n- m

A subsequence of an infinite sequence (x,) is a sequence k + x,, , where k -+ nk is a strictly increasing infinite sequence of integers. It follows at once from (3.13.5) that:

13 LIMITS 51

(3.13.10) r f a = lim x, then a = lim xnkfor any subsequence of(x,,). n-+ m

k-+ Q)

Let (x,,) be an infinite sequence of points in a metric space E; a point b E E is said to be a cluster value of the sequence (x,) if there exists a subsequence (x,,) such that b = lim x n k . k+m

A cluster value of a subsequence of a sequence (x,) is also a cluster value of (x,,). If (x,) has a limit a, a is the unique cluster value of (x,), as follows from (3.13.10); the converse does not hold in general: for instance, the =n ~ (n 2 1) has sequence (x,,) of real numbers such that xZn= l/n and x ~ , , + 0 as a unique cluster value, but does not converge to 0 (see however (3.16.4))

(3.13.11) In order that b E E should be a cluster value of (x,,),a necessary and suficient condition is that, for any neighborhood V of b and any integer m , there exist an integer n 2 m such that x, E V. The condition is obviously necessary. Conversely, suppose it is satisfied, and define the subsequence (x,,,) by the following condition: no = 1 and nk is the smallest integer > nk-l and such that d(b, x,,,) < I/k. ASd(x,,, , b) < I/h for any k 2 h, the subsequence (x,,,) converges to b.

(3.13.12) I f b is a cluster value of (x,) in E, and if the mapping g of E into E' is continuous at b, then g(b) is a cluster value of the sequence (g(x,,)). Clear from the definition and (3.13.6). From (3.13.7) it follows that if b is a cluster value (and afortiori a limit) of a sequence of points x, belonging to a subset A of E, then b E A. Conversely :

(3.13.13) For any point a EA, there is a sequence (x,,) of points of A such that a = lirn x,,. n-

m

For by assumption, the set A n B(a; l / n ) is not empty, hence (by the axiom of choice (1.4.5)) for each n, there is an x, E A n B(a; l/n), and the sequence (x,) converges to a by (3.13.9).

(3.13.14) Let f be a mapping of A c E into a metric space E' and a EA. In order that f have a limit a' E E' with respect to A at the point a, a necessary and sufficient condition is that, for every sequence (x,) of points of A such that a = lim x,, then a' = lim f(x,,). n - r 00

n-r a,

52

Ill METRIC SPACES

The necessity follows from the definitions and (3.13.6). Suppose conversely that the condition is satisfied and that a’ is not the limit off’with respect to A at the point a. Then, by (3.13.2) and (1.4.5), there exists t( > 0 such that, for each integer 11, there exists x, E A satisfying the two conditions d(a, x,) < l / n and d(a’,f’(x,)) 3 a. The sequence (x,) converges then to a, but (,f(x,)) does not converge to a’, which is a contradiction.

PROBLEMS 1. Let

(11,)

be a sequence of real numbers > 0 such that litn

I / . = 0.

Show that there are

n- m

infinitely many indices n such that I / , 3 u,,, for every rn 3 11. 2. (a) Let (x.) be a sequence in a metric space E. Show that if the three subsequences (x2,J, ( . Y ~ , ~ + , )and (x3,J are convergent, (x,,) is convergent. (b) Give an example of a sequence (x,,) of real numbers which is not convergent, but is such that for each k 3 2, the subsequence ( x k J is convergent (consider the subsequencc (x,,~),where ( p r ) is the strictly increasing sequence of prime numbers). 3. Let E be a separable metric space,fan arbitrary mapping of E into R. Show that the liin f(x) exists and is ri‘isfihct from f(rr), is a t most set of points u E E such that I-”, x t a

denumerable. (For every pair of rational numbers p, 4 such that p -I q, consider the set of points a E E such that

no imply d(x,, xy) < 4 2 ; on the other hand, by (3.13.11) there is a p o 2 no such that d(b, x p o )< 4 2 ; by the triangle inequality, it follows that d(b, x,) < E for any n 2 n o . A metric space E is called complete if any Cauchy sequence in E is convergent (to a point of E, of course).

(3.14.3)The real line R is a complete metric space. Let (x,) be a Cauchy sequence of real numbers. Define the sequence (n,) of integers by induction in the following way: no = 1 and n k + l is the sma~~estinteger > n, such that, f o r p 3 and q 3 n k + l , Ix, - xyl < I / 2 k + Z ; the possibility of the definition follows from the fact that (x,) is a Cauchy sequence. Let I, be the closed interval [x,, - 2 - , , x,, 2 - k ] ; we have I , + ~c r,, for IX,~ - xnL+,l < 2 - k - ' ,. on the other hand, for m 2 t ? k , x, E 1, by definition. Now from axiom (IV) (Section 2.1) it follows that the nested intervals I, have a nonempty intersection; let a be in I, for all k . Then it is clear that la - x,I d 2 - h + ' for all m 3 n k r hence a = lim x,.

+

n+ m

(3.14.4)Ij'a subspace F of a metric space E is complete, F is closed in E. Indeed, any point a E F is the limit of a sequence (x,) of points of F by (3.13.13). The sequence (x,) is a Cauchy sequence by (3.14.1), hence by assumption converges t o a point b in F; but by (3.13.3) b = a, hence a E F ; this shows F = F. Q.E.D.

(3.14.5)In a complete metric space E, any closed subset F is a complete subspace. For a Cauchy sequence (x,) of points of F converges by assumption to a point a E E, and as the x, belong to F, a E F = F by (3.13.7). Theorems (3.14.4) and (3.14.5) immediately enable one to give examples both of complete and of noncomplete spaces, starting from the fact that the real line is complete.

Ill

54

METRIC SPACES

The fundamental importance of complete spaces lies in the fact that to prove a sequence is convergent in such a space, one needs only prove it is a Cauchy sequence (one also says that such a sequence satisfies the Cauchy criterion); the main difference between application of that test and of the definition of a convergent sequence is that in the Cauchy criterion one does not need to know in advance the oalue of the limit. We have already mentioned that on a same set E, two distances d , , d , may be topologically equivalent, but the identity mapping of E, into E, (El, E, being the corresponding metric spaces) may fail to be uniformly continuous. This is the case, for instance, if we take E = R, d,(x, y ) = Jx- pJ, d l ( x , y ) being the distance in the extended real line, restricted to R; E, is then complete and not El since El is not closed in 8. When two distances d,, d, are such that the identity mapping of El into E, is uniformly continuous as well as the inverse mapping, d, and d, are said to be unijornily equbalent. Cauchy sequences are then the same for both distances. For instance, if there exist two real numbers a > 0, fl > 0 such that, for any pair of points x,y in E, adl(x, y ) < d,(x, y ) < fldl(x,y ) , then (1, and d, are uniformly equivalent distances. Let E, E’ be two metric spaces, A a subset of E , f a mapping of A into E’; the oscillatioti of,f in A is by definition the diameter 6 ( j ( A ) )(which may be a).Let a be a cluster point of A ; the oscillation o j f at the point a \\tit11 respect to A is Q ( a ; f ) = infii(,f( V n A)), where V runs over the set of

+

V

neighborhoods of a (or merely a fundamental system of neighborhoods). (3.14.6) Suppose E’ is a complete metric space; in order that

lim f(x) %*a, x E A

exist, a necessary arid su-cient condition is that the oscillation of ,f at the point a, with respect to A, be 0. The condition is necessary by (3.13.2). Suppose conversely that it is satisfied, and let (x,) be a sequence of points of A converging to a ; then it follows from the assumption that the sequence ( f ( x , ) ) is a Cauchy sequence in E‘, for, given any E > 0, there is a neighborhood V of a such that d ‘ ( f ( x ) , f ( y ) )< E for any two points x, y in V n A , and we have x,,E V n A except for a finite number of indices. Hence the sequence ( f ( x , ) ) has a limit a’. Moreover, for any other sequence (y,) of points of A , converging to a, the limits of ( f ( x , ) ) and of (f(y,,)) are the same since d’(f(x,),f(y,)) < E as soon as x, and yn are both in V n A. Hence lim f ( x ) = a‘ from the definition of the limit and from (3.13.14).

x-a, x E A

15

ELEMENTARY EXTENSION THEOREMS 55

PROBLEMS 1. (a) Let E be an rdtvametvic space (Section 3.8, Problem 4). In order that a

sequence (x,) in E be a Cauchy sequence, show that it is necessary and sufficient that lim d ( x , , ~ , , + ~ ) = O . n- m

(b) Let X be an arbitrary set, E the set of all infinite sequences x = (x,) of elements of X. For any two distinct elements x = (xJ, y = (y.) of E, let k ( x , y ) be the smallest integer 12 such that x, # yn ; let d(x, y ) = I/k(x,y ) if x # y , d ( x , x) = 0. Prove that d is an ultrametric distance on E, and that the metric space E defined by d is complete. 2. Let 'p be an increasing real valued function defined in the interval 0 < u < +a,and such that q(0) = 0, q(u)> 0 if u > 0, and y(ir u ) < p(ir) q ( u ) . Let c/(x, y ) be a distance on a set E ; then d,(x, y ) = q(d(x, y ) ) is another distance on E. (a) Show that if q is continuous at the point [I = 0, the distances d a n d d, are uniformly equivalent. Conversely, if, for the distanced, there is a point xo E E which is not isolated in E (3.10.10),and if ciand d, are topologically equivalent, then q is continuous at the point I I = 0. (b) Prove that the functions

+

I!'

(0 < r

< l),

log(1

+

I/),

u/(I

+

+

ti),

inf(1, u )

satisfy the preceding conditions. Using the last two, it is thus seen that for any distance on E, there is a uniformly equivalent distance which is borinded. 3. On the real line, let d(x, y ) = Ix - y / be the usual distance, d'(x, y ) = / x 3 - y31 ; show that these two distances are topologically equivalent and that the Cauchy sequences are the same for both, but that they are not uniformly equivalent. 4. Let E be a complete metric space, d the distance on E, A the intersection of a sequence (U,) of open subsets of E ; let F, = E - U,, and for every pair of points x , y of A, write

d h , Y ) =L(x, y)l(l + f ; d x , Y ) ) , and d'(x, Y ) = d(x, Y )

+ C d d x , y)/2". Show that on m

Il=O

the subspace A of E, d' is a distance which is topologically equivalent to d , and that for the distance d',A is a complete metric space. (Note that a Cauchy sequence for d' is also a Cauchy sequence for d, but that its limit in E may not belong to any of the F, .) Apply to the subspace I of R consisting of all irrational numbers.

15. EL EM E N T A R Y E X T E N S I 0 N TH E OREMS

(3.15.1) Lei f and g be two continuous mappings of a metric space E into a metric space E'. The set A of the points x E E such that f ( x ) = g(x) is closed iti E. It is equivalent to prove the set E - A open. Let a E E - A, then f ( a ) # g(a); let d'( f (a), g(a)) = a > 0. By continuity o f f , g at a and from (3.6.3) it follows that there is a neighborhood V of a in E such that for

56

Ill METRIC SPACES

x E V, d’( f (a), f ( x ) ) < 4 2 and d’(g(a),g(x)) < a/2. Then for x E V, f ( x ) # g(x), otherwise we would have d‘(f(a),g(a)) < c( by the triangle inequality. (“ Principle of extension of identities ”) Let f, g be two contin(3 .I5.2) uous mappings of a metric space E into a metric space E’; if f ( x ) = g ( x ) for all points x of a dense subset A in E, then f = g.

For the set of points x where f ( x ) = g(x) is closed by (3.15.1) and contains A. (3.15.3) Let f, g be two continuous mappings of a metric space E into the The set P of the points x E E such that f ( x ) < g ( x ) is extended real line closed in E.

w.

We prove again E - P is open. Supposef(a) > g(a), and let p E R be such that f ( a ) > p > g(a) (cf. (2.2.16) and the definition of R in Section 3.3). The inverse image V b y j o f the open interval ]p, + 001 is a neighborhood of a by (3.11 .I); so is the inverse image W by g of the open interval [ - 00, p[. Hence V n W is a neighborhood of a by (3.6.3), and for x E V n W, f(x) > p > g(x). Q.E.D. (3.15.4) (“ Principle of extension of inequalities”) L e t f , g be two continuous mappings of a metric space E into the extended real line R; i f f ( x ) < g(x) for all points x of a dense subset A of E, then f ( x ) < g(x)for all x E E.

The proof follows from (3.1 5.3) as (3.1 5.2) from (3.1 5.1). (3.15.5) Let A be a dense subset of a metric space E, and f a mapping of A into a metric space E‘. In order that there exist a continuous mapping f of E into E’, coinciding with f in A, a necessary and suficient condition is that, for any x E E, the limit lim f ( y ) exist in E’; the continuous mapping f is Y+X,Y EA then unique.

As any x E E belongs to A, we must have f ( x ) = lim f ( y ) by (3.13.5), Y-X,YEA

hencef(x) = lirn f ( y ) ; this shows the necessity of the condition and the Y+X, Y E A

fact that if the continuous mapping J exists, it is unique (this follows also

16 COMPACT SPACES

57

from (3.1 5.2)). Conversely, suppose the condition satisfied, and let us prove that the mappingfdefined byf(x) = lim f ( y ) is a solution of the extension Y-x, Y FA

problem. First of all, if x E A, the existence of the limit implies by definition f(x) = f(x), hence f extends f, and it remains to see that f is continuous. Let x E E, V' a neighborhood off(x) in E'; there is a closed ball B' of center f (x) contained in V'. By assumption, there is an open neighborhood V of x in E such thatf(V n A) c B' (by (3.13.1)). For any y E V,f(y) is the limit of f a t the pointy with respect to A, hence also with respect to V n A, by (3.13.5); hence, it follows from (3.13.7) that f(y) E f(V n A), and therefore f ( y ) E B' since B' is closed. Q.E.D.

(3.15.6) Let A be a dense subset of a metric space E, and f a uniformly continuous mapping of A into a complete metric space E'. Then there exists a continuous mapping f of E into E' coinciding with f i n A; moreover, f is uniformly continuous.

To prove the existence off, it follows from (3.15.5) and (3.14.6) that we have to show the oscillation of f at any point x E E, with respect to A, is 0. Now, for any E > 0, there is 6 > 0 such that d(y, z) < 6 implies d'(f(y), f(z)) < 4 3 (y, z in A). Hence, the diameter of f ( A n B(x; 612)) is at most ~ / 3 which , proves our assertion. Consider now any two points s, t in E such that d(s, t ) c 612. There is a Y E A such that d(s, y) < 614 and d ' ( f ( s ) ,f(y)) c ~ / 3 and , a z E A such that d(t, z) < 614 and d'Cf(t),f(z)) < 4 3 . From the triangle inequality it follows that d(y, z) < 6, and as y , z are in A, d'(f (y),f (z)) < 4 3 ; hence, by the triangle inequality, d'(f(s),3(t))< E ; this proves that f is uniformly continuous.

PROBLEM

Let n + r. be a bijection of N onto the set A of all rational numbers x such that 0 < x < 1 (2.2.15). We define a function in E = [0, I ] by puttingf(x) = 1/2", the infinite sum being ," 0, there is aJinite covering of E by sets of diameter < E . This

is immediately equivalent to the following property: .for afiy E > 0, there is a finite subset F of E such that d(x, F) < cfor every x E E. In the theory of metric spaces, these notions are a substitute for the notion of “finiteness” in pure set theory; they express that the metric space is, so t o speak, “ approximately finite.” Note that, from the definition, it follows that compactness is a topological notion, but precompactness is not (see remark after (3.17.6)).

(3.16.1) For a metric space E, the following three conditioiis are equivalent: (a) E is compact; (b) any infinite sequence in E has at least a cluster value; (c) E is precompact and coniplete.

(a) * (b): Let (x,) be an infinite sequence in the compact space E, and let F, be the closure of the set {x,, x , + ~ ., . . , x,,+,,, . . .}. We prove there is a point belonging to all F,. Otherwise, the open sets U, = E - F, would form a covering of E, hence there would exist a finite number of them, UnI,. . . , U,,, forming a covering of E; this would mean that F,, n F,, n ... n F,, = 0; but this is absurd, since if n is greater than max(n,, . . . , nk), F, (which is not empty by definition) is contained in all the F,,,(1 < i < k). Hence the intersection

m

n= 1

F, contains at least a point a. By (3.13.11) and the definition of a

cluster point, a is a cluster value of (x,,). (b) * (c): First any Cauchy sequence in E has a cluster value, hence is convergent by (3.14.2), and therefore E is complete. Suppose E were not precompact, i.e. there exists a number c( > 0 such that E has no finite covering by balls of radius a. Then we can define by induction a sequence (x,) in the following way: x1 is an arbitrary point of E; supposing that d(x,, x,) 2 ct for i Zj,1 < i < n - I , I < ,j < n - 1, the union of the balls of center x i (1 < i < n - I) and radius ct is not the whole space, hence there is x, such that d ( x i ,x,) 2 c( for i < 17. The sequence (x,) cannot have a cluster value, for if a were such a value, there would be a subsequence (x,J converging t o a, hence we would have d(a, x n k )< 4 2 for k > ko , and therefore d(x,, , x,J < ct for h 3 k, , k 3 k , , 12 # k , contrary t o the definition of (x,,). (c) * (a): Suppose we have an open covering (U,JiELof E such that no finite subfamily is a covering of E. We define by induction a sequence (B,) of

16 COMPACT SPACES 59

balls in the following way: from the assumption it follows that the diameter of E is finite, and by multiplying the distance on E by a constant, we may assume that 6(E) < 1/2,hence E is a ball B, of radius 1. Suppose the B, have been defined for 0 < k < n - 1, and that for these values of k , B, has a radius equal to 1/2,,and there is no finite subfamily of (U,),eL which is a covering of B, . Then we consider a finite covering (vk),,,,, of E by balls of radius 1/2”; among the balls vk which have a nonempty intersection with B,,-,, there is one at least B, for which no finite subfamily of (U,) is a covering; otherwise, as these V , form a covering of B,-,, there would be a finite subfamily of (U,) which would be a covering of B,-l ; the induction can thus proceed indefinitely. Let x, be the center of B,; as B,-l and B, have a common point, the triangle inequality shows that

x,) Hence, if n

< 1 p - 1 + 1/2” < 1/2,-2

< p < q, we have

This proves that (x,,) is a Cauchy sequence in E, hence converges to a point a. Let 1, be an index such that a E Un0;there is an a > 0 such that B ( a ; a ) c U,, . From the definition of a, it follows there exists an integer n such that d(a, x,) < 4 2 , and l/2”< 42. The triangle inequality then shows that B, c B ( a ; a ) c Ulo. But this is a contradiction since no finite subfamily of (U,) is supposed to be a covering of B,, .

(3.16.2) Any precompact metric space is separable. If E is precompact, for any n there is, by definition, a finite subset A, of E such that for every x E E, d(x, A,,) < l/n. Let A = A,; A is at most denumerable, and for each x E E, d(x, A) d(x, A) = 0, E =A.

< d(x, A,)

u n

< l/n for any n, hence

(3.16.3) Let E be a metric space. A n y two of the following properties imply the third: (a) E is compact. (b) E is discrete (more precisely, homeomorphic to a discrete space). (c) E isfinite.

60

Ill METRIC SPACES

(a) and (b) imply (c), for each one-point set {x} is open, hence the family of sets {x} is an open covering of E, and a finite subfamily can only be a covering of E if E is finite. On the other hand, (c) implies both (a) and (b), for each one point set being closed, every subset of E is closed as finite union of closed sets, hence every subset of E is open, and therefore E is homeomorphic to a discrete space. Finally, as there is only a finite number of open sets, E is compact.

(3.16.4) In a compact metric space E, any infinite sequence (x,) which has only one cluster value a converges to a. Suppose a is not the limit of (x,); then there would exist a number a > 0 such that there would be an infinite subsequence (x,,) of (x,) whose points belong to E - B(a; a). By assumption, this subsequence has a cluster value b , and as E - B(a; a ) is closed, b belongs to E - B(a; a ) by (3.13.7). The sequence (x,) would thus have two distinct cluster values, contrary to assumption.

(3.16.5) Any continuous mapping f of a compact metric space E into a metric space E‘ is uniforndy continuous. Suppose the contrary; there would then be a number a > 0 and two sequences (x,) and (y,) of points of E such that d(x,, y,) < I/n and d’(f(x,),f ( y , ) ) 3 a. We can find a subsequence (x,) converging to a point a, and as d(x,,, y,,) < l/nk,it follows from the triangle inequality that the sequence (ynL)also converges to a. But f is continuous at the point a, hence there is a 6 > 0 such that d ’ ( f ( a ) , f ( x ) ) < a / 2 for d(a, x) < 6. Take k such that d(a, x,) < 6, d(a, y n k )< 6; then d’(f(x,,),f(y,,)) < a contrary to the definition of the sequences (x,) and (y,).

(3.16.6) Let E be a compact metric space, (U,),ELan open covering of E. There exists a number a > 0 such that any open ball of radius a is contained in at least one of the U, (“ Lebesgue’s property ”). For every x E E, there exists an open ball B(x; r,) contained in one of the sets U,. As the balls B(x; r,/2) form an open covering of E, there exist a finite number of points x i E E such that the balls B(x,; r J 2 ) form a covering of E. If a > 0 is the smallest of the numbers rxi/2,it satisfies the required property: indeed, every x E E belongs to a ball B(x,; rJ2) for some i, hence B(x; a) is contained in B(x,; rXi)since a < r x i / 2 ;but by construction B(x,; r X i ) is contained in some U,.

17 COMPACT SETS 61 PROBLEMS 1. Give an example of a precompact space in which the result of (3.16.6) fails to be true. 2. For a metric space E, show that the following properties are equivalent: (a) E is compact; (b) every denumerable open covering of E contains a finite subcovering; (c) every decreasing sequence (F,) of nonempty closed sets of E has a nonempty intersection; (d) for any infinite open covering ( U A ) A eof, ~E, there is a subset H c L, distinct from L and such that (UA),." is still a covering of E; (e) every poititwisefinire open covering (U,) of E (i.e. such that for any point x E E, x E UA only for a finite subset of indices) contains a finite subcovering; (f) every infinite subspace of E which is discrete is not closed. (Using (3.16.1), show that (f) implies (a), and that (d) and (e) imply (f).) 3. Let E be a metric space, d the distance on E, %(E) c $(E) the set of all closed nonempty subsets of E. We may suppose that the distance on E is bounded (Section 3.14, Problem 2). For any two elements A, B of S(E), let p(A, B) = sup d(x, B), h(A, B) = X€A sup(p(A, B), p(B, A)). (a) Show that, on g(E), h is a distance (the "Hausdorff distance"). (b) Show that for any four elements A, B, C, D of g(E), one has

h(A u B, C u D) < max(h(A, C), h(B, D)). (c) Show that if E is complete, 8(E) is complete. (Let (X.) be a Cauchy sequence in s(E); for each n, let Y , be the closure of the union of the sets X,,, such that p 2 0; consider the intersection of the decreasing sequence (Y.) in E.) (d) Show that if E is precompact, %(E) is precompact (use the problem in Section 1.1). Therefore, if E is compact, 8(E) is compact. 4. Let E be a compact metric space. For every E > 0, let N,(E) be the smallest integer n such that there exists a covering of E by n sets of diameter < 2.5; let M,(E) be the largest integer m such that there exists a finite sequence of m points of E for which the distance of any two (distinct) of these points is > 8 . The number H,(E) = log N,(E) is called the &-entropy of E, the number C,.(E) = log MJE) the &-capacity of E. (a) Show that M,,(E) < N,(E) < M,(E), hence C,,:(E) < H,(E) < C,(E). (b) Show that the functions N,(E) and MJE) of E , defined for 8 > 0, are decreasing and continuous on the right (to prove the continuity of N,(E) on the right, use contradiction, and apply problem 3(d)). (c) If A and B are closed nonempty subsets of E, show that NdA

LJ

M A LJ

+N O ) , B) < HAA) + H,(B), B) < N A N

M,(A

LJ

B) < MAA)

CAA LJ B) < CAA)

+ MAB)

+ CdB).

(d) If E is a closed interval of R of length I , show that NJE) = M2,(E) = 1/28 if / / 2 is~ an integer, and N,(E) = M,,(E) = [//2.5] 1 (where [ I ] is the largest integer < t for t > 0) if / / 2 ~is not an integer.

+

17. COMPACT SETS

A compact (resp. precompact) set in a metric space E is a subset A such that the subspace A of E be compact (resp. precompact).

62

111 METRIC SPACES

(3 .I 7.1)

Any precompact set is bounded.

This follows from the fact that a finite union of bounded sets is bounded (3.4.4). The converse of (3.17.1) does not hold in general, for any distance is equivalent to a bounded distance (Section 3.14, Problem 2) (but see (3.17.6)).

(3.17.2)

Any compact set in a metric space is closed.

Indeed, such a subspace is complete by (3.16.1), and we need only apply (3.14.4).

(3 .I 7.3) In a compact space E, every closed subset is compact.

For such a set is obviously precompact, and it is a complete subspace by (3.14.5).

A relatively compact set in a metric space E is a subset A such that the closure A is compact. (3.17.4) Any subset of a relatively compact (resp. precompact) set is relatioely compact (resp. precompact).

This follows at once from the definitions and (3.17.3).

(3.17.5) A relatively compact set is precompact. In a conlplete space, a precompact set is relatively compact.

The first assertion is immediate by (3.17.4). Suppose next E is complete and A c E precompact. For any E > 0, there is a covering of A by a finite number of sets c, of A having a diameter < ~ / 2 each ; ck is contained in a closed ball D, (in E) of radius 4 2 . We have therefore A c D,, the set

u D, k

u

being closed, and each D, has a diameter

k

< E . On the other hand, A is a

complete subspace by (3.14.5), whence the result. A precompact space E which is not complete gives an example of a precompact set which is not relatively compact in E.

17 COMPACT

SETS 63

(3.17.6) (Borel-Lebesgue theorem). In order that a subset of the real line be relatively compact, a necessary and suficient condition is that it be bounded. In view of (3.17.1), (3.17.4), and (3.17.5), all we have to do is to prove any closed interval [a, b] is precompact. For each integer n, let xk = a k(b - a)/n (0 < k d I ? ) ; then the open intervals of center xk and length 2/n form a covering of [a, b]. Q.E.D.

+

Remark. If, on the real line, we consider the two distances d,, d, defined in Section 3.14, it follows from (3.17.1) that E, is not precompact, whereas El is precompact, since the extended real line R, being homeomorphic to the closed interval [- 1, + I ] of R (3.12), is compact by (3.17.6). (3.17.7) A necessary and sufJjcient condition that a subset A of a metric space E be relatively compact is that every sequence of points of A have a cluster value in E. The condition is obviously necessary, by (3.16.1). Conversely, let us suppose it is satisfied, and let us prove that every sequence (x,) of points of A has a cluster value in E (which will therefore be in A by (3.13.7)), and hence that A is compact by (3.16.1). For each n, it follows from the definition of closure that there exists y , E A such that d(x,, y,) d Ijn. By assumption, there is a subsequence (y,J which converges to a point a ; from the triangle inequality it follows that (x,,) converges also to a. Q.E.D. (3.17.8) The union of two relatiuely compact sets is relatively compact. From (3.8.8) it follows that we need only prove that the union of two compact sets A, B is compact. Let (UJdELbe an open covering of the subspace A u B ; each U, can be written (A u B) n V,, where V, is open in E, by (3.10.1). By assumption, there is a finite subset H (resp. K) of L such that the subfamily(A n V,)neb,(resp. (6 n VA)AeK)is a covering of A (resp. B). It is then clear that the family ((A n B) n V,), E H is a covering of A u B. (3.17.9) Let f be a continuous mapping of a metric space E into a metric space E‘. For every compact (resp. relatiuely compact) subset A of E, ,f(A) is compact, hence closed in E’ (resp. relatively cornpact in El).

64

Ill

METRIC SPACES

It is enough t o prove that f ( A ) is compact when A is compact. Let ( U l ) l E Lbe an open covering of the subspace ,f(A) of E'; then the sets A nf -'(UA) form an open covering of the subspace A by (3.11.4); by assumption, there is a finite subset H of L such that the sets A n , f - ' ( U L ) for I E: H still form a covering of A ; then the sets U, =f(A n f - ' ( U L ) ) for ;IE H will form a covering of,f(A). Q.E.D.

(3.17.10) Let E be a nonetiipty compact metric space, f a continuous mapping of E into R; tliett,f(E) is bounded, and there exist tlt'o points a , h in E such that . f @ ) = inf,fW,f(b) = sup.f(.4. xeE

XE

E

The first assertion follows from (3.17.9) and (3.17.1). On the other hand, f(E) is closed in R by (3.17.2), hence supf(E) and inff(E), which are cluster points off(E), belong to,f(E).

(3.17.1 1) Let A be a conipact subset in a metric space E. Then the sets V,(A) (see Section 3.6) f o r m a fundatiiental system of neigliborhoods of A. Let U be a neighborhood of A ; the real function x + d(x,E - U) is > 0 and continuous in A by (3.11.8), hence there is a point xo E A such that d(xo,E - U) = inf d(x, E - U) by (3.17.10). But d(xo,E - U) = r > 0; X E A

hence V,(A) c U.

(3.17.12) I f E is a compact nietric space, f a continuous injectii'e tilapping of E into a metric space E', then f is a lrotiieotiiorphism of' E onto f (E). All we need to prove is that for every closed set A c E, ,f(A) is closed inf(E) (by (3-11.4)); but this follows from (3.17.3) and (3.17.9).

PROBLEMS

1. Let f be a uniformly continuous mapping of a metric space E into a metric space E'. Show that for any precoinpact subset A of E,f(A) is precompact. 3. In a metric space E, let A be a compact subset, B a closed subset such that A n B = 0. Show that d(A, B) > 0.

18 LOCALLY COMPACT SPACES 65 3. Let E be a compact ultrametric space (Section 3.8, Problem 4), d the distance on E. Show that for every xo E E, the image of E by the mapping .Y -+cf(xo, x ) is an at most denumerable subset of the interval [0, +a[in which every point (with the possible exception of 0) is isolated (3.10.10).(For any I’ = d(xo,x ) > 0, consider the 1.u.b. of cl(xo, y ) on the set of points y such that d(x,,, y ) < 1’, and the g.1.b. of d(xo,z) on the set of points z such that c/(x,, z) > r; use Section 3.8,Problem 4). 4. Let E be a compact metric space, d the distance on E,fa mapping of E into E such that, for any pair (x, y ) of points of E, d(f(x),f(y)) 2 d(x, y). Show thatfis an isometry ofE onto E. (Let a, b be any two points of E ; put fn =fn-, o f , a, =fn(a), b, =f,(b); show that for any E > 0 there exists an index k such that d(a, ak) C E and d(b, bk) < E (consider a cluster value of the sequence (a“)), and conclude that f(E) is dense in E and that 4 f ( a ) , f ( b ) )= 4 0 , b).) 5. Let E, E’ be two metric spaces,fa mapping of E into E’. Show that if the restriction off to any compact subspace of E is continuous, thenfis continuous in E (use (3.13.14)). 6 . Let E,E’ be two metric spaces, f a continuous mapping of E into E , K a compact subset of E. Suppose the restriction f l K off is injective and that for every x E K, there is a neighborhood V, of x in E such that the restriction f 1 V, o f f is injective. Show that there exists a neighborhood U of K in E such that the restriction f 1 U is injective (use contradiction and (3-17.11)).

18. LOCALLY COMPACT SPACES

A metric space E is said to be locally conipact if for every point x E E, there exists a compact neighborhood of x in E. Any discrete space is locally compact, but not compact unless it is finite (3.16.3).

(3.18.1) The real line R is locally conipact hut not compact. This follows immediately from the Borel-Lebesgue theorem (3.17.6).

(3.18.2) Let A be a conipact set in a locally compact metric space E. Then there exists an r > 0 such that V, (A) (see Section 3.6) is relatitlely conzpact in E. For cach x E A, there is a compact neighborhood V, of x ; the 9, form an open covering of A, hence there is a finite subset {xl, . . . , x,} in A such that the

TX,(1 < i 6 n) form an open covering of A. The set U =

u n

i= 1

V,: is com-

pact by (3.17.8) and is a neighborhood of A; hence the result, by applying (3.17.11).

66

Ill METRIC SPACES

(3.18.3) Let E be a locally compact metric space. The following properties are equivalent: (a) there exists an increasing sequence (U,) of open relatively conipact c Un+l for every 11, and E = U,; sets in E such that

u

n,

n

(b) E is a denurnerable uriion of compact subsets; (c) E is separable.

u,

It is clear that (a) implies (b), since is compact. If E is the union of a sequence (K,) of compact sets, each subspace K, is separable (by (3.16.2)); if D, is an at most denumerable set in K,, dense with respect to K,,, then D = D, is at most denumerable, and dense in E, since E =

u n

K, c

u u 15, n

c

15; hence (b) implies (c). Let us suppose finally that E

fl

is separable, and let (T,) be an at most denumerable basis for the open sets of E (see (3.9.4)). For every x E E, there is a compact neighborhood W, of x, hence, by (3.9.3), an index M ( X ) such that x E Tnc,) c W,. It follows that those of the T, which are relatively compact already constitute a basis for the open sets of E. We can therefore suppose that all the T, are relatively compact. We then define U, by induction in the following way: U1 = TI, U n + , is the union of Tntl and of V,(U,), where r > 0 has been taken such that V,(u,) is relatively compact (which is possible by (3.18.2)); it is then clear that the sequence (U,) verifies property (a). (3.18.4) In a locally compact metric space E, eriery opeii subspace atid every closed subspace is locally compact. Suppose A is open in E; for every a E E, there is a closed ball B’(a; r ) which is compact (from the definition of a locally compact space and (3.17.3)). On the other hand, there is r‘ Q r such that the ball B’(a; r’) is contained in A ; as it is compact by (3.17.3), A is locally compact. Suppose A is closed in E, and let a E A ; then, if V is a compact neighborhood of a in E, V n A is a neighborhood of a in A by (3.10.4), and is compact by (3.17.3); this proves A is locally compact.

PROBLEMS 1. If A is a locally compact subspace of a metric space E, show that A is locally closed (Section 3.10, Problem 3) in E. The converse is true if E is locally compact (use (3.18.4)). 2. (a) Show that in a locally compact metric space, the intersection of two locally com-

pact subspaces is locally compact (cf. Problem I).

19 CONNECTED SPACES AND CONNECTED SETS 67

(b) In the real line, give an example of two locally compact subspaces whose union is not locally compact, and an example of a locally compact subspace whose complement is not locally compact. 3. (a) Give an example of a locally compact metric space which is not complete. (b) Let E be a metric space such that there exists a number r > 0 having the property that each closed ball B’(x; r ) ( x E E) is compact. Show that E is complete and that for any relatively compact subset A of E, the set V:,,(A) of the points x E E such that d(x, A) < rj2 is compact.

19. CONNECTED SPACES A N D CONNECTED SETS

A metric space E is said to be connected if the only subsets of E which are both open and closed are the empty set /zr and the set E itself. An equivalent formulation is that there does not exist a pair of open nonempty subsets A, B of E such that A u B = E, A n B = /zr. A space reduced to a single point is connected. A subset F of a metric space E is connected if the subspace F of E is connected. A metric space E is said to be locally connected if, for every x E E, there is a fundamental system of connected neighborhoods of+.

(3.19.1) In order that a subset A of the real line R be connected, a necessary and suficient condition is that A be an interval (bounded or not). The real line is a connected and locally connected space. The second assertion obviously follows from the first. Suppose A is connected; if A is reduced to a single point, A is an interval. Suppose A contains two distinct points a < 6. We prove every x such that a < x < b belongs to A. Otherwise, A would be the union of the nonempty sets B = A n ] - c o , x [ and C = A n ] x , +a[, both of which are open in A and such that B n C = From this property, we deduce that A is necessarily an interval. Indeed, let c E A, and let p , q be the g.1.b. and 1.u.b. of A in R ; if p = -a,then for every x < c, there is y < x belonging to A ; hence x E A, so ] - co, c] is contained in A; if p is finite and p < c, for every x such that p x < c there is y E A such that p < y < x, hence again x E A, so that A contains the interval ] p , c]. Similarly, one shows that A contains [c,q[ if q > c ; it follows that in any case A contains the interval ]p,q [ , and therefore must be one of the four intervals in R of extremities p , q (of course, if p = - 00 (resp. q = 00) p (resp. q ) does not belong to A). Conversely, suppose A is a nonempty interval of origin a and extremity b in R (the possibilities a = - co, n 4 A, b = + co, b 4 A being included).

a.

-=

+

68

I l l METRIC SPACES

Suppose A = B LJ C, with B, C nonempty open sets in A and B n C = 0; suppose for instance x E B, y E C, and x < y . Let z be the 1.u.b. of the bounded set B n [x, y ] ; if z E B, then z < y and there is by assumption an interval [ z , z + h [ contained in [x,y ] and in B, which contradicts the definition of z ; if on the other hand z E C, then x < z , and there is similarly an interval ] z - h, z ] c C n [x,y ] , which again contradicts the definition of z (see (2.3.4));hence z cannot belong to B nor to C, which is absurd since the closed set [x, y ] is contained in A. Hence A is connected.

(3.19.2) rf' A is a connected set in a nietvic space E, then any set B such that A c B c A is connected.

For suppose X, Y are two nonempty open sets in B such that X u Y = B, X n Y = @; as A is dense in B, X n A and Y n A are not empty, open in A, and wewouldhave(X n A) LJ (Y n A) = A , ( X n A) n (Y n A) = 0, a contradiction.

(3.19.3) 111 a metric space E, let (A,), be a family of connected sets haoing a nonempty igtersection; theti A = A, is connected.

u

,EL

Let a be a point of

n A,, and suppose A

=B

v C, where B, C are non-

LEL

empty open sets in A such that B n C = @. Suppose for instance a E 0 ; by assumption there is at least one A such that C n A, # 0; then B n A, and C n A, are open in A, and such that (B n A,) u (C n A,) = A,, (B n A,) n (C n A,) = 0, a contradiction since B n A, # @.

(3.19.4) Let ( A i ) l f i s nbeasequence of coniiected sets such that A i n

0for 1 < i d n - 1;

then

u Ai n

#

is connected.

i= 1

This follows at once from (3.19.3)by induction on n. From (3.19.3)it follows that the union C ( x )of allconnected subsets of E containing a point x E E is connected, hence the largest connected set containing x; it is called the connected component qf x in E. It is clear that for any y E C(x), we have C ( y ) = C(x), and if y $ C(x), then C(x) n C ( y ) = 0; moreover, it follows from (3.19.2)that C(x) is closed in E. For any subset A of E, the connected components (in the subspace A) of the points of the subspace A are called the connected coniponents of A ; if every connected component of A is reduced to a single point, A is said to be totally disconnected.

19 CONNECTED SPACES AND CONNECTED SETS 69

A discrete space is totally disconnected; the set of rational numbers and the set of irrational numbers are totally disconnected, by (2.2.16) and (3.19.1). (3.19.5) In order that a nietric space E be locally connected, a necessary and suflcient condition is that the connected coniponents of the open sets in E be open. The condition is sufficient, for if V is any open neighborhood of a point E, the connected component of x in the subspace V is a connected neighborhood of x contained in V, hence E is locally connected. Thecondition is necessary, for if E is locally connected and A is an open set in E, B a connected component of A, then for any x E B, there is by assumption a connected neighborhood V of x contained in A, hence V c B by definition of B, and therefore B is a neighborhood of every one of its points, hence an open set. x

E

(3.19.6) Any nonernpty opeti set A in the real line R is the union of an at most denumerable .family of' open interoals, ti0 two of which have common points. From (3.19.1) and (3.19.5) it follows that the connected components of A are intervals and open sets, hence open intervals. The intersection A n Q of A with the set Q of rational numbers is denumerable, and each component of A contains points of A n Q by (2.2.16); the mapping Y -+ C(r) of A n Q into the set 0.of the connected components of A is thus surjective, and therefore, by (1.9.2), 0.is at most denumerable. (3.19.7) Let f be a continuous niapping of E into E'; for any connected E, f(A) is connected.

subset A of

Suppose f ( A ) = M u N, where M and N are nonempty subsets of f(A), then, by (3.11.4), A n f -'(M) open in f(A) and such that M n N = 0; and A n f -'(N) would be nonempty sets, open in A and such that A = (A nf -'(M))u(A n f-'(N))and(A n f - ' ( M ) ) n ( A n f-'(N)) = contrary to assumption.

a,

(3.19.8) (Bolzano's theorem) Let E be a connected space, f a continuous niapping of E into the real line R. Suppose a, b are two points of f(E) such

70

Ill

METRIC SPACES

tliat a < h. Theti, f o r

anj’

c such tliat a < c < h tkere exists x

E

E such that

f ( x ) = c.

Forf(E) is connected in R by (3.19.7), hence an interval by (3.19.1). (3.19.9) Let A be a subset of a metric space E. If‘ B is a coiinecred siibset of E such that both A n B arid (E - A) n B are not empty, theii (fr(A)) n B is not empty. In particular, if E is connected, any subset A o j E u‘istiiict f i w r r E atid 0has at least oiie froiitier point. Suppose (fr(A)) n B = 0; let A‘ = E - A ; as E is the union of A, and fr(A), B would be the union of U = A n B and V = A’ n B, both of which are open in B and not empty by assumption (for a point of A n B must belong to A n B since fr(A) n B = 0, and similarly for A‘ n B); as U n V 0, =this would be contrary t o the assumption that B is connected.

A’

Remark. If we agree to call “curve” the image of an interval of R by a continuous mapping (see Section 4.2, Problem 5), (3.19.7) shows that a “curve” is connected, and (3.19.9) that a “curve” linking a point of A and a point of E - A meets fr(A), which corresponds to the intuitive idea of connectedness” (see Problem 3 and Section 5.1, Problem 4). “

PROBLEMS

1. Let E be a connected metric space, in which the distance is not bounded. Show that in

E every sphere is nonenipty. (a) Let E be a compact metric space such that in E, the closure of any open ball B(u: r ) is t h e closed ball B’(a; r ) . Show that in E any open ball B(a; 1.) is connected. (Suppose B(a; r ) is the union C D of two noncmpty sets which are open in B(u: r ) and if ci E C , consider a point x F D such that the distance I . Show that E is a locally compact subspace of E, which is not locally connected; the connected components of E are B, C and the A,, 0 7 3 I ) , but the intersection of all open and closed sets containing a point of B is B u C. Let E be a locally compact metric space. (a) Let C be a connected component of E which is compact. Show that C is the intersection of all open and closed neighborhoods of C. (Reduce the problem to the case in which E is compact, using (3.18.2). Suppose the intersection B of all open and closed neighborhoods of C is different from C; B is the union of two closed sets M 3 C and N without common points. Consider in E two open sets U 3 M and V 3 N without common points (Section 3.11, Problem 3). and take the intersections of E - (M u N) with the complements of the open and closed neighborhoods of C.) (b) Suppose E is connected, and let A be a relatively compact open subset of E. Show that every connected component of A has at least a cluster point in A (if not, apply (a) to such a component, and get a contradiction). (c) Deduce from (b) that for every compact subset K of E, the intersection of a connected component of K with E - K is not empty.

c

7. 8.

9.

PRODUCT OF TWO METRIC SPACES 71

c

c

c

c

-_.

20. P R O D U C T O F TWO M E T R I C SPACES

Let El, E, be two metric spaces, d,, dz the distances on El and E, . For any pair of points x = (xl, x,), y = (yl, yz) in E = El x E, , let

4 x ,v>= max(d,(x,, A),4(xZ, yZ)>.

72

111

METRIC SPACES

It is immediately verified that this function satisfies the axioms (I) to (1V) in Section 3.1, in other words, it is a distance on E ; the metric space obtained by taking d as a distance on E is called the product of the two metric spaces El, E, . The mapping (x,,x,) 4 (x, , sl) of El x E, onto E, x El is an isometry. We observe that the two functions d’, d” defined by

d‘(x,Y>= 4(.%Y l )

+ dz(x2

d ” k v) = ((dl(X1, Y l V

9

Yz)

+ (CJ2(x,

9

Y2)>2>”2

are also distances on E, as is easily verified, and are unifornily equioalerit to d (see Section 3.14),for we have

d(x,I’) d d”(x, y ) < d’(x, 4’) < 2 4 % Y ) . For all questions dealing with topological properties (or Cauchy sequences and uniformly continuous functions) it is therefore equivalent to take on E any one of the distances d, d’, d”. When nothing is said to the contrary, we will consider on E the distance d. Open (resp. closed) balls for the distances d, d,, d, will be respectively written B, B,, B, (resp. B’, B’,, B;) instead of the uniform notation B (resp. B’) used up to now.

(3.20.1) For any point a = ( a l , a,) E E a i d arty r > 0, we hare B(a; r ) = Bl(al ; r ) x B,(a, ; r ) and B’(a; r ) = B;(a, ; r ) x B;(a, ; r ) . This follows at once from the definition of d.

(3.20.2) I f A , is an open set in El, A , an open set in E,, then A, x A, is open in E, x E, . For if a = (a,, a , ) E A l x A,, there exists rl > 0 and r, > 0 such that B,(a,; rl) c Al, B,(a,; r z ) c A,; take r = min(r,, r , ) ; then by (3.20.1), B(a; r ) c A, x A,,

(3.20.3) For any pair of sets A, c El, A, c E,, A, x A, = A , x A,; in particular, in order that A, x A , be closed in E, a necessary and suficient condition is that A, be closed in El and A, closed in Ez . If a = (a,, a,) EA, x A,, for any&> 0 there is, by assumption, an x1 E A , and an x, E A, such that d,(a,,x,)< E , d z ( a z ,x,) < E ; hence if x = (x,,x,),

20

PRODUCT OF TWO METRIC SPACES 73

d(a, x) < E . On the other hand, if (a,, a,) $A, x A, then either a, $ A, o r a, # A,; in the first case, the set (El - A,) x E, is open in E by (3.20.2), contains a and has a n empty intersection with A, x A,, hence a 4 A, x A,; the other case is treated similarly.

(3.20.4) Let z f ( z ) = (f,(z),f,(z)) be a mapping of a metric space F into E = El x E,; in order that f be continuous at a point z o , it is necessary and --f

suflcient that both f l and f 2 be continuous at zo . Let xo = ( f i ( z o ) f2(z0)); , then we have

by (3.20.1),and the result follows from (3.11 .I) and (3.6.3).

(3.20.5) Let f = ( , f l , f , ) be a mapping of a subspace A of a metric space F into El x E,, and let a EA; in order that f haoe a limit at the point a

with respect to A, a necessary and suflcient condition is that both limits b, = lirn f,(z), b , = lirn f2(z) exist, and then the limit of f is =-+a,z

z-a, z E A

b = (h,b2).

E

A

This follows at once from (3.20.4)and the definition of a limit. In particular:

(3.20.6) In order that a sequence of points z, = (x,,, y,) in E = El x E, be confiergent, a necessary and suflcient condition is that both limits a = lirn x,, b = lim y,, exist and then lim z, = ( a , b). n+

m

?I-+ m

n+ m

Note that for cluster values of sequences, if (a, b) is a cluster value of a is a cluster value of (x,,) and b a cluster value of (y,,), as follows from (3.20.6)and the definition of cluster values; but it may happen that ((x,, y,)) has no cluster value, although both (x,,) and (y,,) have one: for instance, in the plane R2, take x,, = l / n , y,, = n, x,,+~= n, y2n+l= l/n. However, if (x,) has a limit a, and b is a cluster value of(y,,), then (a, b) is a cluster value of ( ( x , , y,)), as follows from (3.20.6). ( ( x , , y,)),

74

I l l METRIC SPACES

(3.20.7) In order that a sequence of points z, = (x,, y,) in El x E, be a Cauchy sequence, a necessary and sufficient condition is that each of the sequences (x,), (y,) be a Cauchy sequence. This follows at once from the definition of the distance in El x E2 and the definition of a Cauchy sequence.

(3.20.8) Let z -+f(z) = ( f i ( t ) ,f2(z)) be a mapping of a metric space F into El x E,; in order that f be uniformly continuous, it is necessary and sufficient that both fland f2 be uniformly continuous. This follows immediately from the definitions.

(3.20.9) I f E is a metric space, d the distance on E, the mapping d of E x E into R is uniformly continuous. For Id(x, y ) - d(x’, y’)l

< d(x,x’) + d(y, y’) by the triangle inequality.

(3.20.10) Theprojectionspr, andpr,areuniformlycontinuous in E = El x Ez . Apply (3.20.8)to the identity mapping of E.

(3.20.11) For any a, E E2 (resp. a, E El), the mapping x1 +(xl, a,) (resp. x, 4(al, x,)) is an isometry of El (resp. E,) on the closed subspace El x {a,} (resp. {all x E2) of El x E, * This is an obvious consequence of the definition of the distance in El x E,, and of (3.20.3).

(3.20.12) For any open (resp. closed) set A in El x E, , and any point a , E El, rhe cross section A(al) = pr2(A n ({al} x E,)) is open (resp. closed) in E, .

By (3.20.11)it is enough to prove that the set A

n ({a,} x E,) is open

(resp. closed) in {al} x E z , which follows from (3.10.1)and (3.10.5).

(3.20.13) For any open set A in El x E,, pr, A (resp. pr, A) is open in El (resp. Ez).

20 PRODUCT OF TWO METRIC SPACES 75

Indeed, we can write pr, A =

u

XI

A(x,), and the result follows from

EEI

(3.20.12) and (3.5.2). Note that if A is closed in El x E,, pr, A needs not be closed in El. For instance, in the plane R2,the hyperbola of equation x y = 1 is a closed set, but its projections are both equal to the complement of (0) in R, which is

not closed. Let f be a mapping of E = El x E, into a metric space F. I f f is continuous at a point (al,a,) (resp. uniformly continuous), then the mapping f ( . , a,): x, --f f (x,, a,) is continuous at a, (resp. uniformly continuous). (3.20.14)

That mapping can be written x1 4(x,, a,) -+f ( x l ,a,), hence the result follows from (3.20.11), (3.11.5), and (3.11.9). The converse to (3.20.14) does not hold in general. A classical counterexample is the function f defined in R2 by f (x, y ) = xy/(x2 y 2 ) if ( x ,y ) # (0,O)andJ(0,O) = 0;f is not continuous at (0, 0), forf ( x ,x ) = 1/2 for x # 0.

+

Let El, E 2 , F,, F, befour metric spaces,f , (resp. f,)a mapping of El into F1(resp. of E, into F,). In order that the mappingf:(xL,x 2 )+ (f,(x,),f,(x,)) of El x E, into F, x F, be continuous at a point (a,,az) (resp. uniformly continuous), it is necessary and su$cient that f , be continuous at a, and f , at a, (resp. that both f, and f , be uniformly continuous). (3.20.15)

The mapping (x,, x,) +fi(xl) can be written f i 0 prl, hence the sufficiency of the conditions follow from (3.20.4), (3.20.8), and (3.20.10). On the other hand, the mapping fi can be written x, -+ pr,( f ( x , , a,)) and the necessity of the conditions follows from (3.20.14) and (3.20.10). (3.20.16) Let El, E, be two nonempty metric spaces. In order that E = El x E,

be a space of one of the following types: (i) discrete; (ii) bounded; (iii) separable; (iv) complete; (v) compact; (vi) precompact; (vii) locally compact; (viii) Connected; (ix) locally connected; it is necessary and suficient that both El and E2 be of the same type.

76

I l l METRIC SPACES

The necessity part of the proofs follows a general pattern for properties (i) to (vii): from (3.20.11) it follows that El and E, are isometric to closed subspaces of El x E,; and then we remark that properties (i) to (vii) are “inherited” by closed subspaces (obvious for (i) and (ii), and proved for properties (iii) to (vii) in (3.10.9), (3.14.5), (3.17.3), (3.17.4), (3.18.4)). For property (viii), the necessity follows from (3.19.7) applied to the projections pr, and pr,; similarly, if E is locally connected, for any (a,, a 2 )E E and any neighborhood V, of a, in El, V, x E, is a neighborhood of (a,, u2), hence contains a connected neighborhood W of (a,, a,); but then pr, W is a connected neighborhood of a, contained in V,, by (3.19.7) and (3.20.13). The suficiency of the condition for (i) and (ii) is an obvious consequence of the definition of the distance in E, x E, . For (iii), if D,, D, are at most denumerable and dense in E,, E, respectively, then D, x D, is at most denumerable by (1.9.3), and is dense in E by (3.20.3). For (iv), if (z,) is a Cauchy sequence in E, then (pr, z,) and (pr, z,) are Cauchy sequences in El and E, respectively by (3.20.7), hence they converge to a,, a2 respectively, and therefore (z,) converges to (a,, a*) by (3.20.6). For (vi), if (A,) (resp. (B,)) is a finite covering of El (resp. E2) by sets of diameter < c , then ( A i x B,) is a finite covering of El x E, by sets of diameter < E ; and by (3.16.1), the sufficiency of the condition for (iv) and (vi) proves it also for (v). The proof for (v) yields a proof for (vii) if one remembers the definition of neighborhoods in El x E,. For (viii), let (a,, a2), (b,, b 2 ) be any two points of E; by (3.20.11) and the assumption, the sets { a , } x E, and El x {b,} are connected and have a common point (a,, b,). Hence their union is connected by (3.19.3), and it contains both (a,, a,) and (b,, b 2 ) ;therefore, the connected component of (al, a 2 ) in E is E itself. The same argument proves the sufficiency of the condition for (ix), remembering the definition of the neighborhoods in E. (3.20.17) I n order that asubset A of El x E, be relatively conlpacr, a )iecessary and suficient condition is that prl A ai7d pr, A be relatioely compact ii7 El ai7d E, respectively. The necessity follows from (3.17.9) applied to pr, and pr,; the sufficiency follows from (3.20.16), (3.20.3) and (3.17.4).

All definitions and theorems discussed in this section are extended at once to a finite product of metric spaces.

20

PRODUCT OF TWO METRIC SPACES

77

PROBLEMS

1. Let E, F be two metric spaces, A a subset of E, B a subset of F ; show that fr(A x B) = (fr(A) x B) u (A x fr(B)). 2. Let E, F be two connected metric spaces, A # E a subset of E, B # F a subset of F; show that in E x F the complement of A x B is connected. 3. (a) Let E, F be two metric spaces, A (resp. B) a compact subset of E (resp. F). If W is any neighborhood of A x B in E Y F, show that there exists a neighborhood U of A in E and a neighborhood V of B in F such that U x V c W (consider first the case in which B is reduced to one point). (b) Let E be a compact metric space, F a metric space, A a closed subset of E x F. Show that the projection of A into F is a closed set (use (a) to prove the complement of prz A is open). (c) Conversely, let E be a metric space such that for every metric space F and every closed subset A of E x F, the projection of A into F is closed in F. Show that E is compact. (If not, there would exist in E a sequence (x,) without a cluster value. Take for F the subspace of R consisting of 0 and of the points l / n (n integer 2 1) and consider in E Y F the set of the points ( x n , lit?).) 4. Let E be a compact metric space, F a metric space, A a closed subset of E x F, B the (closed) projection of A into F. Let yo E B and let C be thecross section A-’(yo) = { x E E I ( x , y o ) E A]. Show that for any neighborhood V of C in E, there is a neighborhood W of yo in F such that the relation y c W implies A-’(y) C V (“continuity of the “roots” of an equation depending on a parameter”). (Use Problem 3(a).) 5. (a) Let f be a mapping of a metric space E into a metric space F, and let G be the graph off’in E x F. Show that iff is continuous, G is closed in E x F, and the restriction of pr, to G is a homeomorphism of G onto E. (b) Conversely, if F is compact and G is closed in E x F, then f is continuous (use Problem 3(b)). (c) Let F be a metric space such that for any metric space E, any mapping of E into F whose graph is closed in E x F is continuous. Show that F is compact (use the construction of Problem 3(c)). 6. Let E, F, G be three metric spaces, A a subset of E x F, B a subset of F x G, C = B 0 A = {(x, z ) E E x G j3y E F such that (x, y ) E A and (y, z) E B}. Suppose bothA and B are closed and the projection of A into F is relatively compact; showthat C is closed in E x G (use Problem 3(b)). 7. Let (En) ( n 2 I ) be an infinite sequence of nonempty metric spaces, and suppose that for each ti, the distance d,, on En is such that the diameter of E. is < 1 (see Section 3.14,

Problem 2(b)). Let E be the infinite product

PIXI

En. m

(a) Show that on E the function d((xn),( y o ) )=

“=I

d,(x., y.)/2” is a distance.

(b) For any x = (x.) E E, any integer m >, 1 and any number r > 0, let VJx; r ) be the set of all y = (y.) E E such that & ( x k , yk) < r for k < m. Show that the sets V,,(x;r ) (for all m and v) form a fundamental system of neighborhoods of x in E. (c) Let (x‘”’))be a sequence of points x(‘”)= (x,!“’)).~~ of E; in order that ( x c m )converge ) to a = (a,) in E (resp. be a Cauchy sequence in E), it is necessary and sufficient that for each 12 the sequence (x!””).,~I converge to a. in En (resp. be a Cauchy sequence in En). In order that E be a complete space, it is necessary and sufficient that each En be complete.

78

Ill

METRIC SPACES r

(d) For each n , let A, be a subset of E, ; show that the closure in E of A equal to

fi A,.

=

IT A,, is

"=I

(e) In order that E be preconipact (resp. compact), i t is necessary and sufficient that each En be preconipact (resp. compact). (f) In order that E be locally compact, i t is necessary and sufficient that each En be locally compact, and that all En, with the exception of a finite number at most, be compact. ( 8 ) In order that E be connected, i t is necessary and sufficient that each E. be connected. (11) In order that E be locally connected, it is necessary and sufficient that each E,, be locally connected and that all Em,with the exception of a finite number at most, be connected.

CHAPTER I V

ADDITIONAL PROPERTIES OF THE REAL LINE

Many of the properties of the real line have been mentioned in Chapter 111, in connection with the various topological notions developed in that chapter. The properties gathered under Chapter IV, most of which are elementary and classical, have no such direct connection, and are really those which give to the real line its unique status among more general spaces. The introduction of the logarithm and exponential functions has been made in a slightly unorthodox way, starting with the logarithm instead of the exponential; this has the technical advantage of making it unnecessary to define first amin(117, n integers > 0) as a separate stepping stone toward the definition of ax for any x. The Tietze-Urysohn theorem (Section 4.5) now occupies a very central position both in functional analysis and in algebraic topology. It can be considered as the first step in the study of the general problem of exfendirlg a continuous mapping of a closed subset A of a space E into a space F, to a continuous mapping of the whole space E into F; this general problem naturally leads to the most important and most actively studied questions of modern algebraic topology.

1. C O N T I N U I T Y OF ALGEBRAIC OPERATIONS

+ y of R x R into R is uniformly continuous. This follows at once from the inequality

(4.1.I) The mapping (x,y ) -+ x

+ < Ix’ - XI + ly’ - yl

I(x’ + y’) - ( x y)l and the definitions.

79

IV ADDITIONAL PROPERTIES OF THE REAL LINE

80

(4.1.2) The mapping (x, y ) -+ xy of R x R into R is continuous; for any a E R, the mapping x -+ a x of R into R is uniformly continuous. Continuity of xy at a point (xo, yo) follows from the identity XY - XOYO = Xo(Y

- Y o ) + (x - X 0 ) Y O

+ (x - X O ) ( Y

- Yo).

Given any E > 0, take 6 such that 0 < 6 < 1 and 6(1x01 + lyol + 1) < E ; then the relations Ix - xo( < 6, ly - yoJ < 6 imply [xy - ?coyo[< c. Uniform continuity of x -+ax is immediate, since [ax' - ax1 = J a J. Jx'- XI.

(4.1.3) Any continuous mapping f of R irito itsdf such that f(x +f ( y ) is of type x -+ cx, with c E R.

+ y) =

f(x)

Indeed, for each integer n > 0, we have, by induction on n, f ( n x ) = nf(x); on the other handf(0 + x) =f ( 0 ) + f ( x ) , hencef(0) = 0, and f ( x + (-x)) = f ( x ) f ( - x ) = f ( 0 ) = 0, hence , f ( - x ) = -f(s). From that it follows that for any integer n > 0, f ( x / n ) = f ( x ) / n , hence for any pair of integers p , q such that q > 0, f ( p x / q ) =!f'(x)/q; in other words, f ( r x ) = rf(x) for any rational number r. But any real number t is limit of a sequence (r,) of rational numbers (by (2.2.16) and (3.13.13)),hence, fromtheassumptionon,fand (4.1.2) f ( t x ) = f(lim r,x) = limf(r,x) = lim r,f'(.x) = f ( x ) . lim r, = t f ( x ) . Let then,

+

n-1 a0

n-tm

n-tm

c = . f ( l ) , and we obtainf(x) = c x for every

(4.1.4)

x E R.

n+

cc

The mapping x -+ l/x is continuous at ecerj, poitit xo # 0 in R.

For given any E > 0, take 6 > 0 such that 6 < min(lxol/2, ~Ix,[~/2); then the relation Jx-xoJ < 6 first implies 1x1 > Jxol- 6 > 1x01/2, and then Il/x - l/xol = 1x0 - xl/lxx,I Q 21x0 - xl/lxo[2< E .

Any rational function (xl, . . . , x,) -+ P(xl, . . . , x,)/Q(x,, . . . , x,) where Q are polynomials with real coeficients, is continuous at each point (al, . . . , a,,) qf R" where Q(a,, . . . , a,) # 0. (4.1.5)

P

and

The continuity of a monomial in R" is proved from (4.1.2) by induction on its degree, then the continuity of P and Q is proved from (4.1.1) by induction on their numbers of terms; the final result follows from (4.1.4).

1

(4.1.6) The inappings

continuous in R x R.

CONTINUITY OF ALGEBRAIC OPERATIONS

(x,y ) + sup(x, y ) and

81

(x, y ) --+ inf(x, y ) are unifornzly

+ +

Ix - y ( ) / 2 and inf(x, y ) = (x As sup(x, y ) = (x y the result follows from (4.1.I) and (3.20.9).

+ y - (x - y ( ) / 2 ,

(4.1.7) All open intervals in R are homeomorphic to R.

From (4.1 .I .) and (4.1.2) it follows that any linear function x -+ a x + b, with a # 0, is a homeomorphism of R onto itself, for the inverse mapping x - i a - ' x - a-'b has the same form. Any two bounded open intervals ]LY, /?[, ] y , 6[ are images of one another by a mapping x --+ax+ b, hence are homeomorphic. Consider now the mapping x --t xi( I + 1x1) of R onto 1- 1, I[; the inverse mapping is x xi( 1 - 1x1) and both are continuous, since x --t 1x1 is. This proves R is homeomorphic to any bounded open interval; finally, under the preceding homeomorphism of R onto ] - I , + 1[, any unbounded open interval ]a, + a[or ] - co, a[ of R is mapped onto a bounded open interval contained in ] - I , + 1 [, hence these intervals are also homeomorphic to R .

+

--f

+

(4.1.8) With respect to R x R, the ,fuiiction (x, y ) x y has a lilllit at euerypoint (a, b) g f R x R, except at the points (- GO, + co) and ( + co, - 00); that litnit is equal to + cc (resp. - 00) if' one at least of tl7e coordinates a, b is +co (resp. -m). -+

Let us prove for instance that if a # - co, x + y has a limit equal to +co at the point (a, + co). Given c E R, the relations x > b, y > c - b imply x + y > c, and the intervals ]b, + co] and ]c - 6, + co] are respectively

neighborhoods of a and + GO if b is taken finite and < a ; hence our assertion. The other cases are treated similarly.

With respect to R x R, the function (x, y ) xy has a limit at every point (a, 6 ) qf R x 8, except at the points (0, +a),(0, -a),( f a , 0), (- co,0 ) ;that litnit is equal to + co (resp. - 00) if one at least of the coordinates a, b is injnite, and ifthey have the same sigii (resp. opposite signs). (4.1.9)

-+

Let us show for instance that if a > 0, xy has the limit + co at the point (a, co). Given c E R, the relations x > b, y > c/b, for b > 0, imply xy > c, and the intervals ]b, + co] and ]c/b, + GO] are neighborhoods of a and + 00, if b is taken finite and 0

lim

l/x = -co.

x+o,x - co, for any b < c, thereisx E Esuchthatb 0; from which it follows that f is strictly increasing, since if x < y , then y = zx with z > 1 andf(y) = f ( x ) + f ( z ) > f ( x ) . On the other hand, we have the following lemma: (4.3.1.2) For any integer n 2 1, there is a z > I such that Z" 6 a.

Remark that there is an x such that 1 < x < a, hence a = xy with y > I ; if z1 = min(x, y ) , we have z: 6 x y = a and z1 > 1. By induction define z, > I such that z: < z , , - ~ ,hence z:" 6 a, and a fortiori z l < a. The lemma shows that 0 - : x

then I f ( y ) -f(x)I < f ( z ) 6 l / n for ly - XI , ~(s,),~ , is called a series if the elements x, ,s, are linked by the relations s, = xo x1 * . x, for any n, or, what is equivalent, by xo = s o , x, = s, - snV1 for n b 1 ; x, is called the nth term and s, the nth partial sum of the series; the series will often be called the series of general term x,, or simply the series (x,)

+ +

(and even sometimes, by abuse of language, the series is said to converge to s if lim s, n-+ m

and written s = xo + ...

= s; s is

+ x, + . * . or

+

m

1x,).

The series

n=O

then called the sum of the series m

s=

1x,; r, = s - s,

is called the

n=O

nth remainder of the series; it is the sum of the series having as kth term x , + ~ ;by definition lim r, = 0. n+ m

(Cauchy's criterion) Ifthe series ofgeneral term x, is convergent, E > 0 there is an integer no such that, f o r n 3 no and p k 0, IIs,+,, - s,II = I I X , + ~ + X , , + ~ I I < E . Conversely, i f that condition is satis-ed and if the space E is complete, then the series of general term x, is convergent.

(5.2.1)

then f o r any

+

This is merely the application of Cauchy's criterion to the sequence (s,) (see Section 3.14). As an obvious consequence of (5.2.1)it follows that if the series (x,) is convergent, lim x, = lim (s, - s , , - ~ )= 0; but that necessary condition n-t m

n-+ m

is by no means sufficient.

96

V

NORMED SPACES

(5.2.2) I f the series (x,) and are convergent and have sunis s, s', then the series (x, + x@ converges to the siini s + s' and the series (Ax,) to the sum As f o r any scalar A. (XI,)

Follows at once from the definition and from (5.1.5).

(5.2.3) If (x,) and (x;) are ~ M ' Oseries such that xi = x, except f o r a finite number of indices, they are both concergent or both nonconcergent. For the series (x; - x,) is convergent, since all its terms are 0 except for a finite number of indices.

(5.2.4) Let (k,) be a strictly increasing sequence of integers 3 0 with ko = 0 ;

1 xp, then the series (y,) conrerges

kn+i-l

ifthe series (x,) coilverges to s , and ij-y, =

p=k,

also to s. This follows at once from the relation

n

k,, + I - I

i=O

j=O

1yi = 1 x j and from (3.13.10).

PROBLEMS

Let (a,) be an arbitrary sequence in a normed space E ; show that there exists a sequence 0, and a strictly increasing sequence (kJ of (x,) of points of E such that lim x,

+ + + "-0

integers such that (I= , xo x1 . . . x x nfor every n. Let u be a bijection of N onto itself, and for each n , let ~ ( nbe) the smallest number of intervals [a, 61 in N such that the union of these intervals is cs([O, n]) (a) Suppose p is boimcied in N. Let (x,) be a convergent series in a norined space E; show that the series (x0(J is convergent in E and that

m

"=a

m

x,

=

n=o

xSCn).

(b) Suppose p is unbounded in N. Define a series (x,) of real numbers which is convergent, but such that the series (rS(,,Jis not convergent in R . (Define by induction on k a strictly increasing sequence (mk)of integers having the following properties: (1) If n, is the largest element of u([O, i n x ] ) ,then [0, n k ] is contained in u([O, mk+,]). (2) p(md > k 1. Define then x,, for nk < n < n k + , such that x. = 0 except for 2k conveniently chosen values of n , at each of which x,, is alternately equal to Ilk or to - l / k . ) Let (x,) be a convergent series in a normed space E ; let u be a bijection of N onto itself, and let r(n) = lu(n) - nl . sup \\xn,\\.

+

man

3 ABSOLUTELY C O N V E R G E N T SERIES Show that if lirn r ( n ) = 0, the series (xO{"))is convergent in E and that n-

m

c x0("))- c xy for large n.)

m

97

c xO("). m

n=O

x, =

n=O

n

(Evaluate the difference

k=O

k=O

Let (x,J (m> 0, n > 0) be a double sequence of points of a normed space E. Suppose that: ( I ) for each m > 0, the series x , , ,-t ~ x , , , ~ . . . x,, . . . is convergent in E; .. . ; ( 2 ) for each n > 0, the series let y,, be its sum, and let Y,,,,= x,,,, + x ,,,," TO" YI" . . Y,,,, . . is convergent in E; let t . be its sum. x,, . is convergent; let (a) Show that for each n > 0, the series xOn xI, . z, be its sum.

+

+

+

+ +. + +.

(b) In order that

c

+

+ +. +

y,,, =

,n=0

(a) Show that theseries

+..

c z,, it is necessary and sufficient that lim t, 0. 1 is convergent and has a sum equal to c m2-n2

n=o

n-m

=

-

3/4m2

n31,n;fm

(decompose the rational fraction l/(m' - x')). 1 (b) Let u,"" = -if m # n, and N,,= 0; show that m' - nz

5(2

n = o U,")

n,=0

E( f

= - n=o

, = o U",") # 0.

I f f is a function defined in N x N, with values in a metric space, we denote by f ( m , n ) the limit o f f (when it exists) at the point (+to, to) of a x 8, with lirn

+

m-m."-m

respect to the subspace N x N (Section 3.13). Let (x,") be a double sequence of real numbers, and let s,, = C XIk. h 6 m , kQn

(a) If

lim s,,,,exists, then

m-m,n-a,

lirn

m-m,n-+m

xmn= 0. Give an example in which xnn,= x m n ,

~ , ~ , ~ ~ = - x , ~ , ~ ~ + ~ = - ~ , ~ + ~ , ~ ~ f o r m ~ 2 n +that 1 , x lirn ~ . , ~ sm,=O, .=O,such

+

+

+

+ +

+.

m-rm,n-tm

+

+.

and none of the series xmz0 xml . . . x,." . . ,xOn x l , f.. . x,,, . * is convergent. (b) Give an example in which xnvn= 0 except if m = n 1 , m = n or n = rn 1

c x,,,,, m

(hence all series m, n, but

lirn

",+ m , n-

n=O m

,,s

m

,n=o

x,,,, are convergent),

c x,, c m

m

=

"=O

,=O

+

=0

for all indices

does not exist.

3. A B S O L U T E L Y C O N V E R G E N T SERIES

(5.3.1) In order that a series (x,) of positive numbers be convergent it is necessary and sufJjcie17t that for a strictly increasing sequence (k,) of integers

1x, W

3 0, the sequerice (sk,,) of partial sums be majorized, and then the sum s = is equal to sup sk,.

n=O

n

The assumption x, 3 0 is equivalent to s , - ~ < s, follows at once from (4.2.1).

and then the result

98

NORMED SPACES

V

In a Banach space E, an absolutely convergent series (x,) is a series such that the series of general term llxnll is convergent.

(5.3.2) In a Banach space E , an absolutely convergent series (x,) is convergent,

By assumption, for any and anyp 2 0, IIxn+lII

+

E

> 0, there is an integer no such that for n 2 no

+ IIX,,+~II
0, let no be such that IIxn-lll + + I I X , , + ~ \ ~ < E for n 2 no and p 2 0; then if m, is the largest integer in a-'([O, no]), we have (Iyn+lll+ + Ilyn+pll< E for n 2 m,, p 2 0; furthermore the difference sko - s, is the sum of terms xi withj > n o , hence IIsko - sno\l< E ; therefore, for n 2 no and n 2 m,, 11s; - snII < 3 ~ , which proves that

c x, m

n=O

a,

= n=O

y, .

Let A be any denumerable set. We say that a family ( x , ) , ~ of ~ elements of a Banach space E is absolutely summable if, for a bijection cp of N onto A, the series (xlPcn,) is absolutely convergent; it follows from (5.3.3) that this property is independent of the particular bijection cp, and that we can define the sum of the family (x,),,~ as

c xlP(,,),which we also write c x u . As any m

n=O

asA

denumerable set S c E can be considered as a family (with S as the set of indices) we can also speak of an absolutely summable (denumerable) subset of E and of its sum.

3 ABSOLUTELY CONVERGENT SERIES

99

(5.3.4) In order that a denumerable family ( x , ) , , ~of elements of a Banach space E be absolutely summable, a necessary and sufJicient condition is that )Ix,JI (J c A andfinite) be bounded. Then, for any E > 0, the jinite sums

1

U E l

there exists a jinite subset H of A such that, for any jinite subset K of A for which H n K = 0, IIxull < E , and for any jinite subset L 3 H of A,

1

The first two assertions follow at once from the definition and from (5.3.1). Then, for any finite subset L 2 H, we can write L = H u K with H n K = 0, hence 1) x x U - E x a l l < E ; from the definition of the sum aeH

aeL

1xu it follows (after ordering A by an arbitrary

bijection of N onto A)

(5.3.5) Let ( x , ) , , ~be an absolutely summable family of elements of a Banach space E. Then,jor every subset B of A, the family(xa),eBisabsolutely summable, and C IIxaII G E IIxaII. UEA

U E B

If B is finite, the result immediately follows from the definition. If B is infinite, then llxall < 1 Ilx,Il for each finite subset J of B, and the

x

Or01

result follows from (5.3.4).

aeA

(5.3.6) Let (x,),,~be an absolutely summable family of elements of a Banach space E. Let (B,) be an injinite sequence of nonempty subsets of A, such that A= B,, and B, n B, = 0for p # q ; then, if z, = 1 x u , the series (z,)

u n

is absolutely convergent, and convergent series).

k

m

1z, = C x,

n=O

ueB,

(“associativity”

of absolutely

aeA

Given any E > 0 and any integer n, there exists, by (5.3.2), for each a finite subset J, of B, such that llZkI/ < llxull + E / ( H 1); if

J=

n

+

E

< n,

asJk

J k , we have therefore

1 Ilz,(l < 1llxull + n

k=O

k=O

aE l

E

G

1 IIxaII 4-

aeA

E;

(5.3.1)

then proves that the series (z,) is absolutely convergent. Moreover, let H be a finite subset of A such that, for any finite subset K of A such ( ( E , whence, for any finite subset L of A containthat H n K = 0, ( ( x a < ing H,

11 1xu aeA

c x,I aeK

aEL

< 2~ (see (5.3.4)). Let 170

be the largest integer such

100

V

NORMED SPACES

that H n B,, # @, and let n be an arbitrary integer 2 n o . For each k < n, let Jk be a finite subset of B, containing H n B,, and such that for any finite subset Lk of B, containing J,, we have llzk - C x,II < d ( n 1) (5.3.4). Then, if L =

u Lk,we have /I

from the definition of H that

I 1, let B. be the set of x E Bn-l such that IIx - yll < 6(B,-,)/2 for all y E Bn-l(&A) being the diameter of a set A). Show that 6(B,) < 6(B.-,)/2, and that the intersection of all the B. is reduced to (a b)/2. (b) Deduce from (a) that iffis an isometry of a real normed space E onto a real normed space F, thenf(x) = u(x) c, where u is a linear isometry, and c E F. 4. Let us call rectangle in N x N a product of two intervals of N; for any finite subset H of N x N, let $(H) be the smallest number of rectangles whose union is H. Let (H.) be an increasing sequence of finite subsets of N x N, whose union is N x N and such that the sequence ($(H.)) is bounded. Let E, F, G be three normed spaces, (x.) (resp. (y,)) a convergent series in E (resp. F),fa continuous bilinear mapping of E x F into G. Show that

+

+

(*)

5.

lim

n-m

C

0 such that: (1) the relations JIxJJ G 6, IIx'II Q 6 , IIx x'll Q 6 in E implyf(x x') = f ( x ) +f(x'); (2) the relations llxll 6 6, llhxll G 6 in E (with h E R) imply f(hx) = hf(x).

+

+

106

V

NORMED SPACES

(a) Show that iffsatisfies condition (1) and is continuous at the point 0, it is continuous in a neighborhood of 0 and is linear in a neighborhood of 0 (cf. Problem 1). (b) Let g be a mapping of E into F; in order that g be continuous at the point 0 and linear in a neighborhood of 0, a necessary and sufficient condition is that for any convergent series (x.) in E, the partial sums of the series (g(x.) be bounded in F. (To prove sufficiency, first observe that one must have g(0) = 0; if, for every n, there exist three elements u., v., w. of E such that llunll< 2-", llvnll < 2-", llwnll < 2-", u,, v. w. = 0 and g(uJ g(u,) +g(w.) # 0, form a series (x,) violating the assumption. If there are no such sequences u,, v., w., g verifies condition (1); show that it is necessarily continuous at 0.)

+ +

+

6. EQUIVALENT NORMS

Let E be a vector space (over the real or the complex field), llxlll and llxll, two norms on E; we say that llxlll isfiner than llxll, if the topology defined by (IxI(lis finer than the topology defined by IJxJI,(see Section 3.12); if we note El (resp. E,) the normed space determined by llxlll (resp. Ilxl12), this means that the identity mapping x + x of El into E, is continuous, hence, by (5.5.1), that condition is equivalent to the existence of a number a > 0 such that IIxIJ2< a - IJxJI1. We say that the two norms IIxII1, I(xI(, are equivalent if they define the same topology on E. The preceding remark yields at once: (5.6.1) In order that the two norms IIxlll, ((x((, on a vector space E be equivalent a necessary and sufficient condition is that there exist two constants a > 0,

b >O, such that allxll1

for any x E E.

< llxllz d bllxll1

The corresponding distances are then uniformly equivalent (Section 3.14). For instance, on the product El x E, of two normed spaces, the norms sup(IIxlII, IIx~II)~IIxlII + IIx211, (IIxlI12 + I I X ~ I I ~ ) are " ~ equivalent. On the space E = %?&), the norm l l f l l l defined in (5.1.4) is not equivalent to the norm Ilfll, = suplf(t)l (see Section 5.1, Problem 1). re1

7. SPACES O F C O N T I N U O U S MULTILINEAR MAPPINGS

Let E, F, be two normed spaces; the set Y(E; F) of all continuous linear mappings of E into F is a vector space, as follows from (5.1 S),(3.20.4), and (3.11.5).

7 SPACES OF CONTINUOUS MULTILINEAR MAPPINGS

107

For each u E Y(E; F), let IJuJI be the g.1.b. of all constants a > 0 which satisfy the relation IIu(x)II < a * llxll (see (5.5.1)) for all X. We can also write

(5.7.1) For by definition, for each a > IIuII, and llxll < I , IIu(x)I( 0, llxll 6 1

for any b such that 0 < b < IIuII, there is an x E E such that IIu(x)I( > bllxll; this implies x # 0, hence if z = x/llx/l, we still have IIu(z)II > b llzll = b, and as llzll = 1, this proves that b < sup IIu(x)I(,hence I(u(I< sup IIu(x)(I, 9

llxll 6 1

llxll 1

and (5.7.1)is proved. The same argument also shows that if E # (0)

(5.7.2) We now show that /lull is a norm on the vector space Y ( E ; F). For if u = 0, then IJuJJ= 0 by (5.7.1),and conversely if llull = 0, then u(x) = 0 for llxll ,< 1, hence, for any x # 0 in E, u(x) = llxllu(x/llxll)= 0. It also follows from (5.7.1) that IIAull = 111 * IJu/I; finally, if w = u u, we have II4dI IIu(x)II llu(x>II,hence llwll llull llull from (5.7.1).


,

(11)

(V)

f(x, Y

f ( Y >4

=f(% Y ) .

(Observe that (11) and (IV) follow from the other identities; (V) implies that f(x, x) is real.) When E is a real vector space, conditions (I) to (IV) express that f is bilinear and (V) boils down to f ( y , x) = f ( x , y), which 115

116

VI

HILBERT SPACES

expresses t h a t f i s symmetric. For any finite systems ( x i ) ,( y j ) , (ai), (Bj), we have (6.1 .I)

by induction on the number of elements of these systems. From (6.1.1) it follows that if E is finite dimensional and ( a i ) is a basis of E, f is entirely determined by its values a i j = f ( a i , a j ) , which are such that (by V)) aJ,l. = a I,J.’

(6.1.2)

Indeed we have then, for x = (6.1.3)

1 S i a i ,y = i

S ( X .,Y>

viai I

C aij ti

i, i

iij

.

Conversely, for any system ( a i j ) of real (resp. complex) numbers satisfying (6.1.2), the right hand side of (6.1.3) defines on the real (resp. complex) finite dimensional vector space E a hermitian form. Example (6.1.4) Let D be a relatively compact open set in R2, and let E be the real (resp. complex) vector space of all real-valued (resp. complex-valued) bounded continuous functions in D, which have bounded continuous first derivatives in D. Then the mapping

(where a, h, c are continuous, bounded and real-valued in D) is a hermitian form on E. A pair of vectors x, y of a vector space E is orthogonal with respect to a hermitian form f on E if f ( x , y ) = 0 (it follows from (V) that the relation is symmetric in x, y ) ; a vector x which is orthogonal to itself (i.e.f(x, x) = 0) is isotropic with respect to$ For any subset M of E, the set of vectors y which are orthogonal to all vectors x E M is a vector subspace of E, which is said to be orthogorial to M (with respect t o f ) . It may happen that there exists a vector a # 0 which is orthogonal to the whole space E, in which case we say the formfis degenerate. On a finite dimensional space E, nondegenerate hermitian forms f defined by (6.1.3) are those for which the matrix ( a i j ) is invertible.

2

POSITIVE HERMITIAN FORMS

117

PROBLEMS

1. (a) Let f be a hermitian form on a vector space E. Show that if E is a real vector space, then 4f(x, Y> = f ( x

+ Y , x +v) -f(x

-Y, x

-Y)

and if E is a complex vector space

4fh, Y ) =f(x

+ y. x + y ) - f ( x

-Y ,x

-y)

+ i f ( x + iy, x + iy) - if(x - iy, x

- iy).

(b) Deduce from (a) that if f ( x , x ) = 0 for every vector in a subspace M of E, then f ( x , y ) = 0 for any pair of vectors x , y of M. (c) Give a proof of (b) without using the identities proved in (a). (Write that f(x hy, x +Xu) = O for any h.) 2. Let E be a complex vector space. Show that iffis a mapping of E x E into C satisfying conditions (I), (II), (III), (IV), and such that f ( x , x ) E R for every x E E, then f is a hermitian form on E.

+

2. POSITIVE H E R M I T I A N FORMS

We say a hermitian form f on a vector space E is positive iff(x, x) 2 0 for any x E E. For instance, the form q defined in example (6.1.4) is positive if a, b, c are 2 0 in D.

(6.2.1)

(Cauchy-Schwarz inequality)

I f f is a positive hermitian form,

then for any pair of vectors

X,

If(x, Y>12 < f ( x , X)f(Y, Y ) y in E.

Write a =f ( x , x), b = f ( x , y ) , c = f ( y , y ) and recall a and c are real and 2 0. Suppose first c # 0 and write that f ( x I y , x + l y ) 2 0 for any scalar 1, which gives a + b2 + bA + c3J 0 ; substituting I = -b/c yields the inequality. A similar argument applies when c = 0, a # 0 ; finally if a = c = 0, the substitution I = - h yields -266 2 0, i.e. b = 0.

+

(6.2.2) In order that a positii!e hermitian form f on E be nondegenerate a necessary and suficient condition is tliat there exist no isotropic vector for f otlier than 0 , i.e. that f ( x , x) > 0 for any x # 0 in E.

Indeed, f ( x , x) all y E E.

=0

implies, by Cauchy-Schwarz, that f (x, y ) = 0 for

118

(6.2.3)

VI

HILBERT SPACES

(Minkowski's inequality) I f f is a positi1.e hermitian form, tlien (f(x

+y, x +

d ( A x , x))''2 + ( f b ,

for any pair of vectors x, y in E.

+

As f ( x y , x equivalent to

+ y ) = f ( x , x) + f ( x , y ) + A x , y ) + f ( y , y ) , the inequality is

2,gf(x,Y)= f ( x , Y >+f(x,P,)d 2 ( f ( x , x)f(Y,Y))'12 which follows from Cauchy-Schwarz. The function x -+ ( f ( x , x))'12 therefore satisfies the conditions (I), (Ill), and (IV)of (Section 5.1); by (6.2.2), condition (11) of Section 5.1 is equivalent to the fact that the form f is nondegenerate. Therefore, when f is a nondegenerate positive hermitian form (also called a positive dejnite form), ( f ( x , x))'I2 is a norm on E. A preliilbert space is a vector space E with a given nondegenerate positive hermitian form on E; when no confusion arises, that form is written (x I y ) and its value is called the scalar product of x and y ; we always consider a prehilbert space E as a normed space, with the norm llxll = (x I x)''', and of course, such a space is always considered as a metric space for the corresponding distance Ijx - y ( / . With these notations, the Cauchy-Schwarz inequality is written

I(xlu)l d IIxIl

(6.2.4)

IIYII

and this proves, by (5.5.1), that for a real prehilbert space E, (x, y ) -+ (x 1 y ) is a continuous bilinear form on E x E (the argument of (5.5.1) can also be applied when E is a complex prehilbert space and proves again the continuity of (x, y ) (x I y ) , although this is not a bilinear form any more). We also have, as a particular case of (6.1 .I): --f

(6.2.5) (Pythagoras' theorem) onal vectors, IIX

In a prehilbert space E, i f x , y are orthog-

+ YIl2 = l/xll2+ llY/I2.

A n isomorphism of a prehilbert space E onto a prehilbert space E' is a linear bijection of E onto E' such that ( f ( x )If(y)) = (x Iy) for any pair of vectors x, y of E. It is clear that an isomorphism is a linear isonzetry of E onto E'. Let E be a prehilbert space; then, on any vector subspace F of E, the restriction of the scalar product is a positive nondegenerate hermitian form ; unless the contrary is stated, it is always that restriction which is meant when F is considered as a prehilbert space.

3 ORTHOGONAL PROJECTION ON A COMPLETE SUBSPACE

119

A Hilbert space is a prehilbert space which is coniplete. Any finite dimensional prehilbert space is a Hilbert space by (5.9.1); other examples of Hilbert spaces will be constructed in Section 6.4. If in example (6.1.4) we take a > 0, b 2 0 , c 3 0, it can be shown that the prehilbert space thus defined is not complete.

PROBLEMS

1. Prove the last statement in the case a = 1 , b = c = 0 (see Section 5.1, Problem 1). 2. (a) Let E be a real nornied space such that, for any two points ,Y, y of E, 1I.y

+ Yl12 + Ilx - Y1IZ

= 2(11x1I2

+ llvIl2).

Show that f ( x , y) = IIx i-V I -~ Ilxl/* ~ - /IyIl2 is a positive nondegenerate hermitian form on E. (In order to prove that f ( h x , y ) = h f ( x , y ) , use Problem I of Section 5.5.) (b) Let E be a complex nornied space, Eo the underlying real vector space. Suppose there exists on Eo a symmetric bilinear form f ( x ,y) such that f ( x , x) = l1x/l2for every x E E o . Show that there exists a herniitian form g(x, y ) on E such that f ( x , y ) = .jR(g(x, y ) ) , hence g(x, x) = 1/x1l2for x E E. (Using the first formula of Problem 1 of Section 6.1, prove that f ( i x , y ) = - f ( x , i-v).) (c) Let E be a complex normed space such that Ilx - Y1I2

+ Ilx + Y / I 2 < 2(11x1I2 + llvll’)

for any pair of points x , y of E. Show that there exists a nondegenerate positive hermitian fornif(x,y) on E such thatf(x, x) = XI/^. (Use (a) and (b)). 3. Letfbe a positive nondegenerate hermitian form. In order that both sides of (6.2.1) be equal, it is necessary and sufficient that x and y be linearly dependent. In order that both sides of (6.2.3) be equal, it is necessary and sufficient that x and ,y be linearly dependent, and, if both are # 0, that y = Ax, with real and > 0. 4. Let a, b, c, d be four points in a prehilbert space E. Show that

/(a- c ( / . / / b- d ( (< /la - bll . JJc- dll

5.

+ 116

- cii

. / / a- dll.

(Reduce the problem to the case a = 0, and consider in E the transformation x+x/llxl~’, defined for x # 0.)When are both sides of the inequality equal? be a finite sequence of points in a prehilbert space E. Show that Let ( x i ) ,

If /lxi - x,Jl > 2 for i # j , show that a ball containing all the x i has a radius >(2(n

-

1)jn)’”.

3. O R T H O G O N A L PROJECTION O N A COMPLETE SUBSPACE

(6.3.1) Let E be a prehilbert space, F a complete tvctor subspace of E (i.e. a Hilbert space). For any x E E, there is one and only one point y = P F ( x )E F

120

VI

HILBERT SPACES

such that I/x - J~II = d(x, F). Tlze point y = P,(x) is also the only point z E F such that x - z is orthogonal to F. The mapping x P,(x) of E onto F is linear, continuous, and of norm 1 if F # { O } ; its kernel F’ = PF1(0) is the subspace orthogonal to F, and E is the topological direct sum (see Section (5.4)) of F and F‘. Finally, F is the subspace orthogonal to F‘. --f

Let CI = d(x, F); by definition, there exists a sequence (y,) of points of F such that lim 1I.x - ynl/ = a ; we prove (y,,) is a Cauchy sequence. fl-+

co

Indeed, for any two points u, u of E, it follows from (6.1.1) that (6.3.1.1)

hence

112.4

llym

+

+ -

U(l2

1/24

PJ12= 2(llu112

+ 11z~112>

- yn/I2= 2(llx - ~ , ~ ,+l / IIx ~ - ~ , , l l ~-) 4llx - ! d Y m

+ y,)112.

But

+ y,,) E F, hence jlx - )(ym + y,,)lI2 > a 2 ;therefore, if no is such that for n 2 n o , ( / x- y , ( / 2< g2 + E , we have, form > no and n 3 n o , I/ym- Y,;;? < 4e, +(yJ,

which proves our contention. As F is complete, the sequence (y,) tends to a limit y E F, for which IJx- yll = d(x, F). Suppose y’ E F is also such that /Ix - y’ll = d(x, F); using again (6.3.11), we obtain / ( y- y’(I2= 4a2- 4 /Jx- ) ( y y’)1I2, and as j ( y y ’ ) E F, this implies (ly - y’1I2 < 0,i.e. y’ = y. Let now z # 0 be any point of F, and write that (lx - ( y Az)/l* > a2 for any real scalar A # 0; this, by (6.1 .I), gives

+

+

21B(x - y 1 z )

+

+ R 2 / ( 2 / ( 2> 0

and this would yield a contradiction if we had 9 ( x - y 1 z ) # 0, by a suitable choice of 1. Hence B(x - y 1 z ) = 0, and replacing z by iz (if E is a complex prehilbert space) shows that #(x - y I z ) = 0, hence (x - y I z ) = 0 in every case; in other words x - y is orthogonal to F. Let y’ E F be such that x -y’ is orthogonal to F; then, for any z # 0 in F, we have JIx- (y’ z)1I2 = /Jx- y‘/12 j/zIlz by Pythagoras’ theorem, and this proves that y’ = y by the previous characterization of y . This last characterization of y = P,(x) proves that P , is linear, for if x - y and x’- y’ are orthogonal to F, then Ax - Ay is orthogonal to F and so is (x + x’)- ( y + y’) = (x - y ) (x’ - y ’ ) ; as y y’ E F and Ay E F, this shows that y y’ = P F ( x x’)and Ay = P , ( ~ x ) .By Pythagoras’ theorem, we have

+

+

+

(6.3.1.2)

+

llX/12

=

+

IIpF(x)112

+

+ (Ix

- pF(x)l/2

and this proves that IlPF(x)l1< I/x/J,hence (5.5.1) P, is continuous and has norm d 1 ; but as P F ( x )= x for x E F, we have /lP,I/ = 1 if F is not reduced to 0. The definition of P, implies that F’ = PF’(0) consists of the vectors x orthogonal to F; as x = P,(x) + (X - PF(,y)) and x - P F ( x )E F’ for any x E E, we have E = F F’; moreover, if x E F n F‘, x is isotropic, hence

+

3 ORTHOGONAL PROJECTION ON A COMPLETE SUBSPACE

x

= 0, and

121

this shows that the sum F + F' is direct. Furthermore, the mapping

x -iP F ( x )being continuous, E is the topological direct sum of F and F' (5.4.2). Finally, if x E E is orthogonal to F', we have in particular (x I x - P F ( x ) )= 0 ; but we also have (PF(x) I x - PF(x))= 0, hence /Ix - PF(x)(I2= 0, i.e. x = PF(x) E F. Q.E.D. The linear mapping PF is called the orthogonal projection of E onto F, and its kernel F' the orthogonal supplement of F in E. Theorem (6.3.1) can be applied to any closed subspace F of a Hilbert space E (by (3.14.5)), or to any Jinite dimensional subspace F of a prehilbert space, by (5.9.1).

(6.3.2) Let E be a prehilbert space; then, for any a E E, x -i (x I a) is a continuous linear form of norm Ilall. Conversely, if E is a Hilbert space, for any continuous linear form u on E, there is a unique vector a E E such that u(x> = (x I a)for any x E E. ByCauchy-Schwarz,I(x I a)l < llall * IIxII, whichshows(by5.5.1))~+ (x la) is continuous and has a norm < llalj; on the other hand, if a # 0, then for x, = a / ~ ~ we a ~have ~ , (x, I a) = llall; as llxoII = 1, this shows the norm of x - i ( x l a ) is at least Ilall. Suppose now E is a Hilbert space; the existence of the vector a (=O) being obvious if u = 0, we can suppose u # 0. Then H = u-'(O) is a closed hyperplane in E; the orthogonal supplement H' of H is a one-dimensional subspace; let b # 0 be a point of H'. Then we have ( x I b) = 0 for any x E H. But any two equations of a hyperplane are proportional, hence there is a scalar L such that u(x) = L(x 16) = (xI a) with a = Xb (see Appendix) for all x E E. The uniqueness ofa follows from the fact that the form (x I y ) is nondegenerate.

PROBLEMS 1. Let B be the closed ball of center 0 and radius 1 in a prehilbert space E. Show that for each point x of the sphere of center 0 and radius I , there exists a unique hyperplane of support of B (Section 5.8, Problem 3) containing x . 2. Let E be a prehilbert space, A a compact subset of E, 6 its diameter. Show that there exist two points a , b of A such that /la- bll = 8 and that there are two parallel hyperplanesof support of A (Section 5.8, Problem 3)containingaand b respectively, and such that their distance is equal to 6. (Consider the ball of center a and radius 8 and apply the result of Problem 1 .)

3.

Let E be a Hilbert space, F a dense linear subspace of E, distinct from E (Section 5.9, Problem 2). Show that there exists in the prehilbert space F a closed hyperplane H such that there is no vector # 0 in F which is orthogonal to H.

VI

122

4.

HlLBERT SPACES

Let X be a set, E a vector subspace of Cx, on which is given a structure of complex Hilbert space. A mapping ( x , y ) K(x, y ) of X x X into C is called a reproducing kernel for E if it satisfies the two following conditions: ( I ) for every y E X, the function K(. , y ) : x --f K(x, y ) belongs to E; (2) for any function f~ E, and any y E X, f(y) = (fI K ( . , A). (a) Show that K is a mapping of positive fype of X x X into C, i.e., for any integer of points of X , the mapping n 3 1 and any finite sequence ( x i ) , --f

+x W x i , xi)&

((Ad, (pi))

pi

I. J

of C2" into C is a positiue hermitian form. This in particular implies K(x, x ) 3 0 for every x E X, K(y, x ) = K(x, y ) and IK(x, y)12 < K(x, x ) K ( y ,y ) for x, y in X. Show that f0rj.E E, one has lf(y)I < llfll . (K(y, y))"' for y E X. (b) Show that if (f.)is a sequence of functions of E which converges (for the Hilbert space structure) to f~ E, then, for every x E X , the sequence (f.(x)) converges to f ( x ) in C; the convergence is uniform in any subset of X where the function x + K(x, x ) is bounded. n a finite sequence of points of X, (al)lslsn a sequence of n com(c) Let ( x i ) l i 1 Bbe plex numbers. Suppose det(K(x, , x i ) ) # 0, so that the system of linear equations cj K(xi, x J )= al (1

< i < n) has a

J= 1

unique solution (ci). Show that among the func-

tions f~ E such that f(x1) = a , for 1 f i S n, the function fo

n

= cj J= I

K(. , x i ) has the

smallest norm. In particular, among all functions f~ E such that f ( x ) = 1 for a point E X where K(x, x ) # 0, the function K(. , x ) / K ( x ,x ) has the smallest norm. (d) If X is a topological space and if all the functionsf€ E are continu0u.s in X, then the functions K(.,x) (where x takes all values in X, or in a dense subset of X) form a total subset of E (show that there is no element h # 0 in E which is orthogonal to all the elements K(. , x ) ) . In particular, if X is a separable metric space, E is a separable Hilbert space. 5. (a) The notations being those of Problem 4, in order that there exist a reproducing kernel for E, a necessary and sufficient condition is that for every x E X, the linear form f + f ( x ) be continuous in E. The reproducing kernel is then unique. (b) Deduce from (a) that if there exists a reproducing kernel for E, there also exists a reproducing kernel for every closed vector subspace El of E. If K, is the reproducing kernel for E l , show that for every function f~ E, the function y --f (f1 K,(. ,y ) ) is the orthogonal projection o f f i n El. If E, is the orthogonal supplement of El and K2 the reproducing kernel for E, , then K I $- K 1 is the reproducing kernel for E. 6. Let X be a set, E a vector subspace of Cx, on which is given a structure of complex prehilbert space. In order that there should exist a Hilbert space c Cxcontaining E, such that the scalar product on E is the restriction of the scalar product on B, and that there exists a reproducing kernel for B, it is necessary and sufficient that E satisfy the two following conditions: ( I ) for every x f X, the linear form f + f ( x ) is continuous in E; (2) for any Cauchy sequence (f,)in E such that, for every x E X, lim f.(x) = 0, x

one has lim llfnll

n-m

= 0.

(To prove the conditions are sufficient, consider the subspace

f?

20'"

of Cx whose elements are the functionsffor which there exists a Cauchy sequence (fn) in E such that limfn(x) = f ( x ) for every x E X. Show that, for all Cauchy sequences (f.) n-r m

having that property, the number lim llfnll is the same, and if l i f l l is that number, this n-rm

defines on f a structure of normed space which is deduced from a structure of prehilbert

4 HILBERT SUM OF HILBERT SPACES

123

space which induces on E the given prehilbert structure; finally show that E is dense in 8 and that 8 is complete, hence a Hilbert space, and apply Problem 5(a) to 8.) 7. Let X be a set, f a mapping of X into a prehilbert space H; show that the mapping ( x , y ) + ( f ( x )1 f(y)) of X x X into C is of positive type (Problem 4(a)). 8 . Let X be a set, K a mapping of positive type of X x X into C (Problem 4(a)). (a) Let E be the set of mappings u : X -+ C such that there exists a real number a > 0 having the property that the mapping ( x , Y ) +aK(x, Y ) - 4x)UO

is of positive type; let m(u)be the smallest of all real numbersa 3 Ohavingthat property. Show that m(u) is also the smallest number c such that, for every finite sequence ( x i ) of elements of X, the inequality

holds for all complex numbers h, , p (use the Cauchy-Schwarz inequality). For every E X, show that Iu(x)12 < K(x, x)m(u). (b) Show that E is a vector subspace of Cx, that ( m ( ~ ) ) is ” ~a norm on E and that m(u u ) 4- m(u - u ) 2 2(m(u) m(u)).Conclude that there is a nondegenerate positive hermitian form g(u, u) on E x E such that g(u, u ) = m(u), and that for this form E is a Hilbert space; one writes g(u, u ) = (u I u). (Use Problem 2(c) of Section 6.2 to prove the existence of g ; to show that E is complete, use the last inequality proved in (a)). (c) For every x E X, show that the function K ( . , x ) belongs to E and that (K(. , x) I K ( . ,y ) ) = K(x, y ) for all ( x , y ) E X x X (use Cauchy-Schwarz inequality). Prove that if X is a topological space and if K is continuous in X x X, the mapping x + K(. , x) of X into E is continuous. (d) Deduce from (a) that the Hilbert space E defined in (b) has a reproducing kernel, and if F is the closed vector subspace of E generated by the functions K(. ,x), the reproducing kernel for F (Problem 5(b)) is equal to K. 9. Let E be a prehilbert space, N a finite dimensional vector subspace of E, M a vector subspace of E having infinite dimension, or finite dimension >dim(N). Show that there exists in M a point x # 0 such that llxll= d(x, N). (Consider the intersection of M and of the orthogonal supplement of N.) x

+

+

4. H I L B E R T SUM O F HILBERT SPACES

Let (En) be a sequence of Hilbert spaces; on each of the En, we write the scalar product as ( x , ) y n ) .Let E be the set of all sequences x = (xl, x 2 , . . . , x,, . . .) such that x, E En for each n , and the series ( I I x , ~ ~ ~ ) is convergent. We first define on E a structure of vector space: it is clear that if x = (x,) E E, then the sequence (Axl,.. . , Ax,,,. . .) also belong to E. On the other hand, if y = (y,) is a second sequence belonging to E, we observe that IIx, ynl12< 2(/1~,11~ liyn112) by (6.3.1.1), hence the series (llx, + y,I12) is convergent by (5.3.1), and therefore the sequence (xl + y , , ..., x, + y n r ...) belongs to E. We define x + y = (x,+y,,),

+

+

124

V

HILBERT SPACES

Ax = (Ax,), and the verification of the axioms of vector spaces is trivial. On the other hand, from the Cauchy-Schwarz inequality, we have

IIxnII *

I(xn IYn)I G

IIYnIt G

HIIXnII’

+ IIynII’).

Therefore, if x = (x,) and y = (y,) are in E, the series (of real or complex numbers) ((x, I y,)) is absolutely convergent. We define, for x = (x,) and

y = (y,) in E, the number (x 1 y ) =

00

,= 1

(x, 1 y,); it is immediately verified

that the mapping (x, y ) -.+ (x I y ) is a Hermitian form on E. Moreover we have (x Ix) =

1 llxn\1’, hence ( x l y ) is a positive nondegenerate

hermitian

n= 1

form and defines on E a structure of prehilbert space. We finally prove E is in fact a Hilbert space, in other words it is complete. Indeed, let (xc”’)), where x(”’)=(x$)), be a Cauchy sequence in E: this means that for any E > 0 there is an m, such that for p 2 m, and q >, m, , we have m

(6.4.1)

For each fixed n, this implies first

IIxp) - xP)1l2< E ,

( x ~ ~ ) ) ~ = ~is , a~ ,Cauchy ... sequence in

hence the sequence

En,and therefore converges to a

limit y , . From (6.4.1) we deduce that for any given N N

C IIxn(P) - xn(4)II2 < &

n= I

as soon as p and q are 2 m o , hence, from the continuity of the norm, we deduce that

N

n= 1

0, the series

-

A is the set of all the sums of the series

I

C A,, is convergent

129

"= 1

and has a sum equal to I .

"=I

(b) Prove that the diameter of A is equal to d2 but that there is no pair of points A such that /la- b/l = 42 (compare to Section 6.3, Problem 2). Let E be a Hilbert space with a Hilbert basis Let a, = eZn and

a, h of

b, = ezn

1 ++ 1 e z n f lfor every n > 0; letA (resp. B) be the closed vector subspace of E I1

generated by the a. (resp. b"). Show that: (a) A n B = { O } , hence the sum A B is direct (algebraically). (b) The direct sum A B is not a topological direct sum (consider in that subspace the sequence of points h, - a, and apply (5.4.2.)). (c) The subspace A B of E is dense but not closed in E (show that the point m

+

+ +

(b, - an)does not belong to A

+ B).

"=O

Show that the Banach space 9 ( P ; 1 2 ) can be identified with the space of double

sequences U = (a,,,,) such that: (I) the series m

( 2 ) sup n

m

nr= 0

is convergent for every n:

la,,,nlzis finite. The norm is then equal to IlUIj = sup

"

nt=O

method as in Section 5.7, Problem 2(b)). (a) Let u be a continuous linear mapping of show that the series

c m

fl=O

and that their sums are on E and use (6.3.2).)

/cz,,,12 and


0, there is therefore an integer no such that ilf, - 911 < 4 3 for n 2 n o . For any to E E, let V be a neighborhood of to such that ~l.f,,(t) -fflo(to)l\d 4 3 for any t E V. Then, as Ilf,,(t) - g(t)I/ d 4 3 for any t E E, we have /lg(t)- g(t,)ll < E for any t E V, which proves the continuity of g. Well-known examples (e.g. the functions x -+ x” in [0, 13) show that a limit of a simply convergent sequence of continuous functions need not be continuous. On the other hand, examples are easily given of sequences of continuous functions which converge nonuniformly to a continuous function (see Problem 2). However (see also (7.5.6)):

(7.2.2) (Dini’s theorem) Let E be a compact metric space. Ifa n increasing (resp. decreasing) sequence (f f l )of real-aalued continuous functions converges simply to a continuous function g, it conwrges uniformly to g. Suppose the sequence is increasing. For each E > 0 and each t E E , there is an index n(t) such that for rn 2 n ( t ) , g(t) -f,(t) < e/3. As g and f f l ( r ) are continuous, there is a neighborhood V ( t ) oft such that the relation t’ E V ( t ) implies Ig(t) - g(t’)l < 4 3 and 1fnct,(t) - f&,(t’)l d 4 3 ; hence, for any t‘ E V(t) we have g(r’) -Lcr,(t’) d E . Take now a finite number of points r i in E such that the V(ti) cover E, and let no be the largest of the integers n(t,). Then for any t E E, t belongs to one of the V(ti), hence, for n 2 no , d t ) -L(t) -f,,(t) d A t ) -fncr,)(t) 6 8 . Q.E.D.

PROBLEMS

1. Let E be a metric space, F a normed space, (u,,) a sequence of bounded continuous mappings of E into F which converges simply in E to a bounded function u. (a) In order that u be continuous at a point x, E E, it is necessary and sufficient that for any E > 0 and any integer m, there exist a neighborhood V of x , and an index n > m such that Ilu(x) - u.(x)II < E for every x E V. (b) Suppose in addition E is compact. Then, in order that u be continuous in E, i t is necessary and sufficient that for any E > 0 and any integer m,there exist a finite number

136

VII SPACES OF CONTINUOUS FUNCTIONS

of indices n, > m such that, for every x E E, there is at least one index i for which Ilv(.x) - u,,(x>Il < E (use (a) and the Borel-Lebesgue axiom).

2.

For any integer ti > 0, let g. be the continuous function defined in R by the conditions that g,,(t) = 0 for t < O and I 3 2 / n , g,,(I/n)= 1 , and g,(t) has the form orf p (with suitable constants a , p) in each of the intervals [O, l / n ] and [l/n, 2 / n ] .The sequence (gn) converges simply to 0 in R, but the convergence is not uniform in any interval containing t = 0. Let m i r , , be a bijection of N onto the set Q of rational numbers, and let

+

OL.

fn(t) =

C 2-mg.(t - r,,,). The functions f, are continuous (7.2.1 ), and the sequence (A) lfl=O

3.

4.

5. 6.

7.

8.

converges simply to 0 in R, but the convergence is not uniform in any interval of R . Let 1 be a compact interval of R,and ( f n ) a sequence of monotone real functions defined in I, which converge simply in I to a continuous functionf. Show that f i s monotone, and that the sequence (f.)converges uniformly to f i n I. Let E be a metric space, F a Banach space, A a dense subset of E. Let (fJ be a sequence of bounded continuous mappings of E into F such that the restrictions of the functions fn to A form a uniformly convergent sequence; show that ( f n ) is uniformly convergent in E. Let E be a metric space, F a normed space. Show that the mapping (x, N) + u(x) of E x %F(E)into F is continuous. Let E, E' be two metric spaces, F a normed space. For each mappingfof E x E'into F and each y E E', let f, be the mapping x - f ( x , y ) of E into F. (a) Show that if f i s bounded, if each f, is continuous in E and if the mapping y +A, of E' into XF(E) is continuous, thenfis continuous. Prove the converse if in addition E is compact (use Problem 3(a) in Section 3.20). (b) Take E = E' = F = R,and let f ( x , y ) = sin x y , which is continuous and bounded in E x E'; show that the mapping y - f , of E' into '6F(E) is not continuous at any point of E'. (c) Suppose both E and E' are compact, and for any f E KF(Ex E ) , let be the mapping y -f, of E' into KF(E); show that the mapping f-3 is a linear isometry of PF(E x E') onto % ' C ~ ~ ( ~ ) ( E ' ) . Let E be a metric space, F a normed space. For each bounded continuous mappingf of E into F, let G ( f )be the graph of f i n the space E x F. (a) Show that f - G ( f ) is a uniformly continuous injective mapping of the normed space %F(E) into the space a(E x F) of closed sets in E x F, which is made into a metric space by the Hausdorff distance (see Section 3.16, Problem 3). Let r be the image of VF(E) by the mappingf-- G ( f ) . (b) Show that if E is compact, the inverse mapping G - ' of I' onto %,"(E) is continuous (give an indirect proof). (c) Show that if E = 10, 13 and F = R, G - ' is not uniformly continuous. Let E be a metric space with a bounded distance cl. For each x E E let d, be the bounded continuous mapping y d(x, y ) of E into R. Show that x r!, is an isometry of E onto a subspace of the Banach space %g(E). --f

--f

3. T H E STONE-WEIERSTRASS A P P R O X I M A T I O N T H E O R E M

For any metric space E, the vector space %'g(E) (resp.%?E(E)) is an algebra over the real (resp.complex) field; from (7.1.1) it follows that we have in that algebra llfgll < llfll * Ilgll, hence, by (5.5.1), the bilinear mapping (5 g) -fg

3 THE STONE-WEIERSTRASS APPROXIMATION THEOREM

137

is continuous. From that remark, it easily follows that for any subalgebra A of Vg(E) (resp. V,?(E)), the closure m of A in Vg(E) (resp. VF(E)) is again a subalgebra (see the proof of (5.4.1)). We say that a subset A of BR(E) (resp. B,-(E)) separates points of E if for any pair of distinct points x , y in E, there is a function f E A such that f ( x ># f ( Y ) . (7.3.1) (Stone-Weierstrass theorem) Let E be a compact metric space. If a subalgebra A of 'eR(E) contains [he constant functions and separates

points of E, A is dense in the Banach space VR(E).

I n other words, if S is a subset of WR(E) which separates points, for any continuous real-valued function f on E, there is a sequence (g,) of functions converging uniformly t of, such that each g, can be expressed as a polynomial in the functions of S, with real coefficients. The proof is divided in several steps. (7.3.1 .I) There exists a sequence of real polynomials (u,) which in the interval [0, 13 is increasing and concerges uniformly to J t

.

Define u, by induction, taking u1 = 0, and putting (7.3.1.2)

u,+l(t) = u,(t)

+ +(t - u,2(t))

for n 2 1.

We prove by induction that u , + ~2 u, and u,(t) < JTin [0, 11. From (7.3.1.2), we see the first result follows from the second. On the other hand Jt - u , + , ( t ) = J t -

u,(t) - S ( t

= (Jt- u,(l))(l

- U,2(t))

- +(&+

u,(t>>)

and from un(t) < ,/r we deduce f ( J t + u,(t)) < Jt< 1. For each t E [0, 11, the sequence (u,(t)) is thuj increasing and bounded, hence converges to a limit r ( t ) (4.2.1); but (7.3.1.2) yields f - v 2 ( t ) = 0 and as C(f) 2 0, o(t) = Jt. As c is continuous and the sequence (u,) is increasing, Dini's theorem (7.2.2) proves that (u,) converges uniformly to D.

(7.3.1.3)

For any function f

E

A, If 1 belongs to the closure

A of A on VR(E).

Let a = (1 f (1. By (7.3.1.1), the sequence of functions u , ( f 2 / a 2 ) , which belong to A (by definition of an algebra), converges uniformly to(fz/a2)'/2 = If I/a in E.

138

VII SPACES OF CONTINUOUS FUNCTIONS

(7.3.1.4) to A.

For any pair of functions f, g in A, inf(f, g) and sup(f, g) belong

For we can write sup(f, g) = f( f + g + 1f - gl) and inf(f, g) = f ( f + g - If - gl); the result therefore follows from (7.3.1.3) applied to

the algebra

(7.3.1.5) c1,

A.

For any pair of distinct points x, y in E and any pair of real numbers

p, there is a function f E A such that f ( x ) = u, f ( y ) = 8.

By assumption, there is a function g E A such that g(x) # g(y). As A contains the constant functions, take f = a + ( p - a)(g - y)/(S - y), where y = g(x),& = d Y ) . (7.3.1.6) For any function f E %,(E), any point x E E, and any E > 0 , there is a function g E A such that g(x) =f(x) and g(y) 2/(n - I). Similarly, consider the subset Mi of M, consisting of the 2"-' functions of M, which are equal to x - a in the interval [a, a ( h - a)/n],and for each function g E MA, consider the set of all functions f E K such that g(x) - ( 2 / n ) < f ( x ) 6 g ( x ) for every x E I. Use a similar construction when (b - a)/&is not an integer.)

+

6. REGULATED FUNCTIONS

Let 1 be an interval in R, of origin a and extremity b (a or b or both may be infinite), F a Banach space. We say that a mapping f of I into F is a step-function if there is an increasing finite sequence of points of i (closure of 1 in R) such that xo = a, xn = b, and that f is constant in each of the open intervals ] x i ,x i + l [(0 6 i < n - 1). For any mapping f of I into F and any point x E I distinct from b, we lim f ( y ) exists; we then write the say that f has a limit on the right if Y€l,Y>X Y-x

limit f ( x + ) . Similarly we define for each point x E I distinct from a, the limit on the feft off at x, which we write f ( x - ) ; we also say these limits are one-sided limits off. A mapping f of I into F is called a regulated function if it has one-sided limits at every point of I . It is clear that any step-function is regulated.

(7.6.1) In order that a mapping f of a compact interid I = [a, b ] into F he regulated, a necessary and suficient condirion is thatf be the limit of a unformly convergent sequence of step-functions. (a) Necessity. For every integer n, and every x E I, there is an open interval V ( x ) = ] y( x) ,z(x)[ containing x, such that Ilf(s) -f(t)II < I/n if either both s, t are in ]y(x),x[ n I or both in ]x, z ( x ) [n I . Cover I with a finite number of intervals V ( x , ) ,and let ( c ~be the ) strictly ~ ~ increasing ~ ~ sequence consisting of the points a, b, x i , y ( x i ) and z(xi). As each c j is insome V(xi), c j + ]is either in the same V ( x i )or we have c j + , = z(xi), f o r j d m - 1 ; in other words if s, t are both in the same interval ] c j ,c ~ + ~ then [ , IIf(s) - f(t)ll < I/n. Now define g, as the step-function equal to f at the points c j , and at the midpoint of each interval ] c j , c j + ] [ , and constant in each of these intervals. It is clear that Ilf- grill d l/n.

~

VII SPACES OF CONTINUOUS FUNCTIONS

146

(b) Suficiency. Suppose f is the uniform limit of a sequence (f,) of step-functions. For each E > 0 there is an n such that l\f-fnll Q 43; now for each x E I , there is an interval ]c, d[ containing x and such that Ilf,(s) -f,(t)ll < ~ / 3if both s and t are in ]c, x [ or both in ]x, d [ ; hence under the same assumption we have IIf(s) -f(t)II Q E , and this proves the existence of one-sided limits off at x, since F is complete (3.14.6). Another way of formulating (7.6.1) is to say that the set of regulated functions is closed in BF(E), and that the set of step-functions is dense in the set of regulated functions. (7.6.2) Any continuous mapping of an interval I c R into a Banach space is regulated; so is any monotone mapping of I into R.

This follows from the definition, taking into account (3.16.5) and (4.2.1).

PROBLEMS

1. Let f be a regulated mapping of an interval I c R into a Banach space F. Show that for each compact subset H of I, f ( H ) is relatively compact in F; give an example showing that f(H) need not be closed in F. 2. The function f(x) = x sin(l/x) ( f ( 0 )= 0) is continuous, hence regulated in I = [0, I], and the function g(x) = sgn x (g(x) = 1 if x > 0, g(0) = 0,g(x) = - 1 if x > 0) is regulated in R, but the composed function g o f is not regulated in I. 3. Let I = [a, b ] be a compact interval in R. A function of bounded variation in I is a mapping fof I into a Banach space F, having the following property: there is a number V >, 0 such that,for m y strictly increasing finite sequence (fi)ogrgnof pointsof I, the inequality n- 1

C llf(ti+~)-f(ti)ll

i=o

G V holds.

(a) Show that f(1) is relatively compact in F (prove that f(1) is precompact, by an indirect proof). (b) Show thatfis a regulated function in I (use (a) and (3.16.4)). (c) The function g defined in [0, I ] as equal to xz sin(l/xz) for x # 0 and to 0 for x = 0 is not of bounded variation, although it has a derivative at each point of I.

CHAPTER Vlll

DIFFERENTIAL CALCULUS

The subject matter of this chapter is nothing else but the elementary theorems of calculus, which however are presented in a way which will probably be new to most students. That presentation, which throughout adheres strictly to our general “geometric” outlook on analysis, aims at keeping as close as possible to the fundamental idea of calculus, namely the “ local ” approximation of functions by linear functions. In the classical teaching of calculus, this idea is immediately obscured by the accidental fact that, on a one-dimensional vector space, there is a one-to-one correspondence between linear forms and numbers, and therefore the derivative at a point is defined as a number instead of a linear form. This slavish subservience to the shibboleth of numerical interpretation at any cost becomes much worse when dealing with functions of several variables : one thus arrives, for instance, at the classical formula (8.9.2) giving the partial derivatives of a composite function, which has lost any trace of intuitive meaning, whereas the natural statement of the theorem is of course that the (total) derivative of a composite function is the composite of their derivatives (8.2.1), a very sensible formulation when one thinks in terms of linear approximations. This “ intrinsic” formulation of calculus, due to its greater “ abstraction,” and in particular to the fact that again and again, one has to leave the initial spaces and to climb higher and higher to new “function spaces” (especially when dealing with the theory of higher derivatives), certainly requires some mental effort, contrasting with the comfortable routine of the classical formulas. But we believe that the result is well worth the labor, as it will prepare the student to the still more general idea of calculus on a differentiable manifold, which we shall develop in Chapters XVI to XX. Of course, he will observe that in these applications, all the vector spaces which intervene have 147

148

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DIFFERENTIAL CALCULUS

finite dimension; if that gives him an additional feeling of security, he may of course add that assumption to all the theorems of this chapter. But he will inevitably realize that this does not make the proofs shorter or simpler by a single line; in other words, the hypothesis of finite dimension is entirely irrelevant to the material developed below ; we have therefore thought it best to dispense with it altogether, although the applications of calculus which deal with the finite dimensional case still by far exceed the others in number and in importance. After the formal rules of calculus have been derived (Sections 8.1 t o 8.4), the other sections of the chapter are various applications of what is probably the most useful theorem in analysis, the mean value theorem, proved in Section 8.5. The reader will observe that the formulation of that theorem, which is of course given for vector valued functions, differs in appearance from the classical mean value theorem (for real valued functions), which one usually writes as an equality J’(h) - f ( a ) =f’(c)(b - a). The trouble with that classical formulation is that: ( I ) there is nothing similar to it as soon as f has vector values or when there are a finite number of points where f ’ is not defined; (2) it completely conceals the fact that nothing is known on the number c, except that it lies between a and b, and for most purposes, all one need know is that f ’ ( c ) is a number which lies between the g.1.b. and 1.u.b. off’ in the interval [a, b] (and not the fact that it actually is a value off’). In other words, the real nature of the mean value theorem is exhibited by writing it as a n inequality, and not as an equality. Finally, the reader will probably observe the conspicuous absence of a time-honored topic in calculus courses, the “ Riemann integral.” It may well be suspected that, had it not been for its prestigious name, this would have been dropped long ago, for (with due reverence t o Riemann’s genius) it is certainly quite clear to any working mathematician that nowadays such a “ theory” has at best the importance of a mildly interesting exercise in the general theory of measure and integration (see Section 13.9, Problem 7). Only the stubborn conservatism of academic tradition could freeze it into a regular part of the curriculum, long after it had outlived its historical importance. Of course, it is perfectly feasible to limit the integration process to a category of functions which is large enough for all purposes of elementary analysis (at the level of this first volume), but close enough to the continuous functions t o dispense with any consideration drawn from measure theory; this is what we have done by defining only the integral of regulated functions (sometimes called the “ Cauchy integral ”). When one needs a more powerful tool, there is no point in stopping halfway, and the general theory of (“ Lebesgue”) integration (Chapter XIII) is the only sensible answer.

1 DERIVATIVE OF A CONTINUOUS MAPPING

149

1. DERIVATIVE OF A C O N T I N U O U S MAPPING

Let E, F be Banach spaces (both real or both complex) and A a n open subset of E. Let h g be two continuous mappings of A into F; we say that f and g are tangent at a point x, E A if lim lif(x) - g(x)l)/llx- xoII = 0 ; x-xn,x+xo

this implies of course that f ( x o )= g(xo). We note that this definition only depends on the topologies of E and F; for i f h g are tangent for the given norms on E and F, they are still tangent for equivalent norms (Section 5.6). Jff, g are tangent at xo, and g, h tangent at xo, thenf, h are tangent at x, as follows from the inequality 11 f(x) - h(x)(j < lif(x) - g(x)ll + IIg(x) - h(x)/J. Among all functions tangent at x, to a function f , there is at most one mapping of the form x - f ( x , ) + u(x - x,) where u is linear. For if two such functions x +f(xo) + u l ( x - x,), x 4f ( x o )+ uz(x - x,) are tangent at xo, this means, for the linear mapping u = u1 - u 2 ,that lim Ilu(y)ll/llyll = 0. But this implies L' = 0, for if, given E > 0, there is r > 0 such that ljyll < r implies llv(y)ll d E llyll, then this last inequality is still valid for any x # 0, by applying it to y = rx/ilxil; as E is arbitrary, we see that u(x) = 0 for any x. y-+O,yfO

We say that a continuous mapping f of A into F is diflerentiable at the point x, E A if there is a linear mapping u of E into F such that x + f ( x o ) + u(x - x,) is tangent to f at x,. We have just seen that this mapping is then unique; it is called the derivative (or total derivative) o f f at the point x,, and writtenf'(x,) or Df(x,).

(8.1 .I)If the continuous mapping f of A into F is diferentiable at the point xo, the deriuative f '(x,) is a continuous linear mapping of E into F.

Let u = f ' ( x o ) .Given E > 0, there is r such that 0 < r < 1 and that lltll < r implies llf(xo t ) -f(xo)ll < 4 2 and llf(xo + t ) -f(xo) - 4l)ll < & IllIlP; hence IltlI d r implies Ilu(t)ll < E , which proves u is continuous by (5.5.1).

+

The derivative (when it exists) of a continuous mappingfof A into F, at a point x, E A, is thus an element of the Banach space Y(E; F) (see Section 5.7) and not of F. In what follows, for u E 9 ( E ; F) and t E E, we will write u . t instead of u ( t ) ; we recall (Section 5.7) that Ilu .tll < llull . lltll and that

150

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When E has finite dimension n and F has finite dimension m , f ' ( x o ) can thus be identified to a matrix with nz rows and n columns; this matrix will be determined in Section 8.10. Examples

(8.1.2) A constant function is differentiable at every point of A, and its derivative is the element 0 of 9 ( E ; F). (8.1.3) The derivative of a continuous h e a r mapping u of E into F exists at every point x E E and Du(x) = u. For by definition u(xo) + u(x - x o ) = u ( x ) . (8.1.4) Let E, F, G be three Banach spaces, ( x , y ) --* [x . y ] a continuous bilinear mapping of E x F into G. Then that mapping is differentiable at every point (x,y ) E E x F and the derivative is the linear mapping (s, t ) --* [x . tl [s . y l .

+

For we have [(x

+ s) . ( y + t ) ] - [ x . y ] - [ x . t ] - [ s . y ] = [s. t ]

and by assumption, there is a constant c > 0 such that l\[s * t]ll < c . llsll * (It/l (5.5.1). For any E > 0, the relation sup(llsll, Iltll) = II(s, t)l\ Q E / C implies therefore

"I

+ ). . ( y + t>l - [x ul - [x . 11 - 1s - YlII *

,< & IKS, t)11

which proves our assertion. That result is easily generalized to a continuous multilinear mapping. (8.1.5) Suppose F = F, x F, x ... x F, is a product of Banach spaces, and f = (f i , . . . ,f,) a continuous mapping of an open subset A of E into F. In order that f be differentiable at xo , a necessary and suficient condition is that each fi be differentiable at xo , and then f ' ( x o ) = ( f ; ( x o ) ,. . . ,fA(xo)) (when 2 ( E ; F) is identiJied with the product of the spaces Y(E; FJ).

Indeed, any linear mapping u of E into F can be written in a unique way u = (ul, . . . , u,), where u i is a linear mapping of E into F i , and we have by definition \lu(x)II = sup(llul(x)(~, . . . , ~ ~ u , ( x ) ~whence ~), it follows

2 FORMAL RULES OF DERIVATION 151

(by (5.7.1) and (2.3.7)) that llull

= sup(jlu, I/,

identification of 2 ( E ; F) with the product

m

. . . , IIuJ), which allows the

fl Y(E; Fi). From

the defini-

i= I

tion, it follows at once that u is the derivative off at x , if and only if ui is the derivative of f i at x, for 1 < i Q m. Remark. Let E, F be complex Banach spaces, and E,, F, the underlying real Banach spaces. Then if a mapping f of an open subset A of E into F is differentiable at a point x o , it is also differentiable with the same derivative, when considered as a mapping of A it7tO F, (a linear mapping of E into F being also linear as a mapping of Eo into F,). But the converse is not true, as the example of the mapping z + Z (complex conjugate) of C into itself shows at once; as a mapping of R2 into itself, u : z + 5 (which can be written (x,y ) -+ ( x , - y ) ) is differentiable and has at each point a derivative equal to u, by (8.1.3); but u is not a complex linear mapping, hence the result. We return to that question in Chapter IX (9.10.1). When the mapping f of A into F is differentiable at every point of A, we say that f is differentiable in A ; the mapping x + f ’ ( x ) = Df(x) of A into Y(E; F) will be writtenf’ or Dfand called the derivative o f f in A.

2. FORMAL RULES OF D E R I V A T I O N

(8.2.1) Let E, F, G be three Banach spaces, A an open neighborhoodof x, E E, f a continuous mapping of A into F, yo = f ( x , ) , B an open neighborhood of y o in F, g a continuous mapping of B into G. Then iff is differentiable at x, and g differentiable at y o , the mapping h = g o f (which is defined and continuous in a neighborhood of x,) is differentiable at x, , and we have h’(x0) = S’(Y0) o f ‘(xo).

By assumption, given E such that 0 < E < 1, there is an r > 0 such that, for 1)s11 < r and litil Q r, we can write f ( x 0 + s> =f(x,>+f’(xo) . s + 01(s) S(Y0 + t ) = d Y 0 ) + S’(Y0) . t

+ OAt)

with i\ol(s)I\ Q ~ l l s l land l\02(t)II< ~jltll.On the other hand, by (8.1.1) and (5.5.1), there are constants a, b such that, for any s and t, llf’(xo>*

SII

Q a llsll

and

IlS‘(Y0)

. tll

< b lltll

152

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DIFFERENTIAL CALCULUS

hence Ilf’(x0)

*

+ o,(s)ll G (a + 1) llsll

for /Is/l < r. Therefore, for ~~s~~ < r/(a and

Iloz(f’(x0)

+ I),

. s + Ol(S))lI

II9’(Yo)

we have Q (a f

I)&Ibll

. Ol(S>ll G bE llsll

hence we can write

h(x0 + $1 = d Y 0 +f’(xo) * s + OI(S)) = S(Y0) + S’(Y0) . (f’(xo>. s) + ads) with Ilo3(s)ll Q (a

+ b + 1)E I l ~ l l ,

which proves the theorem.

(8.2.1)has of course innumerable applications, of which we mention only the following one: (8.2.2) L e t x g be twto continuous mappings of the open subset A of E into F. I f f and g are differentiable at x, , so are f + g and af ( a scalar), and we have (f+ g)‘(xo)= f ‘ ( X 0 ) + g’(x0) and (af)’(x,>= af‘(x0). The mapping f + g is composed of (u, v) 4u + v, mapping of F x F into F, and of x + ( f ( x ) , g ( x ) ) , mapping of A into F x F; both are differentiable by (8.1.3)and (8.1.5),and the result follows (for f + g ) from (8.2.1).For af the argument is still simpler, using the fact that the mapping u+au of F into itself is differentiable by (8.1.3).Of course, (8.2.2) could also be proved very simply by direct arguments. Let E, F be two Banach spaces, A an open subset of E, B a n open subset of F. If A and B are homeomorphic, and there exists a differentiable homeomorphism f of A onto B, it does iiot follow that, for each x, E A, f ‘ ( x o ) is a linear homeomorphism of E onto F (consider e.g. the mapping 5 + of R onto itself).

c3

(8.2.3) Let f be a homeomorphism of an open subset A of a Banach space E onto an open subset B of a Banach space F, g the inverse homeomorphism. Suppose f is differentiable at the point x, ,and f ’(x,) is a linear homeomorphism of E onto F; then g is differentiable at y o = f(xo) and g‘(y,) is the inverse mapping t o f ‘ ( x o )(cf. (10.2.5)).

2 FORMAL RULES OF DERIVATION 153

+

By assumption, the mapping s + f ( x , s) - f ( x o ) is a homeomorphism of a neighborhood V of 0 in E onto a neighborhood W of 0 in F, and the inverse homeomorphism if t +g(yo + t ) -g(y,). By assumption, the linear mappingf’(xo) of E onto F has an inverse u which is continuous, hence (5.5.1) there is c > 0 such that Iju(t)/I 9 c Iltll for any t E F. Given any E such that 0 < E < 1/2c, there is an r > 0 such that, if we write f ( x o + S ) - f ( x o ) = f ’ ( x , ) a s + ol(s), therelation llsll < r implies Ilol(s)II 9 E IIsII. Let r‘ now be a number such that the ball lltli 9 r’ is contained in W and that its image by the mapping t -+ g(y, t ) - g(y,) is contained in the ball llsll 9 r. Let z = g(y, + t ) - g(y,); by definition, for lltll < r ’ , this equation implies t = f ( x , z ) - f ( x o ) and as llzll 9 r, we can write t = f ’ ( x , ) . z + ol(z), with liol(z)II 9 E llzll. From that relation we deduce

+

+

u * t = u * (f’(x,) * z)

+u

*

ol(z) = z

+u

*

ol(z)

+

by definition of u, and moreover IIu * ol(z)II 9 c llol(z)IJ < CE llzll < 1 1 ~ 1 1 , hence IIu . t/l 2 llzll - 4 llzll = -t llzll; therefore llzll < 2Ilu. tll < 2cIltl1, and finally IIu . ol(z)II < CE llzll < 2c% Iltll. We have therefore proved that the relation lltll < r‘ implies IIg(y, + t ) - g(y,) - u . t 11 < 2cZc Iltll, and as E is arbitrary, this completes the proof. The result (8.2.3)can also be written (under the same assumptions)

PROBLEMS 1. Let E be a real prehilbert space. Show that in E the mapping x + llxli of E into R is differentiable at every point x # 0 and that its derivative at such a point is the linear mapping s+(sIx)/IIx/l. 2. (a) In the space (co) of Banach (Section 5.3, Problem 5) show that the norm x + J/xlj is differentiable at a point x = (8.) if and only if there is an index no such that > I f . 1 for every n # n o . Compute the derivative. (b) In the space I’ of Banach (Section 5.7, Problem I ) , show that the norm x + llxll is not differentiable at any point (use (8.1.1) and Problem l(c) of Section 5.7).

ltnol

3. Let f be a differentiable real valued function defined in an open subset A of a Banach space E. (a) Show that if at a point xo E A,freaches a relative maximum (Section 3.9, Problem 6), then Df(xo) = 0. (b) Suppose E is finite dimensional, A is relatively compact,fis defined and continuous in A, and equal to 0 in the boundary of A. Show that there exists a point x,, E A where D f ( x o )= 0 (“Rolle’s theorem”; use (a) and (3.17.10)).

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3. DERIVATIVES IN SPACES

O F C O N T I N U O U S LINEAR F U N C T I O N S

(8.3.1) Let E, F , G be three Banach spaces. Then the mapping (u, v) -+ v u (also written vu) of 9 ( E ; F) x 9 ( F ; C ) into 9 ( E ; G) is diferentiable, and the derivative at the point (uo , vo) is the mapping (s, t ) -+ vo s + t uo . 0

0

0

If we observe that, by (5.7.5), the mapping (u, v) -+ v u is bilinear and continuous, the result is a special case of (8.1.4). 0

(8.3.2) Let E , F be two Banach spaces, such that there exists at least a linear homeomorphism of E onto F. Then the set 2 of these linear homeomorphisms is open in 9 ( E ; F ) ; the mapping u -+ u - l of iff onto the set Y? of linear homeomorphisms of F onto E is continuous and diferentiable, and the derivative of u -+ u-l at the point uo is the linear mapping (of Y ( E ; F ) into Y(F; E ) )

-'

s --f

-24;'

0

s

0

u;'.

1. We consider first the case F = E, and write 1, for the identity mapping of E. Then:

(8.3.2.1) If llwll < 1 in 9 ( E ; E), the Zinear mapping 1, + w is a homeomorphism, its inverse (1, + w)-' is equal to the sum of the absolutely convergent series

m

C (n=O

l)nw", and we have

(8.3.2.2)

11(IE

We have

N n=O

+ w)-'

- l E + wII

IIwII" = (1 - IIw IIN")/(l -

- Ilwll>.

~~K'~\2/(1

Ilwll)

< l/(l - Ilwll),

hence, by

m

(5.7.5), (5.3.1), (5.3.2) and (5.7.3), the series convergent in 9(E; E). Moreover, we have (1E+W)(1E-W+w2

= (1, - w

(-1)"

is absolutely

n=O

+"'+(-l)NWN)

+ w2 +

* "

+ ( - l ) N W N ) ( l E + w) = 1 E - ( - l ) N + ' W N + l .

and as w N + l tends to 0 with 1/N, we have by definition and by (5.7.5),for the W

element v

= n=O

( - 1 ) " ~ ~ of 9 ( E ; E), (1,

+ w)v = v ( l E + w) = l E , which

proves the first two statements; the inequality (8.3.2.2) follows from the relation (1, + w)-' - 1, + w = w2(lE - IY f M!2 and from (5.7.5) and (5.3.2).

+

. . a ) ,

4

DERIVATIVES OF FUNCTIONS OF ONE VARIABLE

155

2. Consider now the general case; suppose s ~ 9 ( EF); is such that ( ( s ( (. Ilui'll < I ; then the element 1, + u0-'s, which belongs to 9 ( E ; E), has an inverse, due to (5.7.5) and (8.3.2.1);as we can write uo s = uo(1, ui's), the same is true for uo + s, the inverse being (1, u;'s)-'u,'; hence we have

+ +

+

(ug

+ s)-'

Applying (8.3.2.2)to Il(u0

+ s)-'

- t!;' = ( ( 1 E

UJ = ui's,

- u;' Il(u0

- 1E)u;'.

we obtain, for llsll < l/llu;'II

+ u;'su;'ll

Therefore, if we take llsll

+ u,'s)-'

d Ilu,'113. lIsll'/(l - lu;'II

*

Ilsll).

< 1 / 2 ~ ~ u ~we' ~ have l,

+ s)-'

- u;'

+ u,'su,'ll

d c lls11'

' ~ this ~ ~ ends , the proof. with c = 2 ~ ~ u ; and

4. DERIVATIVES OF F U N C T I O N S O F O N E VARIABLE

When we specialize E to a one-dimensional vector space (identified to R or C), we know that 9 ( E ; F) is naturally identified to F itself, a vector b E F being identified to the linear mapping 5 -+ b( of E into F (5.7.6). If f is a differentiable mapping of an open set A c E into F, its derivative D f ( t 0 )at a point toE A is thus identified to a vector of F, and the mapping Df to a mapping of A into F. If F itself is one-dimensional (identified to R or C), we obtain the classical case of the derivative (at a point) as a number. The general results obtained above boil down in that last case to the classical formulas of calculus; for instance, (8.3.2),when E and F are one-dimensional, is simply the formula giving the derivative of 1/4 as equal to - 1/(' for 5 # 0. We explicitly formulate the following consequence of (8.2.1):

(8.4.1) Let E, F be two real (resp. complex) Banach spaces, f a differentiable mapping of an open subset A of E into F, g a diferentiable mapping of an open subset I of R (resp. C) into A; then the derivative at 5 E I of the composed mapping h =f .g of I into F is the vector of F equal to Df(g( 0, k > 0. (b) The real functionfequal to x z sin(l/x) for x # 0, to 0 for x 0, is differentiable in R, but ( f ( x ) - f ( y ) ) / ( x - y ) has no limit when ( x , y ) tends to ( 0 , O ) in the set of pairs such that x > 0, y > 0, x # y . (c) In the interval I = [O, I], the sequence of continuous functions f. is defined as follows: fo(t) = t ; for each n > 1, fn has the form at fl in each of the 3" intervals

+

+

:

+

k 3"

-Sf 0, such that lim t, = 0 and

x

4.

t2)

t2 t:.)

+

n-m

that Ilt;'f(tnan)ll = a. tends to co; one can suppose that the sequences ( I , ) and 1,) are decreasing and tend to 0. Define a continuous mapping g of [0, 11 into E such that g(0) = 0, that g'(0) exists and is equal to 0, and that g ( J i t,) = faan).) 5. (a) Let E, F be two Banach spaces,fa continuous mapping of an open subset A of E into F. Show that if is diffcrentiable at x o t A, it is quasi-differentiable at xo and its quasi-derivative is equal to its derivative.

(din

158

Vlll DIFFERENTIAL CALCULUS

(b) Suppose E has finite dimension. Show that iff is quasi-differentiable at xo E A,

f is differentiable at xo . (Use contradiction: let u be the quasi-derivative o f f at xo , and suppose there is a > 0 and a sequence (x,) of points of A, tending to xo , such that llf(x,) -f(xo) - u (x. - xo)II> aIIx, - xo(I. Using the local compactness of E, show that one may suppose that the sequence (((x, - xo 11) is decreasing, and that the sequence of the vectors z. = (x,- xo)/Ilx,- xo 11 tends to a limit in E; then define a continuous mapping g of [0, 11 into E such that g(0) = xo , that g'(0) exists, but that

-

u(g'(0)) is not the derivative of t + f ( g ( t ) ) at t = 0.)

6. Let I = [0, I], and let E be the Banach space WR(I). In order that the mapping x -+ llxll of E into R be quasi-differentiable at a point xo , it is necessary and sufficient that the function 1 Ixo(t)l reaches its maximum in I at a single point to E I ; the quasi-derivative of x + l/xll at xo is then the linear mapping u such that u(z) = z(to) if -+

xo(to)> 0, u(z) = -z(to)

if xo(to)< 0 (compare Section 8.2, Problem 3). (To prove the condition is necessary, suppose lxol reaches its maximum at two distinct points t o , tl at least; let y be a continuous mapping of I into itself, equal to 1 at t o , to 0 at t l ; examine the behavior of ( I/xo hyll - /IxoIl)/has the real number h # 0 tends to 0. To prove the condition is sufficient,let h + z, be a continuous mapping of I into E, having a derivative a f E at h = 0 and such that zo =O; observe that if r, is the largest number in I (or the smallest number in I) where t Ixo(t) z,(t)l reaches its maximum, then t, tends to to when h tends to 0.) Deduce from that result that the mapping x -+ j/x/I of E into R is not differentiable at any point (compare to Section 8.2, Problem 2). 7. Letfbe a continuous mapping of an open subset A of a Banach space E into a Banach space F. Supposefis lipschitziun in A : this means (7.6, Problem 12) that there exists a constant k > 0 such that llf(xl) -f(xZ)ll < k I/xl - xz/I for any pair of points of A. Let xo E A, and suppose there is a linear mapping u of E into F such that, for any vector a # 0 in E, the limit of (f(xo f a t ) --.f(x0))/t when t # 0 tends to 0 in R,exists and is equal to u(a). Show that f i s quasi-differentiable at xo . llu rb /I of 8. (a) Let u , b be two points in a Banach space E. Show that the mapping t R into itself has a derivative on the right and a derivative on the left for every t E R (prove that if 0 < t < s, then ([la bril- Ilall)/t < (ila bsll- Ilall)/s and use (4.2.1)). (b) Let u be a continuous mapping of an interval I E R into E. Show that if at a point t o E I, u has a derivative on the right, then t Ilu(t)ll has at to a derivative on the right and

+

+

-+

-+

+

+

+

--f

(D+ IlulI)(to)

< IID+u(to)ll

(apply (a)). (c) Let U be a continuous mapping of I into Y(E; E). Show that if at a point to E I, U has a derivative on the right and U(t0) is a linear homeomorphism of E onto itself, then the mapping f l\(U(t))-lII= f ( t ) , which is defined in a neighborhood of t o , has a derivative on the right at t o , and that

-

5. T H E M E A N VALUE THEOREM

Let I = [ct, p] be a compact interval in R, f a continuous mapping of I into a Banach space F, cp a continuous mapping of I into R. We suppose that there is a denumerable subset D such that, for each 5 E I - D, f and cp have both a derivative at 5 with respect to I (8.4), and that (1 f'(5)II < cp'(5j. Then I l f (PI - f(4I1 G cp(P) - c p ( 4

(8.5.1)

5 THE MEAN VALUE THEOREM

159

Let n -+ p n be a bijection of N onto D ; for any E > 0, we will prove that left hand side being independent of E , this will complete the proof. Define A as the subset of I consisting of the points 5 such that, for a < 5 < 5 ,

IIf(p) -f(a)II < cp(p) - cp(a) + E(P - a + 2 ) ; the

It is clear that a E A ; if 5 E A and a < v] < 4, then v] E A also, by definition; this shows that if y is the 1.u.b. of A, then A must be either the interval [a, y [ or the interval [a, y ] ; but in fact, from the definition of A it follows at once that A = [ E , y ] . Moreover, from the continuity of f a n d cp it follows that (8.5.1.1)

Ilf(Y) -f(.>II< d Y ) - cp(4 + 4 Y - 4 + E

c

2-"

Pn 0, ex is strictly increasing (by (8.5.3)), and hence e = e' > e0 = 1. The function ex is also written exp(x) or exp x. The function log, x is written log x and it follows from (8.2.3)and (4.2.2) that D(log x) = I/x for x > 0. Furthermore D(a") = log a * a".

PROBLEM

Study the variation of the functions (1

+

!-)"".

(1

+ $)Y

(1 +!)(I

+ ;):

(I

+

y+'

for x > 0 , p being a fixed arbitrary positive number; find their limits when x tends to

+to.

9. PARTIAL DERIVATIVES

Let f be a differentiable mapping of an open subset A of a Banach space E into a Banach space F; Dfis then a mapping of A into 9 ( E ; F). We say that f is continuously diferentiable in A if Dj i s continuous in A. Suppose now E = E, x E, . For each point (al, a z )E A we can consider the partial mappings x1 +f ( x , , a z ) and x, +f(a,, xz) of open subsets of El and E, respectively into F. We say that at (a,, a,), f is d/ferentiab/e with respect to the first (resp. second) variable if the partial mapping x1 +f(xl, a 2 ) (resp. x2 +f ( a , , xz)) is differentiable at a, (resp. a z ) ; the derivative of that mapping, which is an element of -%'(El ;F) (resp. 9 ( E z ; F)) is called the partial derivative off at ( a l , a,) with respect to the first (resp. second) variable, and written D,f(a,, a,) (resp. D,f(a,, a,)).

PARTIAL DERIVATIVES

9

173

(8.9.1) Let f be a continuous mapping of an open subset A of El x E, into F. In order that f be continuously differentiable in A, a necessary and suficient condition is that f be differentiable at each point with respect to the first and the second variable, and that the mappings (x, , x,) -+ D, f ( X I , x,) and (x,, x,) -+ D,f(x,, x2) (of A into 2’(El; F) and 9 ( E 2 ; F) respectioely) be continuous in A. Then, at each point (x,, x,) of A, the derivative off is given by (8.9.1.1)

Df(x1,

~

2

)(11, t 2 )

=Dif(xi,

~ 2 ) ti.

+ Dzf(x1,

~ 2 ) t. 2 .

(a) Necessity The mapping x, -)f(x,, a,) is obtained by composing f and the mapping x1 -+ (x,, a,) of El into El x E2 , the derivative of this second mapping being r, - + ( t , , 0) by (8.1.2), (8.1.3), and (8.1.5). Then by (8.2.1), x1 f(x,, a,) has at (a,, a,) a derivative equal to t , -+ Df(a,, a,) . (tl, 0). If we call i, (resp. i,) the natural injection t , .+ ( t , , 0) (resp. t2 + (0, t , ) ) , which is a constant element of 2’(El; E, x E2) (resp. 2’(E2; E, x E,)), we therefore see that D, f ( a l ,a,) = Df (al, a,) i,, and similarly D2f (al, a,) = Df(a,, a2) i, (all this is valid iff is simply supposed to be differentiable in A). As the mapping ( 0 , u ) -+ v u of 9(El x E,; F) x 9( E l ; El x E,) into 9 ( E l ; F) is continuous ((5.7.5) and (5.5.1)), the continuity of D, f and D, f follows from that of Df; finally, as ( t , , t , ) = il(tl) + i2(t2),we have (8.9.1 .I). (b) Suficiency Write -+

0

0

0

f ( 0 , + 4 > a2 + f 2 ) - f (a19 a,) = (f(a1

Given

E

+ t , , 0 2 + t 2 ) -f(a, + t , , 0 2 ) ) + ( f b l + t , , a,)

-f(a,, a,)).

> 0, there is, by assumption, an r > 0 such that, for IItlll d r IIf(a1

+ t , , a2) -f(a,,

02)

- D,fta,,a2), fill

E

Iltlll.

On the other hand, we have in a ball B of center (al, a,) contained in A, by (8.6.2) lIf(a1

+ t , , a2 + 12) -f(a1 + t l , 0 2 ) - D,f(a1 + t l , 0 2 )

*

t2II

The continuity of the mapping D, f therefore implies that there is r‘ > 0 such that for /lt,Ij < r’ and Ilt,II < r‘, we have IIf(a,

+ t,, a, +

-f(a,

+ t , , 0,) - D2f(al + I , ,

0,)

- t,Il

and on the other hand

I/ D2f (01 + t17a,) - D2f (al, a,)ll

0 and 6 > 0 such that the relation It1 < 6 implies If.(t) -f.(O)I < Alrl for every n ; this implies that S U P lL(0)l < +a. (b) In order that f be differentiable at 0, it is necessary and sufficient that for every E > 0, there is a 6 > 0 such that the relation It1 G 6 implies Ifn(t)-fn(O) -K(O)tI G E It1 for every n.

10 JACOBIANS 175

(c) In order that the derivativef' exist in a neighborhood of 0 in E and be continuous at 0, a necessary and sufficient condition is that there exist a neighborhood J c I of 0 such that: (1) eachf,' exists in J ; (2) sup Ifk(0)l < + a ;(3) the sequence(f.')isequiconn

tinuous a t the point 0 (Section 7.5). (See Section 8.6, Problem 3.) (d) Let f . ( t ) = e""/n for every n 2 1,Yo(()= I . Show that f is quasi-differentiable at every point x E E; if u(x) is the quasi-derivative o f f at the point x , show that the mapping ( x , y ) u(x) . y of E x E into Fo is continuous, but that f i s not differentiable at any point of E. Let f be a continuous mapping of an open set A of a Banach space E into a Banach space F. Suppose that for any x E A and any y E E, lim ( f ( x t y ) - f ( x ) ) / t = g ( x , y ) --f

+

c-o,r+o

exists in E. If, for y l E E, 1 < i < n, and xo E A, each of the mappings x + g ( x , y l ) is

+ + +

I:

continuous at xo , show that g ( x o , y1 yz . . . y.) = g(xo,y l ) (apply the mean 1=1 value theorem). Let E l , Ez , F be three Banach spaces,fa continuous mapping of an open subset A of El x E2 into F. In order that f be differentiable at ( a l , a 2 )E A, it is necessary and sufficient that: (1) Dlf(al, a 2 ) and D2f(al, a 2 ) exist; (2) for any E > 0, there exists 6 > 0 such that the relations IIfJ < 6, Ilt2i1< 6 imply Ilf(a1

+ t l , a , + t z ) -fh+ t l ,

0 2 ) -f(a1,

a2

+

t 2 ) +f@l,

aJil < E(llflll+ Iltzll).

Show that the secondconditionissatisfied if Dlf(al, az)exists and thereisaneighborhood V of (al, a 2 ) in El x E2 such that D2f exists in V and the mapping ( x I ,x 2 ) - + D 2 f ( x I x, z ) of V into P ( E 2 ; F) is continuous. Let f be the real function defined in R Z by f(x,y ) = (xy/r)sin(l/r) for (x, y ) # (0, O), with r = (x' Y ' ) ' ' ~ , and f ( 0 , O ) = 0. Show that Dlf and D 2 f exist at every point ( x , y ) E R2, and that the four mappings x Dlf(x, b), y Dlf(a, y), x D 2 f ( x , b), y --f D 2 f ( a ,y ) are continuous in R for any (a, b) E R2, but that f i s not differentiable at

+

--f

--f

--f

NO).

Let I be an interval in R , f a mapping of I' into a real Banach space E, such that, for any ( a l , ... , a,) E I,, each of the mappings x , + f ( a ~ ,... , u,.-I, x j . U J + I , . . ,a,) (1 < j < p ) is continuous and differentiable in I, and furthermore, the p functions D j f ( l < j < p ) arc bounded in 1'. Show thatfis continuous in I' (use the mean-value theorem).

.

10. JACOBIANS

We now specialize the general result (8.9.1) to the most important cases. 1. E = R" (resp. E = C"). Iff is a differentiable mapping of an open subset A of E into F, the partial derivative D,f(a,, . . . , a,) is identified to a vector of F (Section 8.4), and the derivative off is the mapping fl

((1,

cn)+

k=l

Dkf(al?

...)'%)lk.

If Dfis continuous, SO is each of the Dk$ Conversely, if each of the mappings DJexists and is continuous in A, then f is continuously differentiable in A. 2. E = R" and F = R" (resp. E = C" and F = C'"). Then we can write

176

Vlll

DIFFERENTIAL CALCULUS

f = (ql,. . . , cp), where the qi are scalar functions defined in E, and by (8.1 -5)f is continuously differentiable if and only if each of the 'pi is continuously differentiable; again, by case I , 'pi is continuously differentiable if and only if each of the partial derivatives D j q i (which is now a scalar function) exists and is continuous. Furthermore, the (total) derivative of f is the linear mapping (Cl,

with

*.*)

Cn)+(Vl,

. . . >V m )

n

Vi

=

C (Djcpi(E1, j=

* * * 3

an>>Cj ;

1

in other words, f',which is a linear mapping of R" into R" (resp. of C" into C"), corresponds to the matrix (Djqi(al, . . . , a,)), which is called the jacobian matrix o f f ( o r of cpl,. . . , cp), at (al, . . . , a,,). When m = n, the determinant of the jacobian (square) matrix off is called thejacobian off(or of cpl, . . ., cp,). Theorem (8.9.2) specializes to Let cpj (1 < j < m ) be m scalar functions, continuously diferentiable in an open subset A of R" (resp. C"); let $i (1 < i < p ) be p scalar functions, continuously diflerentiable in an open subset B of R" (resp. C") containing the image of A by (ql, .. ., cp;), then $ O,(x) = $i(cp,(x), ..., cp,(x)) for x E A and 1 < i < p , we have the relation

(8.10.1)

(Dk Oi) = (Dj II/i)(Dkq j ) between the jacobian matrices; in particular, when m = n = p , we have the relation det(D,Oi) = det(Dj$,) det(Dkcpj) between the jacobians.

a

We mention here the usual notations f;,(tl, . . . , t,,),-f ( t l , . . . , t,,),

at i

for Dif(tl, . . . , tn),which unfortunately lead to hopeless confusion when substitutions are made (what does f,'(y, x) or f J x , x ) mean?); the jacobian det(Djcpi(tl, . . . , t,)) is also written D(cpl, . . ., cp,)/D(t,, . . . , t,) or d(cp1, * * * cp")/a(tl, . . * , t"). 9

11. DERIVATIVE O F A N INTEGRAL DEPENDING O N A PARAMETER

(8.1 1.I)Let I = [cr, /3] c R be a compact interaal, E, F real Banach spaces, f a continuous mapping of I x A into F (A open subset of E). Theti g(z) = ff(t, z ) d( is continuous in A.

11 DERIVATIVE OF A N INTEGRAL DEPENDING ON A PARAMETER

177

Given E > 0 and zo E A, for any 5 E I, there is a neighborhood V(5) of 5 in I and a number r(5) > 0 such that for q E V(5) and ))z- zo))< r(5), 11 f ( q , z) -f(5, zo)II < E . Cover I with a finite number of neighborhoods V(ti), and let r = inf(r(5J). Then IIf ( 5 , z) -f ( 5 , zo)Il < E for llz - zoII < r and any 5 E I; hence, by (8.7.7) for llz - z0II < r. Q.E.D.

It&) - g(z0)II

< E(B - 4

(8.11.2) (Leibniz’s rule) With the same assumptions as in (8.11.1), suppose in addition that the partial derivative D2f with respect to the second variable exists and is continuous in I x A. Then g is continuously diflerentiable in A, and

(observe that both sides of that formula are in Y(E; F)). The same argument as in (8.51.I) applied to D,f, shows that given E > 0 and zo E A, there exists r > 0 such that IID, f ( 5 , z ) - D, f ( 5 , zo)I(< E for ))z- zo)I< r and any 5 E I; hence, by (8.6.2)

+ t ) -At, ZO) - D, f(5, zo) tll < E lltll for any 4 E I and any t such that lltll < r. By (8.7.7)we therefore have IIf(5,

20

*

Butby(8.7.6)and(5.7.4)wehaveJa’(D2f(t;,zo).t)d5 =(fD2f(5,zo)d5) . t for any t , and this ends the proof.

PROBLEMS

1. Let J C R be an open interval, E, F two Banach spaces, A an open subset of E, f a continuous mapping of J x A into F such that D,fexists and is continuous in J x A, 0: and fl two continuously differentiable mappings of A into J. Let

/*(,,”c, O(Z)

g(z) =

4d6.

178

Vlll

DIFFERENTIAL CALCULUS

Show that g is continuously differentiable in A, and that y’(z) is the linear mapping

2.

(apply (8.9.1) and (8.11.2)). Letf,g be two real valued regulated functions in a compact interval [a, b ] ,such thatfis decreasing in [a, b] and 0 < y ( t ) < 1. Show that

s,:

where A

=

s.”

g(1) dt.

1f

( t ) dt

< /abf(t)g(t)dt < JOa+h) dt

When is there equality? (Consider the integrals

Jba’+h”)f(t) dr, where h(y) =

1

g ( t ) df, as

s:

f ( t ) g ( t ) dt and

functions of y , and similarly for the other

inequality.) 3. Let the assumptions be the same as in Problem 1, except that CL and p are merely supposed to be continuous, but not necessarily differentiable, but in addition it is supposed that f(ar(z), z)= 0 and f(P(z), z ) = 0 for any z E A. Show that g(z) is continuously differentiable in A, and that g’(z)

L P(Z)

=

Dzf(E, z ) d[. (Use Bolzano’s theorem (3.19.8)

to prove that if 5 belongs to the interval of extremities p(zo) and p(z), there i3 z’ t A such that llz’ - zo/I< Ilz ZOII and [ = p(z’); if M is the 1.u.b. of llDzfll in a neighborhood of (/3(zo),zo), use the mean value theorem to show that Ilf([, z)ll < M Ilz - zoI/. 4. Let 1 [a, b ] ,A = [c, d ] be two compact intervals in R , f a mapping of I x A into a Banach space E, such that: ( I ) for every y E A, the function x + f ( x , y ) is regulated in I and for every x t 1, the function y -.f(x, y ) is regulated in A; ( 2 ) f is bounded in I x A ; (3) if D is the subset of I i( A consisting of the points (x, y ) where f is not continuous, then, for every xo E I (resp. every y o E A), the set of points y (resp. x) such that (xo,y ) E D (resp. (x,y o ) t D) is finite. ~

Jgb

(a) Show that the function g ( y ) = f ( t , y ) dt is continuous in A. (If

E

> 0 and

y o E A are given, show that there is a neighborhood V of y o in A and a finite number of intervals Jr c I (1 < k < n) such that the sum of the lengths of the Jr is < E and that, if

W =I -

u J, , f i s continuous in W x V; to prove that result, use the Borel-Lebesgue

k = l

theorem (3.17.6).) (b) Deduce from (a) that

/:lrv Jabf(x,Y ) dx = jabdxJcd f ( x , Y ) dy.

s: Jab

(Consider the two functions z + dy

f ( x , y ) dx and z -->

(Cf. Section 13.21). (c) Deduce from (b) that if a = c, b = d , then

(consider the function equal to f ( x , y ) for y

< x , to 0 for y > x).

y ) dy for z E A.)

12 HIGHER DERIVATIVES 5.

179

(a) Let f be a strictly increasing continuous function in an interval [0, a ] , such that

r’

f(0) = 0 ; let g be the inverse mapping, which is continuous and strictly increasing in the

.c,p <
O,

ysR;

for x>O,

y20, p>l,

q>l,

a > 0,

1

1

-+-=1, P Q

b > 0 and (pa)4(qb)p> 1.

12. HIGHER DERIVATIVES

Suppose f is a continuously differentiable mapping of an open subset A of a Banach space E into a Banach space F. Then Dfis a continuous mapping of A into the Banach space 9 ( E ; F). If that mapping is differentiable at a point xo E A (resp. in A), we say thatfis tiisice drfSerentiable at x, (resp. in A), and the derivative of Df at xo is called the second deriuatiue of f a t x,,, and writtenf”(x,) or D2f(xo). This is an element of 2(E; 9 ( E ; F)); but we have seen (5.7.8) that this last space is naturally identified with the space 9 ( E , E; F) (written Y2(E; F)) of continuous bilinear mappings of E x E into F: we recall that this is done by identifying u E 2 ( E ; 9 ( E ; F)) to the bilinear mapping (s, t ) ( u * s) t ; this last element will also be written u . (s, t ) . --f

(8.12.1) Suppose f is tii’ice diferentiuble at x,; then, for any jixed t E E, the derioative of the mapping x-)Df(x) . t of A into F, at the point x o , is s + D2f(x0) . (s, t ) .

If we observe that x -+ Df(x) . t is composed of the linear mapping u + u . t of 9(E; F) into F and of the mapping x Df(x) of E into Y (E; F) the result follows from (8.2.1) and (8.1.3). -)

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180

DIFFERENTIAL CALCULUS

(8.12.2) If f is t,cice diferentiable at xo, then the bilinear mapping ( s , t ) --t D2f(xo) . (s, t ) is symmetric, in other rvords D2f(xO) . (s, t )

= D2f(xO)

Consider the function of the real variable

. ( t , $1.

5 in the interval [0, 11:

g(5) =f ( x o + 5s + 0 - f ( x o + 5s)

where s, t are such that llsll < f r , lltll < f r , the ball of center xo and radius r being contained in A. From (8.6.2) we get

and

II fYx0 + 5s) - f’(x0) - f”(xo> . (5s)/I

E

IIs I1

hence

lls’(5) - ( f ” ( X 0 ) . t >

*

sII G 2E llsll . (llsll

+ lltll)

and therefore

Ildl)- d o )

- (f”(X0) . t ) . sII

+ +

+

< 6.2 llsll (llsll + Iltll).

t ) -f(xo But g(1) - g(0) = f ( x o s t ) - f ( x o in s and 1, hence, by exchanging s and t , we get li(f”(xo)

+ s) +f(xo) is symmetric

< 6~(ilsll+ 11t11)’. < +r‘; but if we replace s and t

. t > . s - (f”(xO>+ s> . tll

Now this inequality holds for ~~s~~ < fr’, lltll by Is and I t , both sides are defined and multiplied by 1/212, hence the result is true for all s and t in E, in particular for llsll = /It11 = 1, which proves (by (5.7.7))that

llf”(xo> . ( 1 , s) -f”(x,> . (s, t>ll < 248 114 . lltll for all s and 1 ; as E is arbitrary, this ends the proof. In particular,

12 HIGHER DERIVATIVES

181

(8.12.3) Let A be an open set in R” (resp. C“); f a mapping f of A into a

Banach space F is tccdce differentiable at xo , then the partial derivatives D if are differentiable at xo, and Di Dj f ( x 0 ) = Dj Di f ( x 0 ) for 1 < i d n , 1 0 such that 11 [x . y]ii d c llxll . lly//in E x F; by definition ofthe norm in Y ( E x F; G) (5.7.1), we have llg(x-3Y)I/ Q C ( l I ~ l l + IlJ~ll)6 2c SUP(ll.~lI> IIJ~II)

hence g is a continuous linear mapping of E x F into 9 ( E x F; G), and therefore (x, y ) + [x . J!] is twice differentiable and its second derivative at (x, y ) is (by (8.1.3) and (8.12.1)) (($1,

tl)?(s2 t2)) - + b l 3

. f 2 1 + i s 2 . tll.

This is a mapping independent of ( x ,y ) , hence the result. (8.12.10) Let E, F, G be three Baiiacli spaces, A an open subset of E, B an open subset of F; iff is a p times continuously diflerentiable inupping of A into B, g a p times continuously clifferetitiable mapping of B into G , tlien 11 = g o f is a p times continuously chflc~rentiahleniappitig of A into G. For p = 1, the result follows from (8.2.1) and from the fact that u is a bilinear continuous mapping of 2 ( E ; F) x 9 ( F ; G) into (u, 11) + I ’ 9 ( E ; C) by (5.7.5). Use now induction on p ; as li’(x) = g ’ ( f ( x ) ).J”(x), and ,f and g’ are p - I times continuously differentiable, the induction hypothesis shows that g’ c , f ’ is p - 1 times continuously differentiable; from (8.12.6) and (8.12.9), it then follows that 11’ is p - 1 times continuously differentiable, hence Ii is p times continuously differentiable by (8.12.5).

184

Vlll

DIFFERENTIAL CALCULUS

Example

Suppose there is a linear homeomorphism of a Banach space E into a Banach space F, and let &? c 9 ( E ; F) be the open set of these homeomorphisms i n Y ( E ; F) (8.3.2). Then the mapping u .+ u-' of &? onto 2- is indejitiitely differentiable. We prove by induction on p that u -+ u-' is p times differentiable, the property being true for p = 1 by (8.3.2). Given v and w in 9(F; E) = M, let f ( v , w) be the linear mapping t - u t IO of L = 9 ( E ; F) into M ; it is clear that f is bilinear (and maps M x M into 9 ( L ; M)) and (5.7.5) proves that Ilf(v, w)II 6 \lull * I\wII, hence f is continuous, and therefore indefinitely differentiable by (8.12.9). Now the first derivative of u -+ u-' is, by (8.3.2), the mapping u - + f ( u - l , u - ' ) ; by (8.12.6) and (8.12.10) it follows that if u-+u-' is p times differentiable, so is u -+f(u-', u - ' ) , and therefore, by (8.12.5), u -+ u-' is p 1 times differentiable. (8.12.11)

-+

+

Remark. When f is a mapping of an interval J c R into a real Banach space F, we have defined earlier (Section 8.4) the notion of derivative off at toE J with respect to J. By induction on p , we define the pth derivative o f f at to,with respect to J, as the derivative at to (with respect to J) of the ( p - 1)th derivative o f f (which is therefore supposed to exist in a neighborhood of to in J); it is an element of F, written again Dpf(to) orf(p)( a n , f ( " ) ( f= ) 0 has at least n - 1 distinct roots in 1- 1 , 1[. (Show by induction on k that there is a strictly increasing sequence xk, < xk. < . . . < x k , of points of 1- I ,I[ such that f(')(xr, i ) f ( k ) ( ~ k , +,) < 0 for I < i < k - 1; use Rolle's theorem.)

,

,

186

4.

Vlll

DIFFERENTIAL CALCULUS

Let E, F be two Banach spaces, A an open subset of E, f a n n times differentiable mapping of A into F. Let xo E A, hi E E ( I < i < n ) be such that xo f o r O < ( , < l , l < i < n . Wedefinebyinductiononk(1 < k < n )

E

A

+ hi) - f ( ~ o )

A'f(xo; h l ) =f(xo

Akf(x,; h i , . . . ,h,)

"

+ tihr

= A"'gr(xo

; h i , ...,h k - i )

with SkW =f ( x

+

-Ax).

hk)

(a) Show that

IlA"f(xo;hi, ..., h " ) l l ~llhill' I l h 2 I l ~ ~ ~ I IlD"f(z)/I lh~Il~~~ Z t P

where P is the set of points x o (b) Deduce from (a) that Ilhlf(xo ; hi,

< llhi I/

"

+ C tih i ,0 < ti< I . (Use induction on n.) i=i

...,h.1-

.

D"f(xo) (hi, . . ,hnN

llhz /I' ' . llhnll SUP IID"f(z)

- D"f(xo)ll.

Z E P

5.

Let f be a continuously differentiable mapping of an open subset A of R2 into a Banach space E. Suppose that in a neighborhood V of (a, h ) E A, the derivative D2(Dlf) exists and is continuous. (a) Let (x,y ) E V; show that for every E > 0, there exists 6 > 0 such that the relations /hi < 6 , lkl < 6 imply lIA2f(x, Y ; h, k ) - Dz Dif(x,

~ W l! P. and the conclusion follows from (5.5.7). 0

I/
0, there exists an indefinitely differentiable mapping f of R P into E such that D"f(0) = ca for every a. (c) Deduce from (a) that if g is an indefinitely differentiable mapping of a closed

14 TAYLOR'S FORMULA

193

interval I c R into E, and J an open interval containing I, there exists an indefinitely differentiable mappingfof R into E which coincides with g in I and with 0 in R - J. 5. Let f b e a mapping of an interval 1 c R into a Banach space E, and supposefis n times differentiable at a point a E I. Show that

(use induction on n and (8.5.1) with v([) = ([ a)"-'). 6 . Let I c R be an interval containing 0, f an n - 1 times differentiable mapping of I into a Banach space E. Write -

which definesf, in I - {O}. (a) Show that iffis n p times differentiable at t = 0, f.can becontinuouslyextended to I and becomes a function which is n p - 1 times differentiable at all points t # 0 in a neighborhood V of 0 in I, and p times differentiable at t = 0; furthermore f:(O) = k! f'"''(0) for 0 < k < p , and lim fAP+'"'(t)tk = 0 f or 1 G k ~ n - l . r-0, t t o . r E v (n k ) ! (Express the derivatives off. with the help of the Taylor developments (Problem 5 ) of the derivatives off, and use Problem 2 of Section 8.6.) (b) Conversely, let g be an n p - 1 times differentiable mapping of I - {0} into E, such that lim g ' p f k ' ( f ) fexists k for 0 < k < n - 1. Show that the function g can

+

+

+

+

r-+O.tto,re~

be extended to a p - I times differentiable mapping of I into E, and that the function g(t)t" is n p - I times differentiable in 1; if furthermore g'p)(0)exists, then g(t)t" is n + p times differentiable at 0. (c) Suppose I = 1- 1, I[, and supposefis euen in I , i.e.f(-t) = f ( t ) . Show, using (a) and (b), that i f f is 2n times differentiable in I, there exists an n times differentiable mapping h of I into E such that f ( t ) = h(r2). 7. (a) Let f b e an indefinitely differentiable mapping of R" into a Banach space E. Show that

+

. .,X") =fa. . . , 0) + xlfI(x1, .. . ,X") + x2 f2(x* ...,x.) + . . . + x. f.(xJ where fk is indefinitely differentiable in R"-k+l (1 < k < n). Write f ( x l , .. . , x,) = (f(xl,.. . , x.) - f ( O , x 2 ,. . . , x.)) +f(O, x 2 ,. . ., x.) and apply (8.14.2) to the first f(x1,.

1

summand, considered as a function of x l ;with a suitable value ofp (depending on k ) , this will prove that ( f ( x l , ..., x,)-f(O, x 2 , . .., x,))/xl is k times differentiable at (0,. . . , O ) ; finally, use induction on n.) (b) Deduce from (a) that for any p > 0,

where all the fa are indefinitely differentiable and f o ( x )= f ( O , . . . , 0). (c) Let f be an indefinitely differentiable mapping from R" into R; suppose that f(0) = 0, Di f ( 0 )= 0 for 1 < i < n, and that the quadratic form ([I,

. . .*t n ) E Dt Dj f(0)ti i.j +

t j

194

Vlll

DIFFERENTIAL CALCULUS

is nondegenerate. Show that, by using a linear transformation in R" one can assume that DlDjf(0) = 0 for i # j and a1 = D:f(O) # 0 for 1 < i < n. Prove that there exists a neighborhood U of 0 and n indefinitely differentiable functions g , defined in U, such that in U

and gi(0) = 1 for 1 < i < n. (Use again (a) applied to each of the functions fl defined in (a), and then apply the usual method of reduction of a quadratic form to a form having a diagonal matrix.) 8. (a) Let S be a metric space, A , B two nonempty subsets of S, M a vector subspace of the space V,(S) of real continuous functions in S, N a vector subspace of M, u +L(u) a linear mapping of M into the space RAof all mappings of A into R.We suppose that: ( I ) there exists a function u0 E N such that L(uo)is the constant 1 on A ; ( 2 )if u E Nand there is a t E B such that u(t) = 0, then there is x E A such that (L(u))(x)= 0. Let u E M such that L(u) = 0; show that for any function u E M such that u - u E N, and any t E B, there exists 6 E A (depending on t) such that u(r) = u ( t ) uo(f)(L(u))(@ (Observe that u o ( t ) # 0, and therefore there is a constant c (depending on t) such that u ( t ) - v ( t ) - cuo(t) = 0). (b) Suppose S is compact, A is connected and dense in S, and all functions u E N vanish on S - B. Suppose that L(u) is continuous in A for every u E M, and that if a function u E N is such that (L(u))(t)> 0 for any t E A, then u has no strict maximum on B. Show that in such a case condition ( 2 ) of (a) is also verified. 9. (a) Let f be an n times differentiable real function defined in an interval 1; let XI < x 2 < ... < xp be points of I, n i (1 < i < p ) integers > O such that nl n2 . . . np = n. Suppose that at each of the points x l , fCX'(x,)= 0 for 0 < k < nl - 1. Show that there is a point 4 in the interval ] x l , x p [ such that f("-')([) = 0 (apply Rolle's theorem iteratively). (b) Let g be an n times differentiable real function defined in I, and let P be the real ) P(k)(x,)for 0 < k < n, - 1 , 1 < i < p. polynomial of degree n - 1 such that g ( k ) ( x i= Show that for any x E I, there exists 6 in the interior of the smallest interval containing x and the xi (1 < i < p), such that

+

+ + +

(Use Problem 8(a), or give a direct proof, using (a) in both cases.) 10. Let g be a real odd function, defined and 5 times differentiable in a symmetric neighborhood I of 0 in R. Show that, for each x E I s(x)

X =3 (g'(x)

+ 2g'(O)) - 180 g y . 9 XS

where [ is a number belonging to the open interval of extremities 0 and x . Deduce from that result that, iffis a real function, defined and 5 times differentiable in [a, b ] , then

with a < 4 < b (" Simpson's formula").

14 TAYLOR'S FORMULA

195

11. Let I = [a, b] be a compact interval, and let Mo be the vector space of real continuous functions defined in I and such that, for any t E ]a,6[, the limit

.

exists in R.All real functions which are twice differentiable in I belong to Mo (a) Let M be the vector subspace of Mo consisting of functions f for which L (f)is continuous in ]a, b[. Show that any function of f~ M is twice differentiable in ]a, b[ and that L(f)=f". (Use Problem 8(a) and 8(b), taking S = I, A = B = ]a,b [ , and for N the subspace of M consisting of functions f for which f ( a ) = f ( b ) = 0.) (b) Show that the function f ( t ) = t cos(l/t) belongs to M o , although it is not differentiable at t = 0. 12. What are the properties of functions with values in a Hilbert space which correspond to the properties of real functions discussed in Problems 9(b), 10, and 1 1 ? (Cf. Section 8.5, Problem 6.) 13. Let f be an indefinitely differentiable mapping of a compact interval I = [a,b] c R (with a 3 0) into a Banach space. (a) Show that, for any two integers p , 4 such that 0 < p < q,

(For p < n < q, express f ( " ) ( a )using Taylor's formula at the point b.) (b) Under the same conditions, show that

(apply Taylor's formula tof'p) and use Problem 4(c) of Section 8.11). (c) Suppose f is indefinitely differentiable in the interval [a, +a[where a 3 0, and that for every integer n 2 0, there is a finite number M. such that Ilf'"'(x)Ilxn/n! < M. for every x 3 a . Show that for 6 > a and n < q, f ( " ) ( b )= -

s,'

(b - x ) l I - n - l f('J)(x)

(4-n-

I)!

dx

and for a < x < b

where the series and the integral are convergent (use Taylor's formula). Conclude that one has

and

(d) Show that, under the assumptions of (c), the function

(whose values are finite and 30,or f a ) is increasing in [a, +m[.

196

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DIFFERENTIAL CALCULUS

(e) Supposefis indefinitely differentiable in [a, +co[ and that lim f(")(x) = 0 x++m

for every n > 0.

Prove that the sequence of numbers (finite and 2 0 or equal to +a)

J.

=jotIlf(")(x)II

x"- 1

dx

(n > 1)

is increasing. (When these numbers are all finite, write f(")(x) =

-jx+mf("+l)(t) dt

and use Problem 4(c) of Section 8.1 1.) (f) Suppose f is indefinitely differentiable in [a, +to [ and that the integrals

are finite. Show that, for x 3 a,

14. Let f b e a n indefinitely differentiable mapping of the interval [a, +a[, where a 3 0,

into a Banach space. (a) Show that, for every p > 0,

where both sides may be equal to equal to both limits (w)

lim

+ co. Show that if both sides are finite, they are also

C llf(")(x)ll 2 m

x++m n = p

X'

and

lim n-rm

x"jCm llf(")(x)ll -dX (n - l)! 1

(I

(Use Problem 13). (b) I f f satisfies the assumptions of Problem 13(c), then the limits (**) always exist and are equal to both sides of (*).

CHAPTER I X

ANALYTIC FUNCTIONS

In this chapter, we have tried to emphasize the most general facts pertaining to the theory of analytic functions, and in particular to state as many results as possible for analytic functions of any number of variables; until Section 9.13, the theorems which concern only functions of one variable are inserted in a context in which they appear as technical intermediates to the general statements; it is only in Sections 9.14 to 9.17, and in many problems in this chapter and the next one, that we really deal with properties special to the one variable case. Furthermore, we have discussed simultaneously the case of analytic functions of real variables and of analytic functions of complex variables as long as it can be done (i.e. until Section 9.5). Finally, we have kept throughout our general principle of dealing from the start with vector valued functions; as usual, this does not require any change in the proofs, and the reader will see in Chapter XI how useful the consideration of such functions can be. Of course, one can only expect to find here the most elementary part of the very extensive theory of analytic functions. The definition is given by the local existence of power series representing the function, and it is by the technique of power series that the differential properties of analytic functions are obtained (9.3.5) (the usual definition of analytic functions by the existence of continuous derivatives only applies, of course, to functions of complex variables, and therefore that characterization is postponed until Section 9.1 0). The fundamental results about power series are Abel’s lemma (9.1.2) -from which is derived the vital possibility of substituting power series into power series (9.2.2)-and the principle of isolated zeros (9.1.5), whose most important consequence is the principle of analytic continuation (9.4.2), which expresses the “solidarity” between the values of an analytic function at different points of the domain where it is defined. From that point on, we have to assume that the variables are complex; 197

198

IX ANALYTIC FUNCTIONS

with the exception of the principle of maximum (9.5.9), all additional properties of analytic functions of complex variables derive from a single new idea, that of “ complex integration,” and from its fundamental features, Cauchy’s theorem (9.6.3), Cauchy’s formula (9.9.1), and its generalization, the theorem of residues (9.16.1).The form of Cauchy’s theorem which we give here is not the best possible, for it expresses the integral along a circuit as an invariant of the homotopy class of that circuit, whereas in fact it is an invariant of its homology class. In most applications, however, this has no inconvenience whatsoever, and in contrast to the fact that the proof of the weak form of Cauchy’s theorem needs almost no topological preparation, the proof of the complete theorem would have required some developments of algebraic topology, which we feel are above the level of the present volume. The interested reader will find the complete Cauchy theorem, together with all the necessary prerequisites, in Ahlfors [l], Cartan [8], and Springer [17]; we shall come back to that question in Chapter XXLV. Instead of using more results from algebraic topology in order to obtain such refinements, we have thought it might interest some readers to see how, by the very simple device introduced by S. Eilenberg, it is possible to obtain quite deep information on the topology of the real plane (including the Jordan curve theorem), using merely the most elementary facts about complex integration; this is the purpose of the Appendix (which, by the way, is not used anywhere in the later chapters and may therefore be bypassed without any inconvenience). As we have announced in Chapter I , the reader will find no mention in this chapter of the so-called “ multiple-valued ” or multiform ” functions. It is of course a great nuisance that one cannot define in the field C a genuine continuous function Jz which would satisfy the relation (Jz)’ = z ; but the solution to this difficulty is certainly not to be sought in a deliberate perversion of the general concept of mapping, by which one suddenly decrees that there is after all such a “function,” with, however, the uncommon feature that for each z # 0 it has titto distinct “values.” The penalty for this indecent and silly behavior is immediate: it is impossible to perform even the simplest algebraic operations with any reasonable confidence ; for instance, the relation 2Jz = Jz Jz is certainly riot true, for if we follow the “definition” of J z , we are compelled to attribute for z # 0, ticso distinct values to the left-hand side, and three distinct values to the right-hand side! Fortunately, there is a solution to the difficulty, which has nothing to d o with such nonsense; it was discovered more than 100 years ago by Riemann, and consists in restoring the uniqueness of the value of \/. by “doubling,” so to speak, the domain of the variable z, so that the two values of 2 / z corre“

+

1

POWER SERIES

199

spond to tit'o different points instead of a single z ; a stroke of genius if ever there was one, and which is at the origin of the great theory of Riemann surfaces, and of their modern generalizations, the complex manifolds which we shall define in Chapter XVI. The student who wishes to get acquainted with these beautiful and active theories should read H. Weyl's classic [19] and the modern presentation by Springer [17] of Riemann surfaces, and H. Cartan's seminar [7] and the recent book of A. Weil [IS] on complex manifolds.

1. POWER SERIES

In what follows K will denote either the real field R or the complex field C ; its elements will be called scalars. In the vector space K" over K, an open (resp. closed) polydisk is a product of p open (resp. closed) balls; in other words it is a set P defined by conditions of the form Izi- ail < ri (resp. Izi - ail < ri), 1 < i < p , on the point z = (zl, . . . , z,), with ri > 0 for every index; a = ( a l , . . . , a,) is the center or P, r l , . . . , r, its radii (a ball is thus a polydisk having all its radii equal). (9.1.1) Let P, Q be trr'o open polydisks in KP siich that P n Q # @; f o r any tn'o points x, y in P n Q , the segment (Section 8.5) joining x and y is contained in P n Q ; in particular P n Q is connected.

+

Indeed, if Ixi - ail < r i , lyi - ail < r i , then ltxi (1 - t ) y i - ail < tlxi - ail (1 - t)lyi - ail < r for 0 < t < 1 ; the last statement follows from the fact that a segment is connected (by (3.19.1) and (3.19.7)) and from (3.19.3).

+

,

We introduce the following notation: for any element v = ( n l , . . . , J?,) in NP ( n , integers 3 0 ) and any vector z = (zl,. . . , z,) E ISP, we write * + n p . If E is a Banach space zv = z1'z;'. . . z i p and IvI = n, + n2 + (over K), ( c , ) , , ~ ~a family of elements of E having NP as set of indices, we say that the family(c,z"), E N P of elements of E is a power series in p variables z i (1 < i < p ) , icsith coeflcients c, . E K P be such that b , # 0 f o r 1 < i Q p , and that the family ( c v b v ) be bounded in E. Then f o r any system of radii (ri) such that 0 < r < lbil f o r I < i < p , the portxer series (c,zv) is normally summable (7.1) in the closedpolydisk ofcenter 0 and radii ri ("Abel's lemma").

(9.1.2) Let b = ( b L ,. . . , 6,)

200

IX ANALYTIC FUNCTIONS

For if (/c,b’l( < A for any v EN!’, it follows from the definition of the norm in KP that if (zil < ri < [hi( (1 < i < p ) , we have ( ( c v z v< ( JAq’, with q = ( q l , , . . , q,), qi = ri/Jbil< 1. It follows from (5.5.3) that the family ( q v ) v p N of p positive numbers is absolutely summable, hence the result by (5.3.1). (9.1.3) Uiider the assumptioris of (9.1.2), the sum of the p o u w series (c, z‘) is continuous in the opeii polydisk of center 0 and radii Ibil.

As every point of that polydisk is interior to a closed polydisk of radii ri < / b i / ,the result follows from (7.2.1). Let q be any integer such that 1 < q < p ; for any v = ( n l , . . . , n,), write v‘ = (ti1, . . . , n,), V “ = (nq+,,. . . , n,); consider KP as identified to the product K 4 x KP-,, and for z = (zl . . . , z,) E KP, write z’ =.(z,, . . . , z,), Z” = (z,,,, . . . , z,). With these notations: (9.1 -4) Suppose the pouter series (c,z’) is absolutely summable in the polydisk P of radii ri aiid center 0 iii KP. Then, f o r any v” E NP-, the series (c(,.,, , ~ ~ ~ z ’ ’ ’ ) is absohrtely suinmable in the polydisk P’, projection of P on K4;let g J z ’ ) be its sum. Then, f o r any z’ E P’, the power series (g,.,(z‘)z“’”) is absolutely summable in the polydisk P”, projection of P 011 Kp-q, and its sum is equal to the sum of the series (c’z’). As z’ = Z”’Z’””, the fact that each of the series (q,,,,, , ~ ~ ~ Z ‘ “ Z ” ~ ” (v” ) fixed) is absolutely summable, and that gV,,(z‘)z”‘”= cvzv,follows from (5.3.5)

c

c

V”

V

and from the associativity theorem (5.3.6) for absolutely summable families. If we take Z” E P” such that zi # 0 for q 1 < i < p , the absolute summability of (C(,., ‘“)Z”’) follows.

+

(“ Principle of isolated zeros ”) Suppose (cnzn) is a power series (9.1.5) in one variable which converges in an open ball P of radius r, aiid

let f ( z )

=

2 c,zn.

Then, unless all the c, are 0 , there is r‘ < r such

n=O

that f o r 0 < ( z ( < r ’ , f ( z ) # 0.

Suppose h is the smallest integer such that cl1# 0; then we can write f( z ) = zh (c,, + c,,+,z + * * . + e l l + m ~+m* . * ) and the series (ch+,,,zm)converges in P; i f g ( z ) = c , + c , + , z + ~ * ~ + c , , + , , , z m +. . . , g iscontinuous in P by (9.1.3) and as g(0) = c,, # 0, there is r‘ > 0 such rhat g(z) # 0 for J z J< r’; hence the result.

2 SUBSTITUTION OF POWER SERIES IN A POWER SERIES

201

(9.1.6) Suppose two p o s e r series (a,zv) and (b,z") are absolutely summable and have the same sum in a polydisk P; then a, = b, f o r every v E NP. Use induction on p ; for p = 1, the result follows at once from (9.1.5). Taking the difference of the two power series, we can assume b, = 0 for m

I

every v ; applying (9.1.4) with q = p - 1, we have n=O

gn(z')z; = 0, hence

gn(z') = 0 for every n and every z' in the projection P' of P on KP-'; the induction hypothesis applied to each gn yields then a, = 0 for every v.

PROBLEMS

..

1. Let (cy z") be a power series in p variables zl (1 < i < p ) ; let a = (al, . ,a,) E Kp. In order that a real number Y > 0 be such that, for any r E K such that If) < Y, the series (c,(fa,)"' . . . (fa,)",) be absolutely summable, it is necessary and sufficient that

for all but a finite number of indices v = (nl, . . . , n,) (apply (9.1.2).) In particular, for p = I , there is a largest number R 2 0 (the "convergence radius," which may be .c) such that the series (c.z") is convergent for 121 < R, and that number is given by I /R = lim (s~p(IIc,+~/l'/("+~))), which is also written lim . supllc.ll'~"

+

n-m

k30

"-+W

(cf. Section 12.7). When in particular lim l/c,l/'/" exists, it is equal to l/R. "-m

2. Give examples of power series in one complex variable, having a radius of convergence R = 1 (Problem 1) and such that: ( I ) the series is normally convergent for 121 = R ; ( 2 ) the series is convergent for some z such that IzI = R , but not for other points of that circle; (3) the series is not convergent at any point of Iz/ = R. 3. Give an example of a power series in two variables, which is absolutely summable at

two points ( a l , a2), (bl, b2), but not at the point

"1.

bl, 2 +

(Replace z by

in a power series in one variable.) Let (c,,~"), ( d , , ~ "be ) two power series in one variable with scalar coefficients; if their radii of convergence (Problem 1) are R and R', and neither R nor R' is 0, then the radius of convergence R" of the power series (c,d,z") is at least RR' (takenequal to cu if R or R' is co).Give an example in which R" > RR'.

z1z2

4.

+

+

2. SUBSTITUTION O F P O W E R SERIES I N A P O W E R SERIES

Let Q be a polydisk of center 0 in Kq, and suppose the p power series in q variables ( b r ' u p ) with scalar coefficients are absolutely summable in Q (with p = ( m l , . . . , mq), u = (ul,. . . , uq), up = u y l . . . u,"~). We write

202

IX ANALYTIC FUNCTIONS

gk(u)=

bF’u”, Gk(u)=

c IbF)

[ u p .On the other hand, let (a,zv) be

a power

B

P

series in p variables with coefficients in E, which is absolutely summable in a polydisk P of KP, of center 0 and radii rk (1 < k < p ) . If, in a monomial zv = z;’ . . . z i p , we replace “formally” each z k by the power series gk(u), we are led to take the formal “product” of n, + n2 + + nP series, i.e. to pick a term in each of the n, + ... + nP factors, to take their product and then to “ s u m ” all terms thus obtained. We are thus led to consider, for each v = (nl, n 2 , . . . , nP) the set A, of all finite families ( p k j ) = p where p k j E Nq, k ranges from 1 to p , and for each k , .j ranges from 1 to n,; t o such a p we associate the element tP(U)

= a,

nn P

nk

bfpkJ.

k = l j=1

With these notations:

Suppose sl, . . . , sq are q numbers > O satisfying the conditions Gk(sl,. . . , sq) < rk for 1 < k < p . Then, f o r each u in the open polydisk S c Kq of center 0 and radii si (1 < i < q), the family (t,(u)) (where p ranges through the denumerable set of indices A = A,) is absolutely summable,

(9.2.1)

u

and i f f ( z ) =

VENP

1avzv,its sum is equal t o f ( g , ( u ) ,g2(u),. . . ,g,(u)). V

In other words, under the conditions G,(s,, . . . , sq) < rk (1 6 k < p ) , “substitution” of the series gk(u) for zk ( 1 < k < p ) in the series f yields an absolutely summable family, even before all the terms t,(u) having the same degrees in u l , . . . , uq have been gathered together. To prove (9.2.1), we need only prove that the family (t,,(u)) is absolutely summable; that its sum is f ( g , ( u ) , . . . , g,(u)) follows by application of the associativity theorem (5.3.6) to the subsets A, of A, and by using (5.5.3), which shows that t,(u) is equal to a,(gl(u))”* . . . (gP(u))“P.To prove the PEA,

family (t,(u)) ( p E A) is absolutely summable, we apply (5.3.4). For any finite subset B of A, we have, by (5.3.5) and (5.5.3)

and by assumption, the right-hand side of that inequality is the element of index v of an absolutely summable family; hence the result. Write t,(u) p k j = (mkj,,.

= cPul,

with ,A

= (,Al,

. . , mkjq)). From (9.2.1)

. , . , ,Aq), Izi

1 c mkji (if P

=

k=l

nk

j=1

we have

and (5.3.5) it follows (taking all the ui

3 ANALYTIC FUNCTIONS 203

to be ZO,u E S), that for each A, the family of the c p , where p ranges over all elements of A which correspond to the same 2, is absolutely summable in E; if d , is its sum, we see, by the associativity theorem (5.3.6),that

(9.2.1.1)

f(g,(u), ' .

. ?

g,(u)) =

1d i ui i

the series on the right-hand side being absolutely summable in the polydisk S. By definition, that power series is the ponier series obtained by substituting g k ( U ) to z k ,for 1 < k < p , in the power series ( a , 2').

(9.2.2) I f the point (gl(0),. . . ,gp(0))of KP belongs fo P, then there exists in Kq an open polydisk S such that, for u E S, the series gk(U) may be substituted to zk (1 < k < p ) in the poii'er series (a, z"). Observe that by definition, Gk(0)= Igk(0)l for 1 < k 0 (1 < i < q ) such that G,(s,, . . . , sq) < rk for 1 < k < p follows at once from the assumption.

3. A N A L Y T I C F U N C T I O N S

Let D be an open subset of KP. We say that a mapping f of D into a Banach space E over K is analytic if, for every point a E D, there is an open polydisk P c D of center a, such that in P, f ( z ) is equal to the sum of an absolutely summable power series in the p variables zk - a, (1 < k < p ) (that series being necessarily unique by (9.1.6)). Suppose K = C, let b be a point of D, and let B be the inverse image of D by the mapping x + b + x of RPinto Cp. Then it follows at once from the definitions that x +f ( b + x) is analytic in the open subset B of RP.

(9.3.1) Let (a,z") be an absolutely summable powjer series in an open polydisk P c KP. Then f ( z ) = a,zv is analytic in P; more precisely, if ri (1 < i d p ) V

are the radii of P, for any point h = (bi)E P, f ( z ) is equal to the sum of an absolutely summable poiver series in the zk - h k in the open polydisk of center b and of radii ri - lbil ( I < i < p ) . This follows at once from (9.2.1)applied to the case q = p , gk(u)= b, + z i k ; we have then G,.(u)= Ibkl + u k , and the conditions Gk(s,,. . . , sp) < rk (1 < k < p ) boil down to s, < rk - lbkl (1 d k O; suppose lim a,/b, = s. n- m

(a) Suppose the series (b.2") is convergent for / z /< 1, but not for z = 1 (which means k

that if ck = C b,, lim ck = n=o

k-m

+ m). Show that the series (a.z") is absolutely convergent

for Iz/ < I , and that, if I = [0, I[,

(Observe that, for any given k ,

lim 2-1,

IEI

( cb,z")

=

+a).

nak

(b) Suppose the series (b,z") is convergent for every z. Show that the series (ant")is absolutely convergent for every z, and that if J is the interval [0, +a[in R, then lim ( F a n z n ) / ( g $ z n ) = s . z + + m . z ~ J n=O

(Same method.)

c c m

(c) Show that if the series (a,) is convergent and

a,

= s,

then the series (a.z") is

"=O

absolutely convergent for /zI < 1, and that

lim 2-1.

*El

m

a,, z" = s. (Apply (a) with b, = 1

"=a

for every n; this is "Abel's theorem ".) (d) The power series ((- 1)"~") has a radius of convergence 1, and its sum l / ( l z ) tends to a limit when z tends to 1 in I, but the series ((-1)") is not convergent (see Problem 2). 2. Let (a.2") be a power series in one variable having a radius of convergence equal to 1 ; letf(z) be its sum, and suppose thatf(1-) exists. If in addition lim nu. = 0, show that

+

n-m

the series (a,)is convergent and has a sum equal to f(1->. ("Tauber's theorem": observe that if 1na.l < E for n 1 k, then, for any N > k , and 0 < x < 1

and

3. Let (a,z")be a power series in one variable having a radius of convergence r > 0, and I) and 191 < r. let (b,) be a sequence of scalars # O such that q = lim ( b n / b n + exists n-ra

Show that, if C.

= sob.

+

lim (c,/b.) exists and is equal to f ( q ) . n- m

bn-i

+ ... + a.60

4 THE PRINCIPLE OF ANALYTIC CONTINUATION

207

4. Let (pnz"),(q.z") be two power series with complex coefficients, and radius of convergence #O, and let f(z) = C p n z " ,g(z) qnznin a neighborhood U of 0 where both

=c "

0

series are absolutely convergent. Suppose qo = g(0) # 0 ;then there is a power series c,z" which is absolutely convergent in a neighborhood V c U of 0 and has a sum equal to f(z)/g(z) in V (remark that the series (z") is convergent for IzI < 1 , and use (9.2.2)). If all the qn are >O, the sequence (q.+&.) is increasing, the pn are real and such that the sequence (p,/qn) is decreasing (resp. decreasing), show that c, 2 0 (resp. c, < 0) for every n >, 1 . (Write the difference

P"

Pn- 1

4.

4"-1

as an expression in the qk and c k , and use induction on n.) Deduce from that result that all the derivatives of x/log(l - x ) are 1x1, we have

hence, for any x E R

and by (9.1.2) the series is normally convergent in any compact interval. Using the remark which follows (9.4.5), we can define in C an entire function ez (also written exp z) as equal to the sum of the power series (z"/n!). We have

(9.5.2)

ez+z'

- eZ& -

for both sides are entire functions in C2 which coincide in R2, and we apply

(9.4.4). For real x , e-'X is the complex conjugate of e'", since (-ix)" is the complex conjugate of (ix)";from (9.5.2)it follows that (eiXI= 1. We define cos x = B(e'"), sin x = 9(eiX) for real x; they are entire functions of the real variable x by (9.3.3), and the relation leixi = 1 is equivalent to cos2x + sin'x = 1, and implies lcos XI < 1 and Isin X I < 1 for any real x . Moreover, we have D(e') = ez

(9.5.3)

since both sides are entire functions (by (9.3.5) in C,which coincide in R. In particular (see Remark following (8.4.1)), D(eix)= ie'" for real x , hence

(9.5.4)

D(cos x)

=

-sin x ,

D(sin x ) = cos x.

The definitions of cos x and sin x for real x can also be written cos x = f(eix+ e -ix), sin x = (e'" - e-'")/2i; these formulas may be used to define cosz and sinz for complex z , replacing x by z in the right-hand sides. With these definitions, formulas (9.5.4)are still valid for complex values of x .

(9.5.5) There is a number II > 0 such that the solutions of the equation ez = 1 are the numbers 2nni (n positive or negative integer).

212

z

IX ANALYTIC FUNCTIONS

If z = x + iy, we have JezI = ex)eiyl= ex, hence ez = 1 implies x We first prove:

= 0,

= iy.

(9.5.5.1)

The set of points x 2 0 such that cos x

=0

is not empty.

One has

(4k

+ 3)(4k + 4)

and obviously the convergent series on the right hand side has all its terms 2 0; therefore

in other words, cos 2 < 0; therefore (3.19.8) the continuous function cos x takes the value 0 in the interval 10, 2[. As cos x is continuous, the set D of the roots of cos x = 0 such that x 2 0 is closed (3.1 5.1) and does not contain 0, hence has a smallest element which we denote by n/2. Then we have sin' 7112 = 1, and as sin x is increasing for 0 < x < 4 2 , sin 7112 = 1, einiz= i. This already shows that e21ri= 1, hence eZnlri= 1 for every integer n, and by (9.5.2) (9.5.6)

ezi2nni

- e'.

To end the proof of (9.5.5) we have only to show that the equation eix = 1 has no root in the interval 10, 2n[. But from(9.5.2) we deduce cos (x + 7~12)= -sin x, hence cos x d 0 for 4 2 d x < 71, and as cos (x + n ) = -cos x, we see that cos x < 1 for 0 < x < 2n, and this ends the proof. (9.5.7) The mapping x+ eix is a continuous bijection of any interval [a, a -t2711 on the "unit circle" U: IzI = 1 in C, and a liomeomorphism of la, a + 2n[ on the complement of eia in U. The mapping is obviously continuous, and it is injective by (9.5.2) and (9.5.5). To prove it is surjective in [a, a + 27-4, we can obviously suppose = ct + ip, a' + p' = 1 ; a = 0, for if [ E U , [e-'" is also in U. Let d 1 and, in the interval [0, 711, cos x is continuous and cos 0 = 1, as cos n = - 1, there is y E [0, rr] such that cosy = c( by Bolzano's theorem (3.19.8). Then sin y = & p ; if sin y = p, we are through; if not we have cos (2n - y ) = c o s y = c( and sin (271- y ) = -sin y = p. Let V be the complement of ei" in U, and TO = eibE V with a < b < a + 2n; if the inverse

EXAMPLES OF ANALYTIC FUNCTIONS 213

5

mapping of the restriction of x eix to ]a, a + 2 4 was not continuous at io, there would be in ]a, a + 2 4 a sequence (x,) whose elements would belong to the complement of a neighborhood of b, and such that lim eiXn= to; but then a subsequence (xllk)would tend to a limit c # b --f

m

in the compact set [a, a + 2x1 by (3.16.1), and as eic # eib we arrive at a contradiction. (For another proof, see (1 0.34.)

n-

(9.5.8)

The unit circle U is connected.

This follows from (9.5.7), (3.19.1) and (3.19.7).

(9.5.9) ("Principle of maximum") Let (c, z") be a porver series with complex coefticienfs,absolutelysummable in an open polydisk P c C p o jcenter 0 and let j ( z ) be its sum. Suppose that there is an open ball B c P ojcenter 0 such that If(z)l < If(0)l for every z E B. Then c, = 0 for every index v # (0, . . . , 0), in other tilords,f is a constant. We first prove that the theorem is true for any p if it is true for p = 1 . Indeed, for any z = (z,, . . . , zp)E P, consider the function of one complex variable g ( t ) =f( tz,, . . . , tz,) which is analytic for It1 < I E with E small enough. As Ig(t)l < Ig(0)J for small values of t, we have g(t) =g(O) by assumption, and in particular f ( z l . . . , z p ) = g ( l ) =f ( 0 ) . For p = I , we can suppose co 0, otherwise the result is obvious by (9.1.6). Suppose there are indices n > 0 such that cn # 0, and let m be the smallest of them. We can write

+

+

+

+

f ( z ) = ~ o ( l bmZm z " ~ ( z ) ) where b, # 0, h is analytic in P and h(0) = 0. Let r > 0 be such that IzI < r is contained in B and lh(z)l d +lb,l for IzI d r (9.1.3). Write b, = lb,lc with [[I = 1 ; by (9.5.7) there is a real t such that emit= [-';for z = reit, we therefore have 11

+ bmzm+ zmh(z)I= 11 + Ibmlrm+ zmh(z)I 3 1 + tlbmlrm

which contradicts the assumption If(z)l 6 IcoJin B. The result (9.5.9) does not hold if C p is replaced by RP, as the example

of the power series l/(l

+ z 2 )=

m

n=O

(- 1 ) " ~(for ~ " IzI < 1 ) shows.

214

IX ANALYTIC FUNCTIONS

(9.5.10) Let f be a complex valued analytic function dejined in an open subset A c CP, and which is not constant in any connected component of A. For any compact subset H c A, the points z E H where If ( z ) l = sup1f(x)I xeH

(which exist by (3.17.10)) are frontier points of H.

Follows at once from (9.5.9) and the principle of analytic continuation (9.4.1).

PROBLEMS

1. Show that if W(z) < 0, then, for any integer n > 0

2.

(use Taylor's formula (8.14.2) applied to t +P). Prove that, for real x

1

COS x -

(

1-

xz

+ x4 - . . . + (-

and the difference has the sign of (-

I)"+l;

)I 1X1Zn+2 +

1)" -

(X22 4n !

0, 1 - s < e-s.)

+

+

similar to the one obtained in (c), observing that

216 9.

IX ANALYTIC FUNCTIONS

(a) Let f i k (1 < j < m, 1 < k < n) be scalar analytic functions defined in a n open connected subset A of Cp; let tcik be real numbers 30. Show that the continuous n

~fi~(~)~a'k~f~~(~)~'zk~f~~(~)~umk cannot reach a relative maximum at a point ofA,unless each of the products ~fIr(z)lu1"~~ Ifmk(z)lumk(l < k < n) is constant in

function u(z) = C

k=1

A. (Observe that if f(z) is analytic in A and f(zo) # 0, then, for every real number A, there is a function gA(z) which is analytic in a neighborhood of zo and such that IgA(z)j= If(z)lA in that neighborhood; use Problem 8(c) to that effect.) Extend the result to the case in which the tcJk are arbitrary real numbers, provided none of the hkvanishes in A. (b) Generalize to u(z) the result of Problem 3(a). 10. Let f ( z ) be a complex function of one complex variable, analytic in the open set A defined by R1< IzI < Rz (where 0 < R1 < Rz). For any r such that Rl < r < Rz,let M(r) = suplf(z)l. Show that if R1 < rl < rz < r3 < R , , then IT1 = r

log M(rz)
0 such that It - t’( d E , 15 - 5’1 < E imply Icp(t, dt

i

Therefore we are reduced to proving the relation r- 1

r- I

which can also be written r- 1

C (gij(yj(ti+

1))

i=O

- gij(yj+ l(ti+ 1 ) )

- gij(rj(ti>)+ g i j ( Y j + I ( t i > ) >

= 0.

But yj(ti) and yj+l(ti) both belong to Q i - l , n Q i j for 1 < i < r, hence, by what we have seen above gij(yj(ti)>- gij(Yj+l(ti>> = gi-1, j(yj(ti)) - gi-1, j(Yj+l(ti)) hence the left-hand side of (9.6.3.1) is reduced to gr-1, j(Yj(tr)) - gr-1, jbj+l(tr))- gOjbj(t0)) + 90j(yj+l(to)). But as y j and yj+l are circuits, we have yj(to)= yj(tr)and yj+l(to)= y j + , ( t r ) ; moreover, these two points belong to Q o j n Q r . - l , j , which is connected; the difference g r - l , - goj is thus constant in that set by (8.6.1), and this ends the proof. (9.6.4) Let yl, y 2 be two roads in an open set A c C,having same origin u and same extremity v , and such that there is a homotopy cp of y1 into y2 in A which leaves u and vfixed (i.e., q(a, I;) = u and cp(b, I;) = v for every I; E [a, /?I if cp is defined in [a, b] x [a,PI). Then, for every analytic function f in A,

j-p) dz j n f ( 4dz. =

+

Let yy be the road opposite to yl, and let y 3 ( t ) = yy(t - b a ) for b < t < 26 - a ; y 3 is a road equivalent to y y . By definition, y1 v y 3 and y 2 v y 3 are circuits. Moreover these circuits are homotopic in A, for if we define $ ( t , I;) as equal to cp(t, 5) for a < t < b, to y 3 ( t ) forb < t < 2b - a, $ is a loop homotopy in A. Applying (9.6.3), we get / y 2 f ( z )dz

+ jJ(4

dz. Q.E.D.

Yl

f ( z ) dz

+ JY, f ( z ) dz =

7 PRIMITIVE

OF

A N ANALYTIC FUNCTION

221

1

7. PRIMITIVE OF A N ANALYTIC F U N C T I O N IN A SIMPLY CONNECTED DOMAl N

A simply connected domain A c C is an open connected set such that any loop in A is homotopic in A to a loop reduced to a point; it is clear that any open subset of C homeomorphic to A is a simply connected domain.

Example (9.7.1) A star-shaped domain A c C with respect to a point a E A is an , segment joining a and z is contained open set such that for any ~ E Athe in A. Such a set is clearly connected ((3.19.1) and (3.19.3)); if y is any loop in A, write q ( t , 5 ) = a + (1 - t ) ( y ( t )- a ) for 0 < 5 < 1 ; 40 is a loop homotopy of y into the loop reduced to a. An open ball is a star-shaped domain with respect to any of its points. (9.7.2) If A c C is an open connected set, for any two points u, u of A there is a road of origin u and extremity u. We need only prove that the subset B c A of all extremities of roads in A having origin u is both closed and open in A (Section 3.19). If x E A n B, there is a ball S of center x contained in A, and by assumption S contains the extremity u of a road y of origin u ; the segment of extremities u, x is contained in S, and if y is defined in [a, b], the road y1 equal to y in [a, b], to y l ( t ) = u ( t - b)(x - v) in [b, b + 11is in A and has origin u, extremity x; hence x E B. On the other hand, if y E B, there is a ball S of center y contained in A ; for any u E S, the segment of extremities y , u is contained in S and we define in the same manner a road of origin u, extremity u, which is in A, hence S c B. Q.E.D.

+

(9.7.3) I f A c C is a simpIy connected domain, any function f analytic in A has a primitive Icjhich is analytic in A. Let a, z be two points of A, yl, yz two roads in A of origin a and extremity z; then J y , f ( x )dx = f ( x ) dx. Indeed, we may suppose, by replacing yz In by an equivalent road, that y1 is defined in [b, c] and y z in [c, d ] ; then y = yI v y y is a circuit in A, which is therefore homotopic to a point in A,

222

IX ANALYTIC FUNCTIONS

hence

j7f (x) dx = 0 by

Cauchy's theofem, and this proves our assertion.

We can therefore define g(z) as the value of f ( x ) dx for any road y in A J-7 of origin a and extremity z, and by (9.7.2), g is defined in A. Now for any z o E A, there is an open ball B c A of center zo in which f ( z ) is equal to a convergent power series in z - z,; by (9.3.7) there is therefore a primitive h off in B which is analytic, and such that h(z,) = g(zo); hence we have for ZEB

h ( z ) - h(z0) =

sd

f ( z o + t(z - z ~ ) ) (z zO) dt.

Jg

But the right-hand side is by definition f ( x ) dx, where CJ is the road t + zo t(z - z o ) defined in [0, 11; as that road is in B c A, we have g(z) - g(z,) = Jo f ( x ) dx by definition of g , and therefore g(z) = h(z) in B. Q.E.D.

+

8. INDEX OF A P O I N T WITH RESPECT TO A CIRCUIT

(9.8.1) Any path y defined in an interval I = [a, b] and such that y(1) is contained in the unit circle U = { z E C 1 ( z I = I}, has the form t -+ ei*(*), where $ is a continuous mapping of I into R; if y is a road, $ is a primitive of a regulated function.

As y is uniformly continuous in I, there is an increasing sequence of points t k (0 < k < p ) in I such that to = a, t, = b, and that the oscillation (Section 3.14) of y in each of the intervals 1, = [tk, tk+l] (0 < k < p - 1) be < 1. This implies that ?(I,) # U ; if 8, E R is such that eiek# y ( I k ) (9.5.7), then x + e'@' e k ) is a homeomorphism of the interval 10,274 on the complement of eiek in U (9.5.7). If q k is the inverse homeomorphism, we can therefore write, for t E I,, y ( t ) = eiJlk(*), where $k(t) = ( P k ( Y ( t ) ) +e, is continuous in I k . By (9.5.5), we have $k+l(tk+l)= $k(tk+l) + 2nkn with nk an integer (0 < k < p - 2 ) . Define now $ in 1 in the following way: $(t) = $ , ( t ) for t E 10; by induction on k, We put $(t) = $ k ( t ) $(tk) - $k(tk) for f k < t < tk+,. By induction on k , it is immediately seen that $ ( t k ) - $ k ( t k ) is an integral multiple of 271 for 0 < k < p - 1 ; therefore y ( t ) = ei*(') for t E I, and $ is obviously continuous in I. Moreover, if y(t) = a ( t ) + iP(t), we have a ( t ) = cos $ ( t ) , P ( t ) = sin $ ( t ) , and one of the numbers cos $ ( t ) , sin $ ( t ) is not 0; from (9.5.4), and (8.2.3) applied to one of the functions cos x, sin x at a point where it has a derivative # 0, we deduce that if y has a derivative at a point t , so has $, and i$'(t) = y'(t)/y(t),which ends our proof.

+

8 INDEX OF A POINT WITH RESPECT TO A CIRCUIT

223

(9.8.2) For any point U E C , and any circuit y contained in C - { a } , Jydz/(z- a ) has the form 2ntri, bvhere n is a positive or negative integer. By a translation, we can suppose a

= 0.

Suppose y is defined in I

=

[b, c ] ;

the function q(t, 5) = 5 y ( t ) + ( 1 - = j ( x ; y ) by Cauchy’s theorem. As the set Z of integers is a discrete space, the conclusion follows from (3.19.7).

+

-=

224

IX ANALYTIC FUNCTIONS

Exainple

(9.8.4) Let c, be the circuit t -+ enitdefined in 1 = [0, 2n], M being a positive or negative integer; we have &,(I) = U ; E, is called “the unit circle taken n times”. We observe that the open set C - U has tii’o connected components, namely the ball B: IzI < 1 and the exterior E of B defined by IzI > 1. Indeed, B is connected as a star-shaped domain (9.7.1); and by Section 4.4and (9.5.7) E is the image of ]I, +a[x [0, 2n] by the continuous mapping (x, t ) -+ xe“, hence the result by (3.19.1), (3.20.16), and (3.19.7) (a similar argument also proves the connectedness of B and of B -{O}); finally in C - U, B and E are open and closed since B is open in C and B = (C- U) n and we have B n E = 0. From the definition and (9.5.3) it follows that j ( 0 ; E,) = n, hencej(z; E,) = n f o r any point z of B. Let us show thatj(z; cn) = 0 f o r any point of E; more generally:

s,

(9.8.5) Ifacivcuit yiscontainedinaclosedballD: Iz - a / < r, thenj(z; y ) = 0 f o r any point z exterior to D. Indeed, suppose y is defined in an interval I = [b, c], and that Iy’(t)l in that interval. By definition,

0 (resp 4 ( z ) < 0). Show that f i s analytic in A . (Suppose the disk Izl < r is contained in A.Let y + (resp. y - ) be the circuit defined in [- I , + I ] by y + ( t ) = (2r 1)r for -1 < t < 0, y + ( t ) = r e f f i rfor 0 < t < 1 (resp. y - ( t ) = re”” for - 1 < t < 0, y - ( t ) = (1 - 2t)r for 0 < t < I.) Show that if Iz/ < r and 9 ( z ) > 0, then

+

using Problem 1 ; hence if y is the circuit t

--f

1 2nr

f ( z ) = -,

rezir in [- 1 , + I ] ,

jfE, x-z Y

Then use (9.9.2).)

3. Show that the conclusion of (9.9.4) still holds whenfis merely assumed to be bounded

in each bounded polydisk contained in A, but not necessarily continuous. (Use Problem 6 of Section 8.9; actually, a deep theorem of Hartogs shows that even this weakened assumption is not necessary; in other words, a function which is analytic separately with respect to each of t h e p complex variables ziis analytic in A,)

9

4.

m

Let f ( z )

THE CAUCHY FORMULA

a, z" be an analytic complex-valued function in the circle IzI

229

< R. Show

"=O

that, for 0 < r < R

Deduce from that result another proof of Cauchy's inequalities. 5.

Let f ( z ) =

m

a.z" be an analytic function in IzI

< R, and let

"=O

m

M ~ ( r ; f )= Let also M(r;f)

=

C llanllr".

"=O

sup Ilf(z)ll. 111 = I

(a) Show that for 0 < r

0. Show that f is analytic in 1. (Use (a), and Problem 3(b) of Section 8.12).

10. CHARACTERIZATION O F ANALYTIC F U N C T I O N S O F COMPLEX VARIABLES

A continuously differentiable mapping f of an open subset A of C p into a complex Banach space is analytic.

(9.10.1)

Applying (9.9.4), we are immediately reduced to the case p = 1. To prove f is analytic at a point U E A ,we may, by translation and homothetic mapping, suppose that a = 0 and that A contains the unit ball B: IzI < 1. For any Z E B, and any I such that 0 Q I Q 1, note that 1(1 - I ) z + Iei'l < 1 - I + I = 1, and consider the integral (9.10.1 . I )

By (8.1 1.I) and Leibniz's rule (8.1 1.2), g is continuous in [0, 11 and has at each point of 10, 1[ a derivative equal to g ' ( I ) = Joznjr(z + I(eit - z))ei' dt

(see Remarks after (8.4.1)). But i y ' ( z + I(eif - z))e" is the derivative of t +f(z + i(ei' - z)), hence, for I # 0, g ' ( I ) = 0, and therefore (remark following (8.6.1 )), g is constant in [0, 11. But asg(0) = 0, g ( I ) = 0 for 0 Q A Q 1. In particular, it follows, for I = 1, that

for any z E B (by (9.8.4)), and the conclusion follows from (9.9.2).

Let f be a continuously dlfferentiable mapping of an open set A c RZPinto a complex Banach space. In order that the function g dejined in A (considered as a subset of C p ) , by f ( x l , x 2 , .. ., x p , y l , . ., y,) = g (xl + iyl, . . . , x, + iy,) be analytic in A, necessary and suflcient conditions are that (9.10.2)

in A for 1 < k

< p (Cauchy's conditions).

10 CHARACTERIZATION OF ANALYTIC FUNCTIONS

231

We are again at once reduced to the case p = 1 by (8.9.1). Let (x,y )

df (x,y ) , b = af (x,y ) ; expressing that the be a point of A, and put a = ax 8Y limits lim (g(x iy 12) - g(x + iy))/h and lim (g(x + iy + ih) - g(x iy))/ih

+ +

+

h-+O

( h real and ZO) are the same, we obtain a + ib = 0. Conversely, if that condition is satisfied, for any E > 0, there is r > 0 such that if (h2 + k 2 ) 1 / 2< r, 11g(x iy h + ik) - g(x iy) - a(h + ik)ll < e(h2 + k 2 ) 1 / 2by (8.9.1 .I)and this proves that z -g(z) has a derivative equal to a at the point z = x iy. The result then follows from (9.10.1). h-tO

+ +

+

+

PROBLEMS

1. Show that a differentiable mapping o f f an open subset A of C p into a complex Banach space is analytic in A ("Goursat's theorem"; f' is not supposed to be continuous). (Given any h in ]0,1[, prove (with the notations of (9.10.1)) that g'(h) exists and is equal to 0. First show that, given E > 0, there are points t o = 0 < t l < . .. < 1, = 2n, a number p > 0, and in each interval [ t k , t k + 11 a point 8, such that, if ( k = z &elek - z), whenever (k x =z ( A h)(e" - z), then x) -f((k) -f'((k)xl < & / X I Ihl < p and tk < t < t k + l (prove this by contradiction, using a compactness argument and the existence off' at each point). Compare then each integral

+

+ +

+

+

to the expression

2.

for Ihl < p.) Let A be an open simply connected subset of C; i f f is a continuous mapping of into a complex Banach space E such that

s,

f ( z ) dz = 0 for any circuity in A, show that f i s

analytic in A. (" Morera's theorem"; show that f h a s a primitive in A.) 3. Let A be an open subset of Cp, y a road defined in I = [a, b], f a continuous mapping of y(1) x A into a complex Banach space E. Suppose that for each x E y(I), the function ( z l , . .. ,z,) + f ( x , zl, . .. , z,) is analytic in A, and that each of the functions

af -(x,

.

z l , . , ,z,) is continuous in y(1) x A (1 < k

conditions, the functions g(zl,. .. , z,)

=

2 ) , f a n analytic mapping of A into a complex Banach space E. Suppose that there is an open polydisk P C A, of center b= < k < p and radii rL (1 < k < p ) such that for every point (ck) of P, there is a

232

IX ANALYTIC FUNCTIONS

+

number p < inf(rl, r 2 )such that the function x1 ixz +f(xl, xz , c 3 ,.. . , cp)is analytic in the open subset /xl ixz - (cI icz)l < p of C (identified to R2).Show that the same property holds for every point (ck) E A (use (9.10.2) and (9.4.2)). 5. Let S be the "shell" in RD( p 2 3) defined by

+

+

+ XI+ '. + x', < (R +

(R - E)' < X :

E)'

(0 < E

< R).

Suppose f is an analytic mapping of S into a complex Banach space E, and suppose that for any u = (x3, .. ,x p ) , the mapping x1 ixz + f ( x l , x 2 , u) is analytic in a neighborhood (in C) of every point of the cross section S(u) (if S(u) is not empty). (a) For any u = ( x 3 ,. .., x,) such that lluIl2 = x: . . . x', < RZ,let y(u) be the road in C defined by f +(RZ - IIuljZ)l/zeiffor -7r < f < T.Let

+ + +

.

+

+

where y = x1 ixz , and f ( y , u ) =f(xl, xz, u ) ; g is defined for lzlz l/ull2< RZ,and z+g(z,u)isanalyticforlzl < (R2 - jl~ll~))'/~.Ontheotherhand,foranyv = (x;,. . . , x i ) such that Ilu II < R. let

+

Show that h,(z, u ) = g ( z , u ) for llull < l l ~ l< l llull E and /z /cc (R2- l l ~ l l ~ ) (apply '/~ Cauchy's theorem (9.6.3)). On the other hand, show that g(z, u ) = f ( z , u) for R - E < llull < R and /zI < (RZ- I I U I ~ ~ ) ' ~ ~Conclude . that f can be extended to a function f which is analytic in the whole ball B : x i . . . x: < (R E ) ~(apply (9.4.2) and Problem 3). Is the theorem still true for p = 2 ? (b) When E = C, show that f(B) c f ( S ) . (Apply the result of (a) to the function l/(f- c), where c $f(S).) In particular, iffis bounded in S , f i s bounded in B. Extend that last property to the case in which E is a complex Hilbert space (method of Problem 6 of Section 8.5).

+

+ +

11. LIOUVILLE'S THEOREM

(Liouville's theorem) Let f be an entire function in C p , with values in a complex Banach space E. Suppose there exists two numbers a > 0, N > 0 such that ]]j(z)I] < a (l ( ~ u p l z ~ lin) ~C". ) Then f ( z ) is the sum ofajinite with ~ c,, . . . n p E E and number of "monomials" C , , , , , ~ . ....~. :2~ Z ~ (9.1 1 . I )

+

nl

Let f ( z ) =

+ n2 + + np < N. **.

cvzv in C p , the power series being everywhere absolutely

convergent. The Cauchy inequalities (9.9.5) applied to the polydisk lzj]< R (1 < j < p ) yields, for any v = ( n l , . . . , np) Q a * (I R ) ~ R - ( " , +... + ' p ) . lc,,,

+

Letting R tend to

+ co, we see that c,, ...",

= 0,

unless n,

+ + np < N .

12 CONVERGENT SEQUENCES OF ANALYTIC FUNCTIONS

233

(9.11.2) (The "fundamental theorem of algebra"). Any polynomial f ( z ) = a,z" + a,zfl-l + * . . + a,, (a, # 0, n 2 1 ) with complex coeficients has at least one root in C. Otherwise, l / f would be analytic in C (9.3.2), hence an entire function (9.9.6). Let r be a real number such that rk 2 ( n t l ) ~ a k / afor o ~ 1 d k dn; then, for IzI 2 r

In other words, l / f is bounded for lzl 3 r. On the other hand, l / f being continuous in the compact set IzI < r, is also bounded in that set (3.17.10), hence l / f is bounded in C.Liouville's theorem then implies l / f is a constant, hence also f, contrary to assumption since If(z)I 2 la,] . Izl"/(n + 1 ) for IzI 3 r.

PROBLEMS

If p > 2, show that a function which is analytic in the complement of a compact subset of Cp is an entire function; hence if in addition it is bounded in the complement of a compact subset of C', it is a constant (use (9.1 1.I) and Problems 4 and 5 of Section 9.10). Is the result true for p = 1 ? L e t f b e a complex valued entire function in CP. Show that the conclusion of (9.11.1) is still valid if it is supposed that

Wf(z)) < a * (sy41,Iz,l)'") for any z in the exterior of a polydisk of CP (use Problem 6 of Section 9.9). Let f(z)

m

= "=O

a, z" be a nonconstant

M(r) = sup Ilf(z)II, so that p(r) IZI =I

entire function. For any r

< M(r);

> 0, let p(r) = sup JJanJIr", n

by Liouville's theorem, lim p(r)= f a . I-m

Suppose there are two constants a > 0 , a > O such that p ( r ) < a.exp(r"); show that there are positive constants b, c such that M(r) < brap(r) c. (Observe that llanIl< a(ea/n)"".)

+

12. CONVERGENT SEQUENCES O F ANALYTIC F U N C T I O N S

(9.12.1) Let (f,)be a sequence of analytic mappings of an open set A c C p into a complex Banach space E. Suppose that for each z E A, the sequence (f,(z)) tends to a limit g(z), and that the convergence is uniform in every compact

234

IX ANALYTIC FUNCTIONS

subset of A, Then g is analytic in A, and f o r each v = ( nl, . . . , np)E NP, the sequence ( D v L ( z ) )converges to Dvg(z)f o r each z E A, the convergence being uniform in every compact subset of A. As g is continuous in A (7.2.1), to prove g is analytic in A, we need only prove that each mapping zk +g(a,, . . . , zk, . . . , up) is analytic in A(a,, .. . , a k - 1 , a k f l , . . . , up), by (9.9.4); in other words we are reduced to the case p = 1. For each a E A c C, let B be a closed ball of center a and radius r contained in A, and let y be the circuit t + a reir (0 < t d2n); then, for each z E fi and each n, we have by Cauchy’s formula

+

But by assumption the sequence ( f , ( x ) ) converges uniformly to g(x) for Ix - a( = r, and as ) z - x( 2 r - IzI, the sequence (L(x)/(x- z ) ) ( z fixed) also converges uniformly to g(x)/(x- z ) for Ix - a ) = r ; hence, by (8.7.8)

which proves g is analytic in

fi by (9.9.2).

Moreover, as

by (9.9.3), the same argument (and (9.9.3) applied to g) shows that f ’ ( z ) tends to g’(z) for every z E 8; furthermore, we have by the mean-value theorem

Returning to the general case ( p arbitrary), let us now show that the sequence ( D k f , ( z ) )converges uniformly to D k g ( z ) in any compact set M c A. There is a number r > 0 and a compact neighborhood V of M contained in A, and containing all points of A having a distance d r to M (3.18.2). For any E > 0, let no be such that \lg(z) -f,(z)l) d E for every n > no and every z E V. Then, applying (9.12.1 . I ) to the sequence of functions zk -+&(al9. . . , ak- zk, ak+ . . . , up), we obtain, for every point z E M, (IDkg(z)- Dkffl(z))I< E/r as soon as n 2 n o . This ends the proof of the theorem when n, + * + np = 1 ; the general case is then proved by induction on nl np.

,,

+ . - -+

12 CONVERGENT SEQUENCES OF ANALYTIC FUNCTIONS 235

Observe again here that the theorem does not hold for analytic functions of real variables, since a sequence of polynomials can have as a limit an arbitrary (e.g. nondifferentiable) continuous function in a compact set, by the Weierstrass approximation theorem (7.4.1).

PROBLEMS

1. (a)

Let

(ak)l, O . 3. An endless road in an open subset A C C is a continuous mapping y of R into A such that in every compact interval I C R, y is the primitive of a regulated function. Iff is a continuous mapping ofy(R) into a complex Banach space E,fis said to be improperly integrable alongy if the improper integral

j. f ( y ( t ) ) y ' ( t )

c//

-an

exists (i.e., if both limits

lim /aof(y(/))y'(r) dt exist in E); the value of that -m

lim Jobf(y(t))y'(t) dt and b-fm

a+

integral is then callled the integral off dong y and written J y f ( z )dz. Let B be an open subset of C p ,g a continuous mapping ofy(R) x B into E; suppose that for each x E y(R), the function ( Z I , . . , z,) -+g(x,zI,. . . , 2,) is analytic in B and % that each of the functions - (x,z , , . . . , z,) is continuous in y(R) x B. Finally

.

suppose that for each (zl, . .. , z,) alongy,and that

j

'"

g(y(t), z,,

E

.

B, x + g ( x , zl, . . , z),

. . .,z,)y'(t)

c//

is improperly integrable

tends uniformly to

-n

s,

y(x, z I , . . . , z),

ctx

when (zI,. . . , z,) remains in a compact subset of B and n tends to i-co. Under these conditions, show that the function ( z I ,. . . , z,) 4.

+

ij

g(x,zl,

. . . , z,)

dx is analytic in B

(compare to (13.8.6)). Extend the result of Problem 2 of Section 9.9 to functions of p complex variables, D, (resp. D-) being defined by S(z,) > 0 (resp. .P(z,) i 0). (Observe that, by (9.12.1), for each z, such that .P(z,) = 0 and the intersection B of A with the set C p - ' x (z,) is not empty, the function (zI, , , z P - J+f(zI, . . . , z , - I , 2,) is analytic in B.)

..

13 EQUICONTINUOUS SETS OF ANALYTIC FUNCTIONS

5.

237

In the plane C, let Q be the square of center 0, defined by lg(z)l < I , l.Y(z)l < I . Let Q o , Q1, Q 2 , Q 3 be the images of Q by the mappings I + i

z

2

2'

z+---+-

ZJ-

-l+i 2

+-2'z

z+-

-1-i 2

I-i 2

z

f-, z+-+-. 2

+ + +

+

z

2

Let m,, = 0 , and for any h > I , let mh = 4 4' . . . 4h; if n = mh 4k + j , with h > I , 0 < k < 4h - I , 0 < j < 3, define inductively Q, as follows: let nl = mh-1 k, and let zn1be the center of Q,,; let p.,(z) z., -t- z/2* and take Q. = y,,,(Q). (a) Let B be the unit disk /zI C I , U the unit circle IzJ= 1. Show by induction on n the existence of three sequence of numbers (a"),(cJ, (t,) defined for I I 2 4, having the following properties:

+

~

(I)

O y o , suclz that

n= I

f ( z ) = g,(z) + g 2 ( z ) in S ('' Laurent series" off). Moreover the poii'er series g l , g , hacing these properties are uiiiqzie, and, .for every circuit y in we h a w

s,

By (9.9.4) we have

the series being convergent for Iz( < r l . On the other hand, for IzI > y o , 1x1 = y o , we have y- 1 1 -z - x - 2 z" where the right-hand side is normally convergent for 1x1 = ro ( z fixed); by (8.7.9), we get

the series being convergent for IzI > r o . This proves the first part of (9.14.2). Suppose next we have in S f ( z ) = C a n z n+ U,

(9.14.2.1)

n=O

C bflz-" n;

n= 1

both series being convergent in S ; let first y be a circuit in S, defined in I; there are points t , t' in I such that y ( t ) = inf y(s) = r and y ( t ' ) = sup y ( s ) = r' ssl

S€l

(3.17.10), hence ro < r < y(s) < r ' < r1 for any s E I . But, for r < JzI< r ' , both series in (9.14.2.1) are normally convergent (9.1.2), hence by (8.7.9), for any positive or negative integer in

+

As zk"/(k I ) is a primitive of z' for k # - 1, we have / / k dz = 0 for any circuit a ; (9.14.2) then follows from the definition of the index. If now y is in 5, we remark that there is an open ring S , : ( 1 - E)ro < IzI < ( I + c)r, contained in A (3.17.11), and we are back to the preceding case.

15 ISOLATED SINGULAR POINTS; POLES; ZEROS; RESIDUES

241

15. I S O L A T E D S I N G U L A R P O I N T S : POLES; ZEROS; RESIDUES

(9.1 5.1) Let A be an open subset of C, a mi isolatedpoint of C - A (3.1 0.1 0), r a number > O such that all points of the ball Iz - a1 < r except a belong to A. I f f is an analytic mapping of A into a complex Banach space E, then f o r 0 < Iz - a1 < r, we have f(z)=

2 cn(z - a)" + c dn(Z - a>-, m

a:

n=O

n= 1

where both series are convergent f o r 0 < Iz - a1 < r, and (X

tvliere y is the circuit t -+ a

+ re"

(0 < t

- u ) " - ~ ~ ( dx, x)

< 2n).

This follows at once from (9.14.2) applied to the ring p where p is arbitrarily small. Observe that the series u ( x ) =

< ( z - a ( < r,

C d n x " is an entire function

such that

n= I

u(0) = 0; we say that the function u(l/(z - a)) is the singular part o f f in the neighborhood of a (or at a). When u = 0, f coincides in the open

set U : 0 < Iz - a / < r with the function g ( z ) =

a:

c,(z - a)", which is

n=O

analytic for Iz - a1 < r ; conversely, iff is the restriction to U of an analytic function fi defined for Iz - a1 < r, then f i = g by (9.9.4) and (9.15.1), hence u = 0. When u # 0, we say that a is an isolated singular point off. If u is a polynomial of degree n 2 1, we say a is a pole of order n o f f ; if not (i.e. if d,,, # 0 for an infinite number of values of m) we say a is an essential singular point (or essential singularity) of j : In general, we define the order w(a; f ) or w(a) off at the point a as follows: o(a) = - co if a is an essential singularity; w(a) = -n if a is a pole of order n 2 1 ; w(a) = rn iff # 0, u = 0 -"

m

and in the power series

1 c,(z - a)"

equal to f ( z ) for 0 < Iz - a1 < r,

n=O

m is the smallest integer for which c,, # 0; finally w(a; 0) = +a.When w ( a ;f ) = m > 0 , we also say a is a zero of order m off. Observe that if both A g are analytic in the open set U : 0 < Iz - a1 < r, and take their values in the same space, then w ( a ; f + g ) 2 min(w(a; f ) , w(a; g ) ) ; if one of the functions f, g is complex valued, then w(a;f g ) = w(a;f ) w ( a ; g ) when one of the numbers w ( a ;f ) , w ( a ; g ) is finite. Any function f analytic in U and of finite order n (positive or negative) can be written in a unique

+

242

IX ANALYTIC FUNCTIONS

way ( z - a)yl, where fl is analytic in U and of order 0 at the point a. Finally, iffis analytic in U and complex valued, and of finite order n7, then it follows from the principle of isolated zeros and from (9.3.2) that there exists a number r' such that 0 < r' < r and that I/f is analytic in the open set 0 < Iz - a1 < r ' ; we have then w(a; IF) = - w(a;f). (9.15.2) Let f be analytic in the open set U : 0 < Iz - a1 < r . In order that o ( a ;f ) 2 n where n is a positive or negative integer, it is necessary and sufficient that there exist a neighborhood V of a in C such that ( z - a)-"f ( z ) be bounded in V n U. The condition is obviously necessary, since a function having order 3 0 at a is the restriction of a function analytic in a ball Iz - a1 < r. Conversely, by considering the function ( z - a)-"f(z), we can suppose n = 0. Then it follows from (9.15.1) and the mean value theorem that if 11 f(z)II d M in U , we have, for any p such that 0 < p < r, lld,,t,ll d Mp" for any m 3 1 ; as p is arbitrary, this implies d,,, = 0 for each m 2 1. Q.E.D. The coefficient d, in (9.15.1) is called the residue off at the point a.

PROBLEMS

1. Show that there are no isolated singular points for analytic functions of p > 2 complex variables (in other words, if A is an open subset of C p ,a E A and a mappingfof A - { a } into a complex Banach space E is analytic, it is the restriction of an analytic mapping of A into E; use Problem 5 in Section 9.10). 2. Letfbe a complex valued analytic function of one complex variable having an essential singularity at a point a E C; show that for any complex number h, it is impossible that the function l / ( f - h ) should be defined and bounded in an open set of the form V - {a},where V is an open neighborhood (use (9.15.2)). Conclude that for any neighborhood V of a such that f i s analytic in V - {a},f(V - { a ) ) is dense in C (" Weierstrass' theorem"; see Section 10.3, Problem 8). 3. An entire function which is not a polynomial is called a trunscendental entire function. Let f be a complex valued entire transcendenial function of one complex variable. (a) Show that for any integer n > 0, the open subset D(n) of C consisting of the points z E C such that If(z)i > n is not empty and cannot contain the exterior of any ball (apply Problem 2 to the function f(l/z)). (b) Let K(n) be a connected component (3.19.5) of D(n). Show that K(n) is not bounded and that If(z)i is not bounded in K ( n ) (if u $ K(n), consider the function f(l/(z - a ) ) and use Problem 14 of Section 9.5). (c) Show that there is a continuous mapping y of [0, K>[into C, such that in every interval [0, a ] ,y is the primitive of a regulated function, and that lim iy(t)i = +a

+

r-ti

IU

1 5 ISOLATED SINGULAR POINTS; POLES; ZEROS; RESIDUES

and lim If(y(t))l r-+m

=

243

+a.(Consider a sequence of open subsets L, c C such that L, is

a connected component of D(n), and L.+l C L, for every n ; the existence of such a sequence follows from (b). Use then (9.7.2)) (d) Extend the preceding results to complex valued entire transcendental functions of an arbitrary number of complex variables. (If f ( z I , . . . , z), = a n l . . “,zl n l . . . z p , there exists at least an index k such that there are infinitely many monomials with non zero coefficient R , ~ ,. . and arbitrarily large n k . On the other hand, prove that if (9,) is a denumerable family of entire complex valued functions of p complex variables, none of which is identically 0, then there exist points (cl, . . . , c,) for which g,,(cl,.. . , c,) # 0 for every m ; to do this, use induction on p , and the fact that for a function h(z) of one complex variable, analytic in A c C and not identically 0, the set of solutions z of h(z) = 0 is at most denumerable (see (9.1.5)).) 4. Let ~ ( xbe) an arbitrary increasing and positive real function defined in [0, cc [. Let (k.) be a strictly increasing sequence of integers such that k1 = 1 , and (n(n - 1))”“ > ~ ( n1 ) for n > 1. Show that the power series

+

f(z)= 1

+

+ 5 (L)k” n- I n=2

is convergent for all z E C, and that for every real x > 2, f ( x ) > ~ ( x(in ) other words, there are entire functions which tend to infinity “faster” than any given real function). 5. For any real numbers a , p such that p > 0, let L=J be the endless road (Section 9.12, Problem 3) defined as follows: for t < - I , L,,p(t) = a - ip - t - 1 ; for - 1 < t < I , L&) = a ipr; for f > 1, L,,&) = a ip r - 1. Let G.,p = LnSp(R). (a) Show that if 77/2 < p < 3rr/2, and if x $ Ga,pthe function z (exp(exp z))/(z - x) isimproperlyintegrablealongL..B. Furthermore,isP1,P2aresuchthat I-”(x)l < < p2 of /4(x)1> p2 > or a ( x ) < a, the integrals along and Llr.ozare the same; similarly, if 9 ( x ) < a1< a 2 ,or g1 < cc2 < B(x), or [.f(x)( > p the integrals along and La2,Aare the same (use Cauchy’s theorem). (b) Deduce from (a) that if L = Lo,11,

+ +

+

--f

PI,

PI

can be extended to an entire function. (c) Show that

(prove that the integral along La,” of exp(exp z ) is independent of cc and p (provided n-/2 < p < 3n-12)). (d) Show that if x belongs to the open set A defined by W ( x ) < 0 or i-”(x)l > rr,

where F(:) is bounded in A (express F(x) by an integral along Lo,pwith p < rr, using (a) and (c)). (e) Show that if x belongs to the open set B defined by W ( x )> 0 and IY(x)l < rr, then 1

E(x) = exp(exp x ) - x

GW +x2

IX ANALYTIC FUNCTIONS

244

where G(x) is bounded in B (prove first, using Cauchy's formula, that if and /9(x)1< x , then

-1

< 9 ( x )< 0

where L'= L - l , n . Show next that that formula is still valid for x E B using (9.4.2) and express G ( x )by an integral along L - l . f lwith p > x ) . (f) Let H(x) = E(x)e-"("'; show that H is an entire transcendental function such that lim H(rejB)= O for every real # (use (d) and (e); compare with the result of v-+m

Problem 3). 6. Let f be a complex valued entire function of p > 2 complex variables. Show that if f ( a l ,..., a,) = b, then for every r > 0 , there exists z = (zI. ..., z,) such that Izp - ukI2= r 2 and f ( z I , . . . ,z,) = b (use Problem 5(b) of Section 9.10). k

7. Letfbe an analytic mapping of an open subset A c C p into a complex Banach space E. A frontier point zo of A is called a regular point forfif there is an open neighborhood V of zo and an analytic mapping of A u V into E which coincides with f i n A. A frontier point of A is said to be singular forfif it is not regular. (a) Let R < +a, be the radius of convergence (Section 9.1, Problem I ) of a power series f ( z ) =C a, z" of one complex variable. There is at least one point zo such that

lzol = R which is a singular point for f. (Otherwise one could cover the circle Iz/ R with a finite number of open balls Bk whose centers hk are on that circle, and such that in each open set B(0; R) u Bx there is an analytic function 6 coinciding with f in B(0; R). Show that for any two indices h, k for which Bh n Bx # @,fh andfx coincide in Bh n Bk, using (9.4.2), and conclude from (9.9.1) and (9.9.2) that the radius of convergence of CI, z" would be > R.)

"

(b) With the notations of (a), suppose a, 0 for every n. Show that the point z = R is singular for f. (One may suppose R = I . Let el" be a singular point for f; then for 0 < I' < I, the radius of convergence of the power series ~ f ( " ' ( r e ' Q ) z " /is t i !exactly 1 - r (9.9.1). Observe that If(")(re")l < f ( " ) ( r ) ,and use (9.1.2).) (c) With the notations of (a), suppose R = I . Let b, c be two real numbers such that 0 < h < 1, c 1 - h, and let p be an integer > I . In order that the point z = 1 be a singular point for f, it is necessary and sufficient that the Taylor series 1 y'")(O)u"/ir! ~

"

+

for the function y(u) = f(bu, c u p + ' )have a radius of convergence equal to I . (Observe that if I { I ~ < I , Ibu" c u P f 1 I < 1, and that the two sides of the last inequality can only be equal for u = 1. The proof for the necessity of the condition has to use (10.2.5), in order to show that there is in the neighborhood of z = I an analytic function h(z) such that z = g(h(z)) in that neighborhood.) (d) Suppose (with the notations of (a)) that a, = 0 except for a subsequence ( t i x ) of integers such that nk+l > (1 8)nk for every k , where 6 > 0 is a fixed number. Show that euery point zo on the circle IzI = R is a singular point for f(" Hadarnard's gap theorem "; the circle /zI = R is called a mrlmlboundary forf). (One may suppose

+

+

R = I . Use criterion of (c), taking p > l/8, and let g(u) =

m

C dnu" be

the Taylor

"=O

development of g at u = 0. By assumption, for given E > 0 there is a subsequence (mi) of integers such that llun,,~~ 2 (I- E ) ' " ~ (Section 9.1, Problem I). On the other hand the function F(u) (bup cu,+l)"" =C e.u" has I I = 1 as a singular point, by (b),

=c J

+

"

hence there is a subsequence (4,)of integers such that jeqll > (1 - ~ ) " l . l:dq~ll 2 (1 - E ) 2 4 9 .

Prove that

16 THE THEOREM OF RESIDUES

245

16. T H E T H E O R E M OF RESIDUES

We first recall that any subset S c C the points of which are all isolated is at most denumerable, for the subspace S of C is then discrete and separable (by (3.9.2), (3.20.16), and (3.10.9)), hence S is the only dense subset of S (3 .I 0.1 0).

(9.16.1) Let A c C be a simply connected domain, (a,,) a (jinite or injinite) sequence of distinct isolated points of A, S the set of points of that sequence. Let f be an analytic mapping of A - S into a complex Banach space E, and let y be a circuit in A - S. Then I c v have

where R(a,,) is the residue off at the point a,, and there are only aJinite number of terms # O on the right-hand side (" theorem of residues"). We can obviously suppose each a, is a singular point for f, for we can extend f by continuity to all nonsingular points a,, which does not change both sides of the formula (since R(a,) = 0 if a, is not singular). Under that assumption, for any compact set L c A, L n S is jinite, for L n S is closed in L, as A - S is open in C by definition; hence L n S, being compact and discrete, is finite (3.16.3). Let I be the interval in which y is defined, and let P be the set of points x E A such that j ( x ; y) # 0. We know (9.8.6) that the closure P of P in C is compact, and P does not contain any frontier point of A, for such a point cannot be in y(I), nor have index # O with respect to y , by (9.8.7); as the set of points x in C - y(1) where the index ,j(x;y) takes a given value is open (9.8.3), any point in P which does not belong to y(1) is in P, hence P c A. On the other hand, let q(t, 5 ) be a loop homotopy in A of y into a one-point circuit ( t E I, ( E J, where J is a compact interval). Then M = cp(I x J) is a compact subset of A. Let H c N be the finite set of the integers n such that a,€ M u P; for each n E H, let zr,(l/(z - a,,)) be the singular part off at the point a,. Let B be the complement in A of the set of points a, such that n 4 H ; then B is open, for a compact neighborhood of a point of B, contained in A, has a finite intersection with S. By definition of the singular parts, there is a function g, analytic in B, and which is equal to f ( z ) -

at every point z # a,, ( n E H).

246

IX ANALYTIC FUNCTIONS

As M c B by definition, y is homotopic in B to a one-point circuit; hence, by Cauchy’s theorem, g(z) dz = 0, in other words

si

the result then follows from (9.14.2), applied to each of the functions u,, in an open

“

ring ” of center a,, , containing y(1).

17. M E R O M O R P H I C F U N C T I O N S

Let A be an open subset of C, S a subset of A, all points of which are isolated. A mapping f of A - S into a complex Banach space E is said (by abuse of language) to be meromorphic in A if it is analytic in A - S and has order > -co at each point of S. By abuse of language, we will always identifyfto its extension by continuity to all points of S which are not poles off; the argument used in (9.16.1) then shows that we can always suppose that for any compact subset L of A, L n S is jinite. Iff, g are two meromorphic functions in A, taking their values in the same space, and %whosesets of poles in A are respectively S, S‘, then S u S’ has all its points isolated, due to the preceding remark; f + g is defined and analytic in A - (S u S’), and has order > --GO at each point of S u S‘, hence is meromorphic in A (note that some points of S u S’ may fail to be singular for f + g). Similarly if f and g are meromorphic in A, and g is complex valued, f g is meromorphic i n A. Iff is meromorphic in A, S is the set of its poles, T the set of its zeros, then all the points of S u T are isolated; for if a E A and w(a) =h, then f ( z ) = ( z - a)hfl(z) in 2 set 0 < Iz - a1 < r, where A is analytic in ( z - a ( < r and f i ( a ) # 0; the principle of isolated zeros (9.1.5) shows that there is a number r‘ such that 0 < r‘ < r, and f ( z ) # 0 for 0 < Iz - a1 < r’. This proves our assertion, and shows moreover that L n ( S u T) is finite for any compact subset L of A (same argument as in (9.16.1)). In particular, iffis complex valued, l/fis meromorphic in A, S is the set of its zeros and T the set of its poles. Moreover, with the same notation as above, we have f ’ ( z ) = h(z - ~ ) ~ - l f , ( z () z - a)hf;(z) for 0 < Iz - a1 < r, hence f‘/f, which is analytic for 0 < I z - a1 < r‘, has order 0 at the point a if h = 0, order - 1 and residue h at the point a if h # 0.

+

(9.17.1) Let A be a simply connected domain in C , f a complex valued meromorphic function in A , S (resp. T) the set of its poles (resp. zeros), g an analytic function in A . Then, for any circuit y in A - ( S u T ) , we have

17 MEROMORPHIC FUNCTIONS

247

a$nite number of terms only being # 0 on the right-hand side.

This follows at once from the theorem of residues, for the residue of gf ’If at a point a E S u T is the product of g(a) by the residue off’/ at the point a. (9.17.2) W i t h the assumptions of (9.17.1) let t --t y ( t ) ( t E 1) be a circuit in A - ( S u T). Zfr is the circuit t +f ( y ( t ) ) ,then

For it follows at once from (8.7.4) that

hence the result is a particular case of (9.17.1) for g = 1. (9.17.3) (Rouchk’s theorem) Let A c C be a simply connected domain, f , g two analytic complex valued functions in A. Let T be the (at most denumerable) set of zeros o f f , T’ the set of zeros o f f + g in A, y a circuit in A - T, dejned in an interval 1. Then, if Ig(z)l < 1 f ( z ) l in y(I), the function f g has no zeros on y(l), and

+

+

The first point is obvious, since f ( z ) g(z) = 0 implies If ( z ) l = Ig(z)l. The function h = ( f + g)/f is defined in A - T and meromorphic in A ; we have

(-f + s > ’- -f’ h’ f+s f+h

in A -(T u T’).

Using (9.17.2), all we have to prove is that the index of 0 with respect to the circuit r : t+h(y(t)) is 0. As gif is continuous and finite in the compact set y(I), it follows from (3.17.1 0) and the assumption that r = sup Ig(z)/f(z)I < 1. In other words, is in the ball Iz - 1 I < r, and as SY(1)

0 is exterior to the ball, the result follows from (9.8.5).

248

IX ANALYTIC FUNCTIONS

(9.17.4) (Continuity of the roots of an equation as a function of parameters) Let A be an open set in C,F a metric space, f a continuous complex valued function in A x F, such that for each a E F, z +f ( z , a) is analytic in A. Let B be an open subset of A, whose closure B in C is compact and contained in A, arid let E F be such that no zero o f f ( z , ao) is on the frontier of B. Then there exists a neighborhood W of a. in F such that: (I) for any a E W, f ( z , a ) has no zeros on the froritier of B ; (2) for any a E W, the sum of the orders of the zeros o f f ( z , a ) belonging to B is independent of a. The number of distinct zeros of f ( z , ao) in B is finite; let a,, . . . , a, be these points. For each frontier point x of B, there is a compact neighborhood U, of x , contained in A, such that f ( z , ao) has no zero in U, (9.1.5); if we cover the (compact) frontier of B by a finite number of sets U x j , the union U of 8 and the U,, is a compact neighborhood of B, contained in A and such that f ( z , ao) has no zero in U n (A - 8).Let r be the minimum of the numbers la, - ajl (i # j ) , and for each i (1 < i < n), let Di be an open ball Iz - ail < ri of radius r, < r/2, contained in B ; then D i n Dj= @ if i # j . Let H =U Di); this is a compact set; let m be the minimum value of I f ( z , ao)l in H ; we have m > 0 by (3.17.10). Now, for each X E 8, there is a neighborhood V, of x contained in A and a neighborhood W, of a. in F, such that I f ( y , a)- f ( x , a,)l < m/2 for y E V, and a E W,. As 8 is compact it can be covered by a finite number of sets VXk (1 < k G p ) ; let W = WXk;this is a neighborhood of a. in F, and by definition, for any

-(u

k

W and any y E 8, we have If ( y , a ) -f ( y , ao)l < m. As a first consequence, it follows that f ( y , a) # 0 for y E H and a E W ; on the other hand, as I f ( z , a ) - f ( z , a0)l < If ( z , ao)l in H, RouchC’s theorem, applied to each circuit t + a, rieit (0 < t G27r) shows that the sum of the orders of the zeros off (z, a ) in Di is independent of a E W, hence the theorem. CY

E

+

PROBLEMS

1. Let A c C be an open simply connected set,fa nieroniorphic complex valued function in A, such that each pole off is simple and the residue of f a t each of these poles is a positive or negative integer. Show that there is in A a meroinorphic function g such thatf= g’/g. (If zo is not a pole off, show that for any point z1 E A which is not a pole off, and any road y in A, defined in I c R,of origin zo and extremity zl, and such that y(1) does not contain any pole off, the number exp 2.

0)

v)dx onlydependsonzoandzl,

and not on the road y satisfying the preceding conditions (use the theorem of residues).) Let f be an entire function of one complex variable, such that for real x , y , llf(x i-iy)ll < e l y 1 .Show that, for any z distinct from integral multiples nn of n,

17 MEROMORPHIC FUNCTIONS

249

(- 1)"f(nn)

+

(z

-m

-

m)'

where the series on the right-hand side is normally convergent in any compact subset of C which does not contain any of the points nn (n integer). (Consider the integral

's f(4 2ni

Yn

dx -sin x ( x - z)'

+

where yn is the circuit t + (n $ ) r e " for -n < t < n.Observe that for every E > 0, there is a number C ( E ) > 0 such that the relations Iz - n7r 2 E for every integer n E Z imply lsin zI > c(c)el,F(z)l;and use the theorem of residues.) 3. (a) Show that for z # nn (n integer) 1 cotz=-+C z

22 z2-n2nz ~

"=I

where the right-hand side is normally convergent in any compact subset of C which does not contain any of the points nn. (Use Problem 2 and the relation lim (cot z - l/z) = 0.) I-0

(b) Deduce from (a) that

, the convergence being

where the product is defined as the limit of k=l

uniform f(z)

in

=n(1 +

01)

-m

every

4

--

compact

subset

of

C.

(Consider

the

entire

function

e - z / n n (Section 9.12, Problem l), and use (a) to prove that the

function (sin z ) / z f ( z ) is a constant.) (c) Deduce from (b) the identity

4.

(see Section 9.12, Problem 2). Let f be a complex valued function analytic in an open neighborhood A of 0 in C p ; for convenience we write w instead of zp and z instead of ( z l , . . . , z ~ - ~Suppose ). that f ( 0 , O ) = 0 and that the function f ( 0 , w ) , which is analytic in a neighborhood of w = 0 in C , is not identically 0. Then there exist an integer r > 0, r functions h,(z) analytic in a neighborhood of 0 in C P - l ,and a function g(z, w ) analytic in a neighborhood B c A of 0 in C p and # O in that neighborhood, such that f ( z , w ) = (w'

+ h,(z)w'-1 + .. . + h,(z))s(z, w )

in a neighborhood of 0 in Cp (the " Weierstrass preparation theorem"). (If f ( 0 , w) has a zero of order r at w = 0, use (9.17.4) to prove that there is a number E > 0 and a neighborhood V of 0 in C p - l such that for any Z E V, the function w 4 f ( z , w ) has exactly r zeros in the disc lwI < E and no zero on the circle Iwi =- E . Let y be the circuit t + w i t for -n < t < n;using the theorem of residues, show that there are

250

IX ANALYTIC FUNCTIONS

functions h,(z) (1 < j < r) analytic in V and such that the polynomial F(z, w) = wr+ hl(z)w'-' ... h,(z) satisfies the identity

+ +

for Z E V and IwI > E). Let (f.) be a sequence of complex valued analytic functions in a connected open subset A of C.Suppose that for each z E A, the sequence ( f . ( z ) )tends to a limit g(z), and the convergence is uniform in every compact subset of A. Suppose in addition that each mapping z -.f.(z) of A into C is injective. Show that either g is constant in A or g is injective (for any zo E A, consider the sequence (f.(z) -f.(zo)) and apply (9.17.4) and the principle of isolated zeros). Let be a real valued twice differentiable function in the interval 10,11. Suppose ) q#)cos x = 0 in 1-77, n [ . Iq~(0)l< Iv(l)l, and let xo be one of the zeros of ~ ( 0 Show that the entire function F(z) =

IO1

p(r) sin zt df

has a denumerable set of zeros; furthermore it is possible to define a surjective mapping n -.z, of Z onto the set of zeros of zF(z), such that each zero corresponds to a number of indices equal to its order, and that lim ( z ~ ~ - x "2n7r)=

+

n-fm

lim ( z ~ ~ xo+ -~2n7r) = 0. (Integrating twice by parts, show that one can

n-fm

+

zF(z) = ~ ( 0) p(1) cos z G ( z ) , where IG(z)l < u e l f l ( z ) l / l z / ;minorize j ~ ( 0) ~ ( 1 )cos zI outside of circles having centers at the zeros of that function, in

write

the same was as lsinzl was minorized in Problem 2 ; and use RouchC's theorem in a suitable way.) Treat similarly the cases in which I ~ ( 0 )> l Ip(1)l or ly(0)l = /q~(l)j.

APPENDIX TO CHAPTER IX

APPLICATION OF ANALYTIC FUNCTIONS TO PLANE TOPOLOGY (Eilenberg’s Method)

1. I N D E X OF A P O I N T WITH RESPECT TO A L O O P

(Ap.1.1) If t + y(t) (a < t < b) is a path in an open subset A of C , there is in A a homotopy cp of y into a road yl, such that cp is defined in [a, b] x [0, 13 and ~ ( a5), = y(a) and q(b, t) = y(b)for every 5 E [0, I ] .

Let I = [a, b ] ;as y(1) is compact, d(y(I), C - A) = p is > 0 (3.17.11). Asy is uniformly continuous in I (3.16.5), there is a strictly increasing sequence (tk)06ksmof points of I such that to = a, t, = b, and the oscillation (Section 3.14) of y in each of the intervals [tk, t,,+l] (0 f k f m - 1) is < p. Define (0 < k < m - 1); it is clear that y1 is a road, with yl(a) = y(a), y,(b) = y(b), and yl(I) is contained in A, since y l ( [ t k , tk+,])is contained in the open ball of center Y(tk) and radius p. Define then cp(t, t) = ty,(t) (1 - 0. Let A‘, H’ be the images of A, H under the homeomorphism Z - + S , , ~ ( Z ) of C - {b} onto C - { 1) ; H’ is compact and A‘ is a connected open subset of C - H’, which is bounded and contains 0. Moreover, the frontier points of A‘ in C are points of H’ and (possibly) 1 ; hence A’ is compact and so is A’ u H’. In addition, if 1 belongs to the boundary of A’, this means that A is unbounded, hence has points in common with the exterior of a ball containing H ; but as that exterior is connected (9.8.4), it is contained in A by definition of a connected component (Section 3.19). This shows that there is a ball V of center I , such that V - (1) c A’, hence 1 is not a frontier point of C - A‘, which proves that the frontier of C -A’ is always contained in H’. We have to show that the mapping u -+ u/lul of H’ into U is essential (Ap.2.2). Suppose the contrary; then there would exist a continuous mapping f of H’ into R such that u/lul = eif(”) for u E H’. By the Tietze-Urysohn theorem (4.5.1), f can be extended to a continuous mapping g of A’ u H’ into R. Define a mapping h of C into U by taking h(u) = u/lul for u E C -A‘, h(u) = eig(”)for u E A’; it follows at once from the definition of g that h is continuous in C. Let r > 0 be such that A‘ is contained in the ball B : IzI < r ; the restriction of h to B is inessential (Ap.2.6), and so is therefore the restriction of h to S : IzI = r. But the identity mapping [ -+ [ of U onto itself can be written h, gl,where h, is the mapping z -+ z/lzl of S onto U, and g1 the mapping -+ rc of U onto S. However, h, is the restriction of h to S, hence inessential, and therefore h, 0 g1 would be inessential (Ap.2.2), contradicting (Ap.2.9).

c

0

(Janiszewski’s theorem) Let A, B be two compact subsets of C, a, b two distinct points of C - (A u B). If neither A nor B separates a and b, and if A n B is connected, then A u B does not separate a and b.

(Ap.3.2)

From the assumption and (Ap.3.1) it follows that the restrictions of b(z)/Is,, b(Z)I to A and B are inessential; by (Ap.2.7) the restriction of that mapping to A u B is also inessential, hence the conclusion by (Ap.3.1).

z -+ s,,

4. SIMPLE ARCS A N D SIMPLE C L O S E D C U R V E S

An injective path t -+ y(t) in C, defined in I = [a, 81, is also called a simple path; a subset of C is called a simple arc if it is the set of points y(I) of a simple path. A loop y defined in I is called a simple loop if y(s) # y ( t )

256

APPENDIX TO CHAPTER IX

for any pair of distinct points (s, t ) of I, one of which is not an extremity of 1. A subset of C is called a simple closed curue if it is the set of points of a simple loop. Equivalent definitions are that a simple arc is a subset homeomorphic to [0, 11, and a simple closed curve a subset homeomorphic to the unit circle U (9.5.7).

The cornplernerit in C of a simple arc is corinected (in other words, a simple arc does not cut the plane).

(Ap.4.1)

Let y be a simple path defined in I, and let f be the continuous mapping of y(1) onto I, inverse to y . Let a, b be two distinct points of C - ?(I). By (Ap.3.1), we have to prove that the restriction cp of z -+ sa, b ( ~ ) / I ~ , , &z)I to y(1) is inessential. But we can write cp = (cp y) o f ; the continuous mapping cp y of I into U is inessential (Ap.2.6), and so is therefore cp by (Ap.2.2). 0

0

(The Jordan curve theorem) Let H be a simple closed curue in C. Then : (a) C - H has exactly two connected components, one of which is bounded and the other unbounded. (b) The frontier of every connected component of C - H is H. (c) I f y is a simple loop dejned it7 I and such tl7at y(1) = H, then j ( x ; y) = 0 i f x is in the unbounded connected cotnponent of C - H, and j ( x ; y ) = f.1 i f x is in the bounded connected component of C - H . (Ap.4.2)

The proof is done in several steps.

(Ap.4.2.1) We first prove (b) izithocrt any assumption on the number of components of C - H. Let A be a connected component of C - H ; as C - H is open, we see as in (Ap.3.1) that the frontier of A is contained in H. Let z E H , and let f be a homeomorphism of U onto H ; let [ = eie E U be such that f ( [ ) = z. Let W be an arbitrary open neighborhood of z in C, V c W a closed ball of center z ; then there is a number w such that 0 < w < n and that f ' ( e " ) E V for 0 - w < t < 0 + w ; let J be the image of that interval by t + f ( e " ) ; then the complement L of J in H is the image by t +f'(eit) of the compact interval [0 + w - 27t, 0 - w ] (9.5.7), and is a simple arc by (9.5.7). It follows from (Ap.4.1) that the open set C - L 3 C - H is connected. Therefore (9.7.2) for any x E A c C - L, there is a path y in C - L, defined in I = [a, b ] , such that y(a) = x,y(b) = z.

4

SIMPLE ARCS A N D SIMPLE CLOSED CURVES

257

The set y(l) n J is compact and contained in V ; let M be its inverse image by y, which is a compact subset of I , such that a 4 M ; let c = inf M > a. Then the image by y of the interval [a, c[ is a connected set P ((3.19.7) and (3.19.1)), which does not meet J nor L, hence is contained in C - (J u L) = C - H ; as P contains x, it is contained in A by definition. But when t < c tends to c, y ( t ) E A tends to y(c) E V, hence y ( t ) E W as soon as c - t is small enough; this shows that z E A. Q.E.D. (Ap.4.2.2) We next prove the theorem under the additional assumption that H contains a segmtvit S with distinct extremities. Applying to C a homeomorphism z +Az I- ,u, we can suppose S in an interval [ - a , a ] of the real line R. Let p = d(0, H - S) d a, and consider an open ball D: IzI < r , with r < p ; then D n (C - H) = D n (C - S), and it is clear that D n (C - S) is the union of the two sets D,: IzI < r , 4 z > 0 and D,: Jz)< r , f z < 0, which have no common points. It is immediately verified that the segment joining two points of D, (resp. D2) is contained in D, (resp. DJ, hence (3.19.3) that D,, D, are connected. On the other hand, we have seen in (Ap.4.2.1) that every connected component of C - H meets D, hence meets D, or D.,; but if two connected components of C - H meet D, (resp. D2), they are necessarily identical, since D, (resp. D,) is connected and contained in C - H (3.19.4). This proves that C - H has at most t ~ v oconnected components. We prove next that C - H is not conriected, hence has exactly fbvo components. Suppose the contrary, and let x E D,, y E D,; as D is connected, C - D does not separate x and y ; on the other hand, if C - H is connected, H does not separate x and y . But H n (C - D) is the complement in H of the open interval 1-r, r [ ; by (9.5.7), this complement is therefore a simple arc, hence connected. By Janiszewski's theorem (Ap.3.2), the union H u (C - D) does not separate x and y ; but this is absurd, since the complement of H LJ (C - D) in C is D, u D,, and D,, D, are open sets without common points, hence D, LJ D, is not connected. As H is compact, it is contained in a ball of center 0, whose complement in C is connected, hence contained in a connected component of C - H ; this shows one of these components A is unbounded, and the other B is bounded. Moreover, it is clear that j(x;y) = 0 when x E A (9.8.5). On the other hand, D, is contained in one of the components of C - H, D, in the other; all we need to prove therefore is that j ( x l ; y) - j ( x 2 ;y) = 5 1 for one point x, E D, and one point x, E D, (9.8.3). Supposing the origin of y to be the point a E S, let J c I be the inverse image by y of H - S, which is a compact interval [s(, p] and let y, be the path t + y ( t ) defined in J, of extremities - a and a. By (Ap.1.1) there is a homotopy 'p, in C - D of

258

APPENDIX TO CHAPTER IX

y, into a road y 2 , such that cp, is defined in J x [0, I ] and cp,(cc, t) = y(cc), q l ( p , t) = y(P) for any 5. Define cp in I x [0, I ] as equal to cpl in J x [0, I ] and to y ( t ) for any ( t , 5 ) ~ ( -1 J) x [0, I ] ; then for any x1 E D , (resp. x2 E D2), cp is a loop homotopy in C - {x,} (resp. C - {x2})of y into a circuit y 3 . We can therefore limit ourselves to proving that j ( q ; y) - j ( x 2 ;y ) = +_ 1 when y is a circuit defined in I, having the following properties: (1) S c ?(I) and if T is the inverse image y - ’ ( S ) , then T is a subinterval of I and the restriction of y to T is a homeomorphism of T onto S; ( 2 ) y(I - T) is contained in C - D (note that perhaps this new y is not a simple loop). Then the inverse image by y of the interval [ - r , r ] is a subinterval [A, p ] of T ; suppose for instance that ?(A) = - r , y ( p ) = r . We can suppose (replacing y by an equivalent circuit) that I = - 7 1 , p = 0 , and moreover that - r is the origin of y, so that I = [ - 7 1 , w ] with w > 0. Take xl = i t , x2 = -it with 0 < 4 < r ; let cr be the road t + y ( t ) , - 7 1 d t < 0, 6, the road t -+ rei‘, --n b t b 0, 6, the road t -+ r e - i t , - n < t < 0. Then, Cauchy’s theorem applied in the halfplane 4 ( z ) < [ (resp. 9 ( z ) > - 4 ) which is a star-shaped domain (9.7.1) yields dz

dz

and

dz z

+ it‘

Hence

Now the left-hand side is independent of 5 , and when t tends to 0, the righthand side tends to 2ni, using the fact that I y ( t ) l > r for 0 ,< t b w, the mean value theorem (to majorize the last integral), and (8.11.1).

(Ap.4.2.3) We now turn to the case in which H contains no segment with distinct extremities. Let a, b be two distinct points of H, S the segment of extremities a, b ; we may again suppose that S is a closed interval in R. By assumption, there is at least one point x E S n (C - H); let J be the connected component of x in S n (C - H), which is an open interval ] y , z [ since S n (C - H) is open in R ((3.19.1) and (3.19.5)); moreover its extremities y , z are in H. Let g be a homeomorphism of H onto the unit circle U, and let g(y) = e”, g(z) = eid, where we may suppose that c < d < c 271 (9.5.7). Let U,, U, be the simple arcs, images of t -+ e i f , c < t b d , and t -+ eit, d d t < c + 27r, and let H,, H, be their images by the homeomorphism f of U onto H, inverse to g. Using (9.5.7), we see immediately that there is a

+

4 SIMPLE ARCS AND SIMPLE CLOSED CURVES

259

homeomorphismf, (resp.f,) of U, (resp. U,) onto the closed interval J = [ y , z ] , such that f,(eic) =f2(ei‘) = y , f i ( e i d )= f 2 ( e i d )= z. Let h, (resp. I?,) be the mapping of U into C, equal to f in U, (resp. in U,), tof, in U, (resp. to& in U,); the definition of J implies that h,, h, are homeomorphisms of U onto two simple closed curves GI = HI u J, G , = H, u J, each of which contains the segment J. Let w E H,, distinct of y and z ; there is an open ball D of center w, which does not meet the compact set G, . From (Ap.4.2.1), each connected component of C - G, has points in D; moreover, if id, wf’ are two points of D in a same connected component of C - GI, iv’ and M”’ are not separated by GI ; they are not separated either by G, , since they belong to D c C - G, which is connected. But GI n G, = J is connected, hence, by Janiszewski’s theorem (Ap.3.2), iv’ and W” are not separated by GI u G , , nor of course by H c G, u G, . In other words, id and w” belong to the same connected component of C - H. But as C - G, has exactly two connected components, and each connected component of C - H has points in D by (Ap.4.2.1), it follows that C - H has at most two connected components. On the other hand, it follows from (Ap.4.2.2) that there are two points NJ’, iv” in D which are separated by G,. We show they are separated by H. Otherwise, as they are not separated by G, , and G, n H = H, is connected, they would not be separated by G, u H 3 GI (Ap.3.2), contrary to assumption. We have thus shown that C - H has exactly two connected components; the same argument as in (Ap.4.2.2) proves that one of them, A, is unbounded and the other, B, is bounded. Finally, we can suppose y is the origin of the loop y, and, if I = [a, PI, that HI = y([a, A]), H, = y ( [ A , /I]). Define the loops y1 and y, as follows: yl(t) = ( t - a + l)(y - z ) + z for a - 1 < t < M, yl(t) = y ( r ) for a < t < A; y 2 ( t ) = y ( t ) for A < t < /I, y 2 ( t ) = y ( t - P)(z - y ) for P < t < P 1. Using (Ap.1 .I) it is immediately verified that for any point x # G, u G , , j ( x ; y ) = ,j(x; yl) + j(x; y,). With the same meaning as above for D, let again M“, w’’ be two points of D separated by GI; then we have j ( d ; y,) = j ( w ” ; y,) since iv’ and KJ“ are not separated by G, (9.8.3), and j ( s ’ ;y , ) - j ( w ” ; 7,) = k 1 by (Ap.4.2.2). From this it follows that j ( w ’ ; y) - ,j(it,”; y ) = k I , which ends the proof.

+

+

(Ap.4.3) Let H be a simple closed curve in C, D the bounded connected component of C - H. Then, for any loop y in D, j ( x ; y ) = 0 f o r any x E H.

Let U be an open ball of center x, having no common points with the set y(1) of points of y. There exists in U a point z E C - (D u H) = C - D (Ap.4.2), and as U is connected, j ( x ; y) = j ( z ; y ) (9.8.3). But j ( z ; y) = j ( y ; y)

260

APPENDIX TO CHAPTER IX

for all points y of the unbounded connected component C - of H (9.8.3), and there are points y E C - r> which are exterior to a closed ball containing y(1); for such points, j ( y ; y) = 0 (9.8.5),hence the result.

PROBLEMS

1. Let A be a connected open subset of C; show that for any two points a , h of A, there is a simple path y contained in A , having a and b as extremities, and whose set of points is a broken line (Section 5.1, Problem 4; this amounts to saying that y is piecewise linear). (Use a similar argument as that in (9.7.2).If a “ square” Q = 1 x 1 c A (I closed interval with nonempty interior in R) is such that a @ Q, and there is a simple path t + y l ( t ) in A, defined in J c R, with origin a and extremity c E Q, consider the E Q, and observe that the segment of extremities smallest value f o E J such that rl(ro) y l ( t o )and any point of Q is contained in Q.) 2. Is Janiszewski’s theorem still true when A and B are only supposed to be closed subsets of C, even if A n B is compact (and connected)? Show that the statement of the theorem remains true in the two following cases: (1) A, B are two closed sets, one of which is compact; (2) A and B are two closed sets without a common point. (If c is a point sufficiently close to a , consider the mapping z+ l/(z - c), and the images of a, b, A and B under that mapping.) 3. For any simple closed curve H in C, denote by P(H) the bounded component of C H. (a) Let A be a connected open subset of C, H a simple closed curve contained in A. Show that A - H has exactly two connected components, which are the intersections of A and of the connected components of C - H (use Problem 2). (b) More generally, if H i (1 < i < r ) are r simple closed curves contained in A, and such that no two of them have common points, the complement of H i in A has ~

u

+

I

exactly r 1 components (use induction on r ) . (c) If H, H’ are two simple closed curves without a common point in C, show that or the closure of one of the sets P(H), P(H’) is contained in either P(H) n P(H’) = 0, the other. (Observe that if H c P(H’), the unbounded component of C - H’ has no common point with P(H), using (3.19.9).) (d) Suppose a connected open subset T of C has a frontier which is the union of r simple closed curves H, ( I < i < r ) , no two of which have common points. Show that there are only two possibilities: (1) T is unbounded and no two of the sets P(H,) have common points, their union being the complement of T; (2) there is one of the H , , say are contained in P(H,) for 1 < i < r - 1, no two of the -H , , such that the /$Hi) (1 < i < r - 1) have common points, and T is the complement of the union of the ,&H,) (for 1 < i < Y - 1) in P(H,). (If y , is a simple loop whose set of points is H i (1 < i < v ) , observe that the indices j ( x ; y , ) are constant for x E T, and that at most one of them may be # 0; otherwise, using (c), show that one at least of the H i would not be contained in the frontier of T.) 4. Let A be a bounded open connected subset of C, such that for any loop y in A and any z E C - A , j ( z ; y ) = 0. (a) Show that for any simple closed curve H c A, the bounded component P(H) is contained in A. (Observe that otherwise it would contain points of C - A, using (3.19.9) and part (b) of the Jordan curve theorem.)

B(H,)

4

SIMPLE ARCS A N D SIMPLE CLOSED CURVES

261

(b) Let ( z 12’) be the euclidean scalar product xx’ 4 yy’ in the plane C = R2 (with z = x -t iy, z’ = x’ -t if). Let Po be the open hexagon defined by the relations

l(zl I ) I < 4,

l(zl eini3)1< 4,

I(zl e-ini3)l< 3.

For any number a > 0, the set of all hexagons aP,. deduced from aPo by all translations of the form me'"'^ I- n), where m and n are arbitrary integers in Z , is called an hexagonalnet of width a ; the sets aP,,,, (resp. aFn,,,)are called the open (resp. closed) meshes of the net; the boundary of aP,,,, is the union of 6 segments (the sides of UP,,), whose extremities are called the vertices of ctP,,,; the nodes of the hexagonal mesh are all the vertices of the meshes. Every node is a vertex of three meshes, and the other extremities of the three sides issuing from that node are called its neighboring nodes. Let B be the union of a finite number of closed meshes of an hexagonal net. Show that if a node belongs to fr(B), there are exactly two neighboring nodes which also belong to fr(B); conclude that fr(B) is the union of a finite number of disjoint simple closed curves, each of which is a union of sides of meshes of the net. (Start with two neighboring nodes a , , a , in fr(B), and show that one may define by induction a finite sequence (a,) of nodes belonging to fr(B), such that a, and a,,, are neighboring nodes for every n.) (c) Let B be the union of a finite number of closed meshes of an hexagonal net of width a , such that fr(B) is a simple closed curve (union of sides of meshes of the net). Let a , h be two neighboring nodes on fr(B); prove that there exists a continuous mapping (z,t ) --z ~ ( zt .) of B x [O, I] into B such that: ( I ) ~ ( z0). z in B; (2) ~ ( zt ,) = z for every t E [O, I ] and every z on the segment S of extremities a and b ; ( 3 ) ~ ( z1), c S for any z E B. (Use induction on the number N of meshes contained in B; consider one side of extremities c, d, contained in fr(B), such that c and d have the same first coordinate, equal to the supremum of pr,(B), and that d has the largest second coordinate among all the nodes in fr(B) having that supremum as first coordinate; show that if N > I , B = B, u P, where P is the unique mesh contained in B and having c and d as vertices, B, is the union of N 1 meshes, and has in common with P one, two or three sides; examine the various possibilities.) Prove that the interior of B is simply connected. (d) Let B be the union of all closed meshes of an hexagonal net of width a , which are contained in A; a is taken small enough for B to be nonempty. Let D be one of the (open) connected components of h ; show that fr(D) is a simple closed curve. (Use (a) and Problem 3(d), to prove that if fr(D) was the union of more than one simple closed curve, there would be simple loops y in A and points z E fr(A) such that j ( z ; y ) = 1 . ) (e) Conclude that A is simply connected, and is the union of an increasing sequence (D,) of open simply connected subsets, each of which is the bounded component of the complement of a simple closed curve (use (c) and (d)). Conversely, such a union is always simply connected. (f) Extend the result of (e) to arbitrary simply connected open subsets of C (for each n, consider the closed hexagons of the hexagonal net of width I / n which are contained in the intersection of A and of the ball B(0; n ) ) . (g) Let A be an open connected subset of C such that the complement C - A has no bounded component; show that A is simply connected (use (9.8.5)). (h) Prove that any connected component of the intersection of a finite number of open simply connected sets in C is simply connected. 5. Show that the following open subsets of C are simply connected but that their frontier is not a simple closed curve: (I) ThesetA, ofpointsx-I i y s u c h t h a t O < x < l , -2 O such that Iy(t)l < p. for (tl < an,and p n + I= inf

(. -; 1

8 . 3 9

where 6,, is the distance of 0 to the set of points y(r) such that (11 2 c(.. (a) Prove that if z, z’ are two points of P(H) such that IzI < P . + ~ and J z ’ 1)

which is defined in a neighborhood of 0 and such that lim(f(x)-L(x))/llxll = O . X-0

Furthermore, that solution is an entire function in Kp. (Apply Problem 10 in a neighborhood of 0; reduce the problem to the case K = C, and apply (9.4.2) and (9.12.1) to prove that f i s an entire function.) (b) Show that there is no solution of the equationf(x) - hf(x/h)= x (h > 1 ) defined in a neighborhood of 0 in R and such thatf(x)/x is bounded in a neighborhood of 0. 12. Let I = [O, a ] , H = [ - b , b ] , and let f be a real valued continuous function in I x H; put M = sup If(x, y)l, and let J = [0, inf(a, b / M ) ] . (X.Y) E

1X H

(a) For any x E J, let E(x) be the set of values of y E H such that y = xf(x, y). Show that E(x) is a nonempty closed set; if g l ( x ) = inf(E(x)), g2(x) = sup(E(x)), show that gl(0) = gz(0)= 0, and that lim gl(x)/x= lim g2(x)/x =f(O, 0). x-o.x>o

X+O.X>O

If g1= gz = g in J, g is continuous (cf. Section 3.20,Problem 5). (b) Suppose a = b = 1 ; let E be the union of the family of the segments S. : x = 1/2", 1/4"+'< y 6 1/4" (n 2 0), of the segments S. : y = 1/4, 1/2"< x C 1/2"-' (n 3 ])and of the point (0,O). Define f ( x , y ) as follows: f ( 0 , y ) = 0; for 1/2"< x C 1/2"-' and y S 1/4", takef(x, y ) = ( ( y / x ) d((x, y), E))+ ;for 1/2"< x C 1/2"-' and 1/4" < y < xz, take f ( x , y ) = ( y / x ) - d((x, y), E) and finally, for 1/2"< x < 1/2"-' and y C x 2 , takef(x, y ) = x - d((x, xz), E) (n 3 I). Show thatfis continuous, but that there is no function g, continuous in a neighborhood of 0 in I and such that g ( x ) = xf(x, g(x)) in that neighborhood. (c) Let u0 be a continuous mapping of J into H, and define by induction u.(x) = x f ( x , U " - ~ ( X ) ) for n 3 1 ; the functions u, are continuous mappings of J into H. With the notations of (a), suppose that in an interval [0, c ] C J, lim ( U , , + ~ ( X )- u.(x)) =

+

"-m

0 for every x ,

and g l ( x ) =gz(x); show that lim u,(x) = g l ( x ) for 0 < x C c. n-ro

Apply that criterion to the two following cases: (1) there exists k > O such that If(x, zl) - f ( x , zz)[ C k Izl - z21 for x E I, zl, zz in H (compare to (10.1.1)); (2) for 0 < x < y < a and zl, zz in H, ]f(x,ZI)-f(x, Z Z ) ~< I Z I - JZ(/X. (d) Whenfis defined as in (b), the sequence (u,(x)) is convergent for every x E I, to a limit which is not continuous. (e) Take a = b = 1, f ( x , y ) = y / x for O < x C 1,lyl < x 2 , f ( x , y ) = x for 0 < x < 1,y 2 x',f(x,y) = - x for 0 C x < I , y < -xz. Any continuous function g in I such that [g(x)l < xz is a solution of g ( x ) = xf(x, g(x)) although If(x, zl) - f ( x , zz)/ < Izl - zzI/x for 0 < x < 1, zI, zZ in H ; for any choice of u O ,the sequence (u.) converges uniformly to such a solution. (f) Definefas in (e), and let f l ( x , y ) = - f ( x , y). The function 0 is the only solution of g(x) = x f l ( x ,g(x)), but there are continuous functions uo for which the sequence (u,(x)) is not convergent for any x # 0, although Ifl(x, zl) - f i ( x , z2)I < [zI- zzI/x for O < x < l,zl,zzinH. 13. Generalize the results of Problems 12(a) and 12(c) when H is replaced by a disk of center (0,O) in R2 (use the result of Problem 3 of Section 10.2).

270

X

EXISTENCE THEOREMS

2. IMPLICIT FUNCTIONS

(The implicit function theorem) Let E, F, G be three Banach (10.2.1) spaces, f a continuously di'erentiable mapping (Section 8.9) of an open subset A of E x F into G. Let (xo ,yo)be a point qf A such that f (x,,yo) = 0 and that the partial derivative D , f (xo,yo) be a linear homeomorphism of F onto G. Then, there is an open neighborhood Uo of xo in E such that, for every open connected neighborhood U of xo , contained in Uo , there is a unique continuous mapping u of U into F such that u(xo) = y o , (x,u(x)) E A and f (x,u(x)) = 0 for any X E U. Furthermore, u is continuously di'erentiable in U, and its derivative is given by (10.2.1 .I ) u'(x) = - ( D , f (x, 4 x 1 ) ) - W l f (x,W ) ) . Let To be the linear homeomorphism D, f ( x o ,yo) of F onto G, T i 1 the inverse linear homeomorphism; write the relation f ( x , y ) = 0 under the equivalent form (1 0.2.1.2) y =y - Ti1 . f ( x .y ) and write g(x, y ) the right-hand side of (10.2.1.2). We are going to prove that it is possible to apply (10.1.1) to the mapping

(x'9 Y'>

+

S(X0

+ x', Yo + Y ' ) - Y o

of E x F into F, in a sufficiently small neighborhood of (0,O).As T;'. To = 1 by definition, we can write, for ( x , y,) and ( x , y,) in A,

-

(Dzf(x0 Y o ) ( Y , - Y 2 ) - (f(x, Y1) -f(& Y,))). Let E' > 0 be such that E(IZ';'(~ G 3; as f is continuously differentiable in A, it follows from (8.6.2) and (8.9.1) that there is a ball Uo (resp. V,) of center x , (resp. y o ) and radius c1 (resp. p) in E (resp. F) such that, for x E U, ,y1 E V, ,y 2 E Vo , we have g(x9 Y , )

- g(x,

Y2)

I l f ( x ,Yl)

=Ti'

*

3

-m,YZ) - D2f(xo Y o )

- Y2)ll EllYl - Y2II ; whence Ildx, Y1) - g(x, Y2)Il G E I I T i ' II * llYl - Yzll G -511Yl - Y2lI for any x E U, ,y , E Vo ,Y,EV,. On the other hand, g(x, yo) - yo = - Ti1- f ( x , yo); 3

*

(Y1

as f ( x o , yo) = 0 and f is continuous, we can suppose E has been taken small enough to have 11g(x, yo) - yoI) G p/2 for x E Uo . We can then apply (10.1.1 ), which yields the existence and uniqueness of a mapping u of Uo into V, , such that f ( x , u(x)) = 0 for every x E U,; asf(xo ,yo) = 0, this gives in particular u(xo)= y o ; finally u is continuous in U, . Next we prove that if U c Uo is a connected open neighborhood of xo , u is the unique continuous mapping of U into F such that u(xo) = y o , ( x , u(x))E A and f ( x , u(x)) = 0. Let v be a second mapping verifying these

2 IMPLICIT FUNCTIONS

271

conditions, and consider the subset M c U of the points x such that u(x) = u(x). This set contains x, by definition and is closed (3.15.1); we need therefore only prove M is open (Section 3.19). But by assumption, x -+ DZf(x, u(x)) is continuous in U, , hence (replacing if necessary U, by a smaller neighborhood), we can suppose that D2f(x, u(x)) is a linear homeomorphism of F onto G for x E U,, by (8.3.2). Let a E M ; the first part of the proof shows that there exists an open neighborhood U, c U of a and an open neighborhood V, c V of b = u(a) such that, for any x E U,, u(x) is the only solution y of the equation f(x, y ) = 0 such that y E V,, . However, as u is continuous at the point a, and u(a) = u(a), there is a neighborhood W of a, contained in U , and such that ~ ( x E) V, for x E W ; the preceding remark then shows that u(x) = u(x) for x E W, and this proves M is open, hence u = u in U. Finally we show that u is continuously differentiable in U, , provided E has been taken small enough. For x and x + s in U,, let us write t = u(x + s) - u(x); by assumption f(x s, u(x) + t ) = 0, and t tends to 0 when s tends to 0. Hence, for a given X E U , , and for any 6 > 0, there is r > 0 such that the relation llsll < r implies Ilf(x s, u(x) t ) -f(x, u(x)) - S(x) . s - T(x) tll < S(llsll Iltll) where S(x) = D,f(x, u(x)) and T ( x ) = D2f(x, u(x)) (8.9.1). This is equivalent by definition to

+

+

+

+

IIS(x) * s + T(X) * 41 < 6(llsll

+ Iltll)

and as T(x) is a linear homeomorphism of F onto G , we deduce from the preceding relation (10.2.1.3)

ll(T-'(x) 0 S(x)) * s + tll < ~ l l ~ ~ 1 ~ x ~+lIltll). l~~141

Suppose 6 has been taken such that 6llT-'(x)ll 0 such that Ilf'(xo) . s l l 3 cI/sI/ for all s E E (5.5.1)J 2. Letf= (ft,f2) be the mapping of RZ into itself defined byf,(xl, x2) = xl;f2(x1,x2) = x2 - x: for x: < x 2 , f2(xl, x 2 ) = (x: - x:x2)/x: for 0 < x2 < x:, and finally fz(x1. - XI) = -fz(xt, x2) for x2 0. Show that f is differentiable at every point of RZ;at the point (0, 0), Df is the identity mapping of RZ onto itself, but Df is not continuous. Show that in every neighborhood of (O,O), there are pairsof distinct points x', x" such that f ( x ' ) = f ( x " ) (compare to (10.2.5)).

274

X

EXISTENCE THEOREMS

+

3. Let B be the unit disc IzI f 1 in R2 and let z -f(z) = z g(z) be a continuous mapping of B into R2 such that lg(z)l < Izl for every z such that JzI= 1. Show that f(B) is a neighborhood of 0 in R2 (“Brouwer’s theorem” for the plane, cf. Chapter XXIV). (Let y be the loop t - f ( e “ ) defined in [0,27r]; show thatj(x; y ) = 1 for all points x in a neighborhood V of 0 (see proof of (9.8.3)); using the fact that, in B, y is homotopic to 0, deduce that there is no point of V belonging to the complement off(B).) 4. Let E, F be two Banach spaces, B the unit open ball /lx/l< 1 in E; let uo be a continuously differentiable homeomorphism of B onto a neighborhood of 0 in F, such that uo(0)= 0; suppose uo’ is continuously differentiable in a ball Vo : llyll< Y contained in uo(B), and Duo is bounded in B and Duo’ is bounded in V,. Let V be a ball llyll < p, with p < r. (a) Show that for any a < 1, there is a neighborhood H of uo in the space 9 P ) ( B ) (Section 8.12, Problem 8) such that for any u E H, the restriction of u to U: llxll< a is a homeomorphism of U onto an open set of F containing V, such that the restriction of u-’ to V is a continuously differentiable mapping @(u) of V into E. (Use (lO.l.l).) (b) Show that the mapping u @(u) of H into 9(EI)(V)is differentiable at the point uo ,and that its derivative at uo is the linear mappings + -(uk 0 cD(uo))-’ . (s 0 cD(uo)). 5. Let E, F be two Banach spaces, f a continuously differentiable mapping of a neighborhood V of xo E E into F. Suppose there are two numbers /?> 0, A > 0 such that: (I) Ilf(x0)II < /3/2h;(2) in the ball U: IIx - xoIJ< p, the oscillation off’ is < l/2A; (3) for every x E U,f’(x) is a linear homeomorphism of E onto F such that ll(f’(x))-’ I1 < A. Let (2,) be an arbitrary sequence of points of U ; show that there exists a sequence (x.).,~ of points of U such that x.+’ = x. - (J’(zn))-’ .f(x,) for n 2 0. Prove that the sequence (x,) converges to a point y E U, such that y is the only solution of the equationf(x) = 0 in U. (“Newton’s method of approximation.” Use (8.6.2) to prove by induction on n that IIx, - x.-’ I/ < 2-”p and Ilf(xn)II < /3/2“+‘X). 6. Let E, F be two finite dimensionalvector spaces over K, A a connected open subset of E, f a continuously differentiable mapping of A x F into F. Suppose that the set r of pairs (x, y ) E A x F such f(x, y ) = 0 is not empty, and that for any (x, y) F I?, D2f(x, y ) is an invertible linear mapping of F onto itself. (a) Show that for every point (xo ,yo ) E r there is an open neighborhood V of that point in r such that the restriction of the projection prl toVis a homeomorphism of V onto an open ball of center xo contained in A. (Use the fact that there is an open ball U of center xo in A and an open ball W of center yo in F such that for each x E U, the equationf(x, y) = 0 has a unique solution y E W, and apply (10.2.1).) (b) Deduce from (a) that every connected component G of r (Section 3.19) is open in I’ and that pr,(G) is open in A. It is not necessarily true that pr,(I’) = A (as the example A = E = F = R,f(x,y) = xy2 - 1 shows), nor that if prl(r) = A, prl(G) = A for every connected component G of r (as the example A = E = F = R,f ( x , y ) = xy2 - y shows). Prove that if pr2(I’) is bounded in F, then pr,(G) = A for every connected component G of F. (If xo is a cluster point of pr,(G) in A, show that there is a sequence (x,, y.) of points of G such that lim x, = xo and that limy. exists in F; n-1 m n-m apply then (a).) (c) The notations of path. loop, homotopy, and loop homotopy in A are defined as in Section 9.6, replacing C by E. Suppose there is a connected component G of r such that prl(G) = A; if y is a path in A, defined in I = [a, b] c R,show that thereexists a continuous mapping u of I into G such that prl(u(r)) = y ( t ) for each t E I (consider the 1.u.b. c in I of the points 6 such that there exists a continuous mapping uc of [a, 61 into G such that prl(uc(r)) = y(f) for a < t < 5 and use (a)). Is that mapping always unique? (Consider the case E = F = C, A = C - { O } , f ( x ,y ) = y 2 - x . ) --f

2 IMPLICIT FUNCTIONS

275

Show that if two continuous mappings u, u of I into G are such that prl(u(t)) = prl(u(t)) = y(r) for each t E 1,and if they are equal for one value of t E I, then u = u (use a similar method). (d) Under the same assumptions as in (c), let v be a continuous mapping of I x J into A, where J = [c, d ] C R. Let u be a continuous mapping of J into G such that prl(u([)) = ~ ( a5), for 6 E J ; and for each 5 E J, let uc be the unique continuous mapping of I into G such that prl(uc(t)) = ~ ( t , tfor ) t E I and u&) = ~(6).Show that the mapping ( t , 5) + u,(t) is continuous in I x J. (Given 5 E J, there is a number r > 0 such that for any t E I, the intersection V, of I' and of the closed ball in E x F, of center u,(t) and radius r, is contained in G and such that prl is a homeomorphism of V, onto the closed ball in E of center y ( t ) and radius r. If L = u{(I), let M be the supremum of ll(D2f ( x , y))-' 0 ( D , f ( x , y))/I for all points ( x , y) E G at a distance < r of L. Let E > 0 be such that E < 1-14and E M < r/4. Show that if 6 is such that the relation 15 - 51 < 6 implies Ilq(t,5) - q ( t , 1) of complex numbers (which may be empty) such that /n,l < lanfl/,f(n.) = 0 and for every c E C such thatf(c) = 0, the number of indices n for which a, = c is equal to the order w ( c ; f ) ;when the sequence (an) is infinite, lim lanl = +co (9.1.5). Show (with the notations of Section 9.12, n-m

Problem 1 ) that there exists an entire function g such that

fi E (?, n a,

f ( z ) = egc2)

n=1

-

I),

(" Weierstrass decomposition ").

9. Let A and B be two open neighborhoods of 0 in E = C p ,A being connected; let ( x , y) + U(x,y) be an analytic mapping of A x B into Y(E; E) (identified to the space of p x p matrices with complex elements).

276

X

EXISTENCE THEOREMS

(a) Suppose there exists a sequence Cun) of analytic mappings of A into B such that u0 (x) = 0 in A and unCx) = U(x, lln-1(x)) · x in A for n ~ L Suppose in addition that for every compact subset L of A, the restrictions of the Un to L form a relatively compact subset of i- e(L). Prove that the sequence (un) converges uniformly in any compact subset of A to an analytic mapping v of A into B such that v(x) = U(x, v(x)) · x in A; furthermore, vis the unique mapping satisfying that equation (use (1 0.2.1) and (9.13.2)). (b) Suppose that in E, A and B are the open balls of center O and radii a and b. Let

p at every point x # xo of that neighborhood, as the example of the mapping (-Y, J,) (x2 - y 2 , xy) shows at the point (0,O). --f

(10.3.1) (Rank theorem) Let E be an n-dimensional space, F an m-dimensional space, A an open tieigliborlioocl of a point a E E, , f a continuously dtfferentiable mapping (resp. q times contiiiuously differeiitiable mapping, resp. indefinitely differentiable mapping, resp. analj!tic mapping) of A into F, such that in A the rank of f ' ( x ) is a constant number p , Then there exists: ( 1) an open neighborhood U c A ofa, and a homeomorphism u of U onto the unit ball I" : Ix,I < 1 ( 1 i t i ) in K", ttAic1i is continuously differentiable (resp. q times cotitinuously differentiable, resp. indefinitely differentiable, resp. analytic) as well as its inivrse ; (2) an open neighborhooci \I I> j ( U ) of b = f ( a ) and a homeomorphism u of the unit ball 1" : 1y,1 < I ( 1 i m> of K" onto V, which is continuously cliffi~rentiahle( resp. q times continuously differentiable, resp. indefinitely dtfferentiable, resp. analj~tic)as 1t.ell as its itirersc I. -such that f ' = I' f b u, n h w f o is the mapping

<
p . Let P be the image of E (and of N ) by the linear mappingf'(0); it is a p-dimensional subspace of F, having the elements di = f ' ( O ) * ci (1 Q i Q p ) as a basis; we take a basis ( d j ) l s j i , of F, of which the preceding basis of P form the first p elements, and we write y

m

=

$,(y)dj for any y

E

F, the

j=l

tjj being linear forms. We denote by y - + H ( y ) the linear mapping

y+

P

C tjj(y)ejof F onto the subspace KP of K" generated by the ei of index

j=l

i 0 be such that the ball lxil < r (1 < i < n) is contained in g(U,), and let U be the inverse image of that ball by g, which is an open neighborhood of 0 ; our mapping u will be 1

1

the restriction to U of the mapping x -+ - g(x). r

Up to now we have not used the assumption that the rank off'(x) is constant in A ; this implies that the image P, of E byf'(x) has dimension p for any x E A. Now we may suppose U, has been taken small enough so that g'(x) is a linear bijection of E onto K" for x E U, (8.3.2); as we have g'(x) * s = H ( f ' ( x ) . s) for s E N, the restriction of f ' ( x ) to N must be a bijection of that p-dimensional space onto P,, and the restriction of H to P, a bijection of P, onto KP. Denote by L, the bijection of KPonto P,, inverse of the preceding mapping; we can thus writef'(x) = L, H o f ' ( x ) . 0

3 THE RANK THEOREM

279

Now consider K" as the product El x E, with El = KP, E, = K"-P. we are going to prove that the mapping (zl, z 2 )+fl(zl, 2 , ) = f ( u - ' ( z l , z2)) of 1" into F does iiof depend 017 z2 , i.e. that D, fl(zl, z , ) = 0 in I" (8.6.1). By defini-

rf'(x> . t

'

i:

= f, - H(J'(x)),-

tion, we can writef(x)

r

= Dlfl(;

+ D2 f i for any t E

(10.3.11)

E. This yields D, f I

G(,4), hence by (8.9.2)

; ;w).

H ( f ( x ) ) , G(x)) . W f ' ( x ) . t )

(;

H(S(x)),

G(0

;

(i

H ( f ( x ) ) , G(x)) . G ( t ) = S, * H ( f ' ( x ) . t )

' 1

(i

, G(x) is a linear mapping of KP = El where S, = rL, - Dlfi - H ( f ( x ) ) r

into F. We prove that S, = 0 for any x E U,. Indeed, if t E N, we have G ( t )= 0 by definition, hence S, . H ( f ' ( x )* t) = 0 by (10.3.1.1). But t -+ H ( f ' ( x ) . t ) = g'(x) * t is a bijection of N onto El for x E U,, and this proves S, = 0. From (10.3.1.1) we then deduce

D, for any

f E

fl

(' r

I

H(J'(x)),- C(x)) r

. G(t) = 0

E; but G maps E onto E,, hence by definition,

which is a linear mapping of E, into F, is 0 for any x E U,, . The relation D,f,(z,, z 2 ) = 0 in I" then follows from the fact that

is a homeomorphism of U, onto an open set containing I". We can now write fl(zl) instead of f l ( z l ,z , ) and consider fi as a continuously differentiable mapping of El = KP into F; we then have f(x) = fl

(!r H(,f(,x-))) for

proves that y

x E U ; in other words, y

I

-+

=fl

- H ( y ) is a homeomorphism r

z1 -+fl(zl)the inverse homeomorphism.

(1. 1

- H ( y ) for y E ~ ( U )This . *

of f(U) orito Ip c El, and

280

X

EXISTENCE THEOREMS

Consider now K"' as the product El x E, with E, = K"-". Let T be the linear bijection of E, onto the supplement Q of P in F generated by cl,,,, . . . , d,,, , which maps the canonical basis of K'"-" onto dD+,,. . . , d,, . We define P ( Z ~ z,) , =,f,(z,) + T(z,) for z , E Ip, z 3 E it is obviously (8.9.1) n continuously differentiable mapping. By definition, we have H(P(z,,z 3 ) )= H(f,(z,)) = rz, ;hence the relation i ( z I , z,) = r ( z ; ,z ; ) implies z ; = z , , and then boils down to T(z,) = T(z;),which yields z3 = z ; ; therefore 2: is it?jectire. The relation S , = 0 proved above shows that for any z , E Ip,fS;(zl)= rL, where s is any point i n U such that f ( x ) =f,(z,); the derivative of I' at ( z , , 2 , ) is therefore the linear mapping ( t , , t,) + rL, . t , + T ( t 3 )((8.9.1) and (8.1.3)). But as the restriction of H to P, is injective, P, is a supplement of Q in F, hence r ' ( z l ,z,) is a litiear iiomeotiior~~hist~i of K"' onto F. For any point ( z , , z 3 ) E I"', there is therefore an open neighborhood W of that point in I" such that the restriction of to W is a homeomorphism of W onto an open subset r(W) of F, by (10.2.5). As in addition is injective, it is a homeomorphism of I"' onto the open subset V = r( I"'), whose inverse is continuously differentiable in V. The relation f = o f o I I then follows from the definitions. 11

11

11

0

(10.3.2) I f t l i e raiik of j ' ( a ) is equal to ti (resp. to n i ) , thn7 the coiiclusioti of (10.3.1) holds li'itiz p = n (resp. p = n i ) . Indeed, at the beginning of this section we have seen that there exists then a neighborhood of a in which the rank of ['(.Y) is > / I (resp. > t ? i ) , hence equal to ti (resp. to 171) since it is always at most equal to inf(in, t i ) (A.4.18).

PROBLEMS

1. Let E, F be two Banach spaces, A an open neighborhood of a point x o E E, f a continuously differentiable mapping of A into F. (a) Suppose f ' ( x o ) is a linear homeomorphism of E onlo its image in F; show that there exists a neighborhood U c A of .yo such that f is a homeomorphism of U onto f(U) (use Problem 3 of Section 10.1). (b) Supposef'(x,) is surjective; then there exists a number a .r0 having the following property: if Nisthekerneloff'(xo),for every s c E,one has 1 : f ' ( x O.)s I l 3 ( 1 . inf,It 1 .yii IS

N

(12.16.12). Show that thereexists a neighborhood V c A of xo such thatf(V) is a neighborhood o f f ( x o )in F (use Problem 8 of Section 10.1). 2. Let A be an open subset of Cp, and f a n analytic mapping of A into C". Show that if fis inrjecrioe, then the rank of Df(s) is equal t o p for every .Y F A. (Use contradiction, and induction on p ; for p = I, apply Rouche's theorem (9.17.3). Assume Df(o) has a rank < : p for some u t A ; show first that after performing a linear transformation in F, one may assume that, if f ( z ) ( f i ( z ) ,. . . , f p (z)), then D,/,(N)- 0, and if g ( z ) (f2(z),...,f, ( z ) ) , the rank of Dg(o) is exactly p - I; then there is a neighborhood ~

3 THE RANK THEOREM

281

U c A of a such that Dg(x) has rank p - 1 for x E U. Using the rank theorem (10.3.1), reduce the proof to the case in which n = O,X(z) zk for 2 < k < p.) Is the result still true when C is replaced by R ? 3. (a) Let A be a simply connected open subset of C, distinct from C, and let a, b be two distinct points of fr(A). (Appendix to Chapter IX, Problem 6 . ) There exists a complex-valued analytic function h in A such that ( h ( ~ )=) (z ~ - a ) / @- b) (Section 10.2,Problem 7); h is an analytic homeomorphism of A onto a simply connected open subset B of C (Problem 2 and (10.3.1)); furthermore, B n (-B) = 0, hence there are points of C exterior to B. (b) Deduce from (a) that there exists an analytic homeomorphism of A onto a simply connected open subset of C contained in the disc U : Iz/ < 1 , and containing 0. 4. (a) Let A be a simply connected open subset of C contained in the unit disk U : / z / < 1, containing 0, and let H be the set of all complex valued analytic functions g in A, such that g is an injecliue mapping of A into C, Ig(z)J < 1 , g(0) = 0 and g’(0) is a real number >O. For each compact subset L of A, the set H,. of the restrictions to L of the functions of H is relatively compact in Kc(L) (9.13.2). Show that the set of real numbers g’(0) (forg E H) is bounded (cf. proof of (9.13.1)); let be the 1.u.b. of that set. Show that there is a function go e H such that g,,(O) = h (use the result of Section 9.17, Problem 5). (b) Suppose g E H is such that g(A) # U, and let c E U -g(A). Replacing g by g1 defined by g,(z) = e-‘@g(ze‘@), one can assume, for a suitable choice of 8, that c is real and >O. There exists a function h which is analytic in A and such that 7

h W 2 = (c - g(z))/(l - CdZ)) and h(0) = 2 / c > 0 (same argument as in Problem 3(a)); show that the function g2 defined by

h(z) =

(dY-gz(z))/(l -dc92(z))

belongs to H, and that g;(O) > g’(0). (c) Conclude from (a) and (b) that the function go defined in (a) is an analytic homeoniorphism of A onto U; using Problem 3(b), this implies that for any simply connected open subset D of C, distinct from C, there is an analytic homeomorphism of D onto U (“Riemann’s conformal mapping theorem ”). 5. (a) Let f be a complex valued analytic function in the unit disk U: (z[< 1 such that f ( O ) = 1 and If(z)l < M in U; show that for ( z /< I/M, I f @ - I1 < Mlz( (apply Schwarz’s lemma (Section 9.5, Problem 7) to the function g(z)= M(f(z) - 1 )/(M2 -f(z))). (b) Let f be a complex valued analytic function in U such that f ( 0 ) = O,f’(O) = 1, if’(z)l < M in U ; show that for 1z/ < I/M, i f ( z ) - zI S MIz12/2 (apply (a) tof’). (c) Show that under the assumptions of (b), the restriction o f f t o the disk B(0; I/M) is an analytic homeomorphism of that disk onto an open subset containing the disk B(0; 1/2M) (apply Rouchk’s theorem ((9.17.3), using the result of (b)). (d) For any complex number a E U, let [ ( ( z )= ( z - n)/(Zz - I ) ; for any complex , Jg’(z)l(l - lzl’) = valued function f analytic in U, show that, if g(z) = = f ( i t ( z ) )then If’(u(z))l(l ~ U ( Z ) ( for ~ ) any z E U . (e) Show that there is a real number (“Bloch’s constant”) b > 1/32’3 having the following property: for any complex valued function f analytic in U and such that Y(0) = 1, there exists zo E U such that, if xo = f ( z o ) , the open disk B of center xo and radius h is contained in f ( U ) and there is a function g, analytic in B and such ~

282

X EXISTENCE THEOREMS

that g(B) c U and f(g(z)) = z for z E B. (Consider first the case in which f is analytic in a neighborhood of 0 , and take for zo a point where lf’(z)l(l - I z / ’ ) reaches its maximum; use then (d) to reduce the problem to the case in which zo = 0, and apply in that case the result of (c) to a function of the form a +f(Rz), where a and R are suitable complex numbers. In the general case consider the functionf((1 - E)z)/(~- E ) , where E > 0 is arbitrarily small.) 6. (a) Let 9)) be the set of all complex valued functions f analytic in the unit disk U: /zl < I , such that f ( U ) does not contain the points 0 and I . For any function f~ 91), there is a unique analytic function g in U such that exp(2n(q(z)) =f(z) in U and (.P(g(O))I< rr (Section 10.2, Problem 7); g(U) does not contain any positive or negative integer. Furthermore (same reference) there is an analytic function h in U such that g(z)/(g(z) - 1 ) = ((I h(z))/(l - h ( ~ ) ) ) h(U) ~ : does not contain any of the points 0, 1, and c : = (dn v’n(n integer > 1). Finally, there is an c; = (dnf .\/n-analytic function g, in U such that exp(y(z)) = h(z): y(U) does not contain any of the points log c; 2k7ri, log c t 2krri (k positive or negative integer, n > I). Show that no disk of radius > 4 can be contained in g,(U); using Problem 5(e), deduce from that result that Ig,’(x)l < 4 / N I - 1x1)

+

+

+

+

for 1x1 < I (consider the function t cy(x (I - I x / ) t ) , for a suitably chosen constant c). Conclude that there is a function F(if, v), finite and continuous in (C - (0, I } ) x [0, I[, such that for every function f~ 9X,loglf(z)/ < F(f(O), r ) for any IzI < r < 1 . (b) Let f~ 9) be such that either If(O)! < 1/2 or ! f ( O ) - I 1 < 1/2. Given r such that 0 < r < 1, show that either If(z)i < 5/2 for jz/ < Y, or there exists a point x such that 1x1 < Y and If(x)l > 1/2, l f ( x ) - 11 > 1/2 and Ilif(x)l 2 1/2. Applying the result of (a) to the function f((z - x)/(az - I)), conclude that there is a function F,(u, u), continuous and finite in [0, co[ x [0, 1[, such that for any function f~ 911, the relations If(0)I < s and JzJ< r imply If(z)l < Fl(s, r ) (“Schottky’s theorem”). 7. Let A be an open connected subset of C, and ( f , ) a sequence of functions of the set 911 (Problem 6 ) . Show that for any compact subset L of A, there exists a subsequence (fJ such that either that subsequence is uniformly convergent in L, or the sequence (I/f,,.) converges uniformly to 0 in L. (Using Schottky’s theorem, prove that the points x E A such that lim ( l / f , ( x ) )= 0 form an open and closed subset of A, hence equal to --f

+

“-1

m

A or empty: in the second case, show, using the compactness of L, that there is a subsequence of (6) which is bounded in a compact neighborhood of L, and apply (9.13.1); in the first case, use similarly (9.73.7) applied to the sequence (lif.).) 8. (a) Let f be a complex valued function, analytic in the open set V : 0 < / z - a1 < r , and suppose a is an essential singularity off(Section 9.15). Show that C - f ( V ) is empty or reduced to a single point (“Picard’s theorem”. Let W be the open subset of V defined by r / 2 < Iz - a1 < r and consider in W the family of analytic functions L(z) =f(z/2”); if there are at least two distinct points in C -f(V), apply Problem 7 to the sequence (fJ,and derive a contradiction with Problem 2 of Section 9.15, using (9.1 5.2).) (b) Deduce from (a) that if g is an entire function in C, which is not a constant, then C - y(C) is empty or reduced to a single point (consider g(l/z) in C - (0)). 9. (a) Show that there is an entire function f(x,y ) in C2 satisfying the identity

f(4x, 4Y) - 4 m Y ) = - S W X ,

+

- 2 ~ ) ) ~ 2 ~ ( 2 x -, 2 m

and such that the term of degree < I in the Taylor development o f f a t the point (0,O) are x y (Section 10.1, Problem 11).

+

4 DIFFERENTIAL EQUATIONS

283

(b) Let g ( x , y) = f ( 2 x , -2.~4, and let J(x, y ) = a(J g ) / a ( x ,y ) ; show that J(2x, -2y) = J ( x , y ) , and conclude that J ( x , y ) = -4 in C2 (express f ( x , y ) and g ( x , y ) in terms of f ( 2 x , -2y) and g ( 2 x , -2y)). Prove that the analytic mapping II : ( x , y) + ( f ( x , y ) , g ( x , y ) ) of C2 into itself is injective (if it was not, it would not be injective in a neighborhood of (0, 0), owing to the preceding expressions). (c) Show that there is a neighborhood of (1, 1) which is not contained in u(C2). (Observe that there exists E such that 0 < E < 1 and that the relations I f ( 2 x , - 2 y ) - l I < ~ , l g ( 2 x , - 2 ~ ) - - < ~ i m p l y J f ( x , y ) - I 1 < e a n d l g ( x , y ) - 11 < E ; conclude that the relations / f ( x ,y ) - 1 < F and lg(x, y) - I I < E would imply IfCO, 0) - 11 < E and (g(0,O)- 11 C F , a contradiction) (compare to Problem 8(b).)

4. DIFFERENTIAL E Q U A T I O N S

Let E be a Banach space, I an open set in the field K, H an open subset of E, f a continuously differentiable mapping of I x H into E. A differentiable mapping u of an open ball J c I into H is called a solution of the differential equation (10.4.1 )

x’ = f ( t , x )

if, for any t E J , we have (10.4.2) u’(t) = f ( t , u ( t ) ) . It follows at once from (10.4.2) that u is then continuously differentiable in J (hence analytic if K = C, by (9.10.1)). (10.4.3) In order that, in the ball J c I of center t o , the mapping u of J into H be a solution of (10.4.1 ) such that u(to) = xo E H, it is necessary and sufJjcient that u be continuous (resp. analytic) in J if K = R (resp. K = C), and such tllat

(10.4.4)

u ( t ) = xo

+

J:

f(s, u ( s ) ) ds

(where, if K = C, the integral is taken along the linear path 5 + to -t- c(t - to), 0 d 5 < 1). This follows from the definition of a primitive, for (when K = C) iff is continuously differentiable and u is analytic, then s + f ( s , u(s)) is analytic (9.1 0.1 ). (10.4.5) (Cauchy’s existence theorem) / f ’ f is continuously differentiable in I x H, for any to E I and any x o E H there exists an open ball J c I of center to such that there is in J a solution u of (10.4.1) such that u(t,) = x,, .

We first prove a lemma:

284

X

EXISTENCE THEOREMS

(10.4.5.1) Let A be a conipact iiietric space, F a metric space, B a coinpact subset of F, g a conti~iuousniappiiig of A x F into a metric space E. Tlieii there is a neighborhood V of B in F such that g(A x V) is hounded irz E.

For any t E A and any z E B, there is a ball S , , z of center t i n A and a ball UI,zofcenter z in F such thatg(S,,: x U , , z ) is bounded, sincey iscontinuous. For any z E B, cover A by a finite number of balls S , , , z and let V, be the ball Ur,,:of smallest radius. Then g(A x V,) is bounded (3.4.4). Cover now B by finitely many balls Vz,; the union V of the Vz, satisfies the requirements (3.4.4.).

(a) Suppose first K = R. Let J , be a compact ball of center to and radius a, contained in I . By (10.4.5.1) there is an open ball B of center xo and radius b, contained in H, and such that M = sup llf’(t, x l l and k =

sup

(f.xkJo x

( i , x k J c tx B

B

]lDz.f(t,x)l\ are finite. Let J, for r < a be the closed ball of

center to and radius r, and let F, be the space of continuous mappings y of J, into E, which is a Banach space for the norm Ilyll = sup Ily(t)ii ISJ,

Let V, be the open ball in F,, having center xo (identified to the constant mapping f -+ xo) and radius 6. For any y E V,, the mapping (7.2.1)

t

-+

xo

+ Ii: f ( s , y ( s ) )ds is defined

and continuous in J,, since

J(S)

E

B by

definition, for y E V,; let g ( y ) be that mapping; g is thus a mapping of V, into F,. We will prove that for r small enough,g verifies the conditions of (10.1.2); applying that theorem and (10.4.3) will then end the proof, with J = j,. Now, for any two points y l , in V,, we have, by (8.5.4) j q 2

Ilf’(s9 Y l W )

- . f h vz(s))II

1.

- IlYl(4 -

L’2(s)//

d k . IIYI - Yzll

for any s E J, ; therefore, by (8.7.7), for any t E J,,

hence llg(yl) - g(y2)I(d krlIy,

-

J ~ ~ I I . On

the other hand, for any

lljr:

J’ E

V,,

d M for any s E J,, hence f ( s , .As)) dsI1 d M r by (8.7.7) and therefore I/g(x,) - xo/Id Mr. We thus see that in order to be able to apply (10.1.2),we should have k r < I and Mr < b(I - kr), and both inequalities will be satisfied as soon as r < b/(M + kb). Ilf(s, y(s))II

(b) Suppose now K = C; define J , , J,, B, M , and k as above, and let F, be the space of mappings y of J, into E which are continuous it7 J, and analj~tic in j,. This is again a Banach space for the norm ((yll= suplly(t)ll, by (7.2.1) and (9.12.1). For y

E

V,, the mapping t + xo +

i E

J,

f ( s , y(s))cls again belongs

5

COMPARISON OF SOLUTIONS OF DIFFERENTIAL EQUATIONS

285

to F,, for it is analytic in 9, since s - t f ( s , y(s)) is (9.7.3); and its continuity in g of V, into F,, and the end of the proof is then unchanged.

J, at once follows from (8.11.1). We therefore have defined a mapping

(10.4.6) Remark. The proof of (10.4.5) shows that the result is still valid when K = R and when f satisfies the following weaker hypotheses: (a) for every continuous mapping t + w ( t ) of I into H, t + f ( t , w ( t ) ) is a regulated function in I (Section 7.6); (b) for any point ( t , x) E I x H, there is a ball J of center t in I and a ball B of center x in H such thatfis bounded in J x B, and there exists a constant k 2 0 (depending on J and B) such that IIf(s, Y , ) - f ( s , Y J I I ,< 4 y l - y,II for s E J, yl, y , in B. Such a function f is said to be locally lipschirzian in I x H ; equation (10.4.2) is then to be understood as holding only in the complement of an at most denumeruble subset of J. This last remark also enables one to replace the open intervals I and J by any kind of interval in R.

5. C O M P A R I S O N O F S O L U T I O N S O F D I F F E R E N T I A L E Q U A T I O N S

We say that a differentiable mapping u of an open ball J c I into H is an approximate solution of (10.4.1 ) with approximation E if we have llu’(t) - f ( f ,

for any t

u(t))ll

d&

E J.

(10.5.1) Suppose lID,f(t, x)ll d k it7 I x H. I f u, u are two approximate solutions of (10.4.1 ) in an open ball J of center t o , with approximations E ~ E, ~ then, for any t E J , we h m e

(For k = 0, (eklt-tol- l ) / k is to be replaced by It - tol).We immediately are reduced to the case K = R, to = 0 and t 2 0 by putting t = to a t , la1 = 1, t 2 0 ; then if ul(t) = u(to a t ) , 01(5) = v(t, + a t ) , u1 and v1 are approximate solutions of x’ = af(ro + at, a - l x ) , whence our assertion. From the relation ilu’(s) - f ( s , u(s))/I < in the interval 0 < s 9 t , we deduce by (8.7.7)

+

+

,

286

X

EXISTENCE THEOREMS

and similarly

l

whence

o(r)

- v(0) -

s:

1

f(s, u ( s ) ) d s = E2 t

From the assumption on D,fand from (8.5.4) and (8.7.7) this yields (10.5.1.2.)

w(t) d w(0)

+ + (El

E2)t

+k

J:

w(s) d s

where w ( t ) = llu(t) - v(t)(l. Theorem (10.5.1) is then a consequence of the following lemma:

u,

(10.5.1.3) in an inferual [0,c ] , cp and $ are two (Gronwall's lemma) regulated functions 2 0, thenf o r ar7y regulateLlfiinction MI 0 in [0,c] satisfying the inequality (10.5.1.4)

"(tj

< cp(t) +

we have in [0, c]

(10.5.1.5) Write y ( t ) ) =

s'

0

1:

4 0 G Y(t) + j'cp(s)$(s)

Ic/(s)w(s) ds

e x P ( [ s k ) di) ds.

0

$(s)it(s)

ds; y is continuous, and from (10.5.1.4) it follows

that, in the complement of a denumerable subset of [0, c ] , we have Y ' ( t ) - $ ( f ) Y ( f ) < cp(t)$(t)

(10.5.1.6)

by Section 8.7. Write z ( t ) = y ( t ) exp( - \' $(s) ds); relation (10.5.1.6) is -0

equivalent to

By (8.5.3) and using the fact that z(0) = 0, we get, for t E [0, c ]

whence by definition Y(t>

~ ; c p ( s l wexP(jsZ$(41d t )

and (10.5.1.5) now follows from the relation

ds

w ( t ) G cp(t)

+y(t).

5

COMPARISON OF SOLUTIONS OF DIFFERENTIAL EQUATIONS

287

(10.5.2) Suppose f is continuously diflerentiable in I x H . If u, c are tn’o solutions of (10.4.1), dejiied in an opeti ball J of center to and such that u(t,) = u(f,), then u = 2’ in J.

It is enough to prove that u and z‘ coincide in every compact ball L of center to contained in J. This follows from (10.5.1) applied to u and u, provided we know that D, f is bounded in some set L x H’, where H’ is an open subset of H containing both u(L) and z(L). But the existence of such a set follows at once from (10.4.5.1 ).

(10.5.3) Suppose E isjiiite dimensional a n d f i s analytic in I x H. Then any solution of (10.4.1) in an open hall J c I is analytic.

This is immediate by definition if K = C. Suppose K = R, and let E = R”; then for any point ( t o ,xo) E I x H there is a ball Lo c C of center to and a ball P c C” of center x, such that L, n R c I and P n R” c H, and an analytic mapping g of Lo x Pinto C” whose restriction to (Lo n R) x (P n R”) coincides with f (9.4.5). There is by (10.4.5) an open ball L c Lo of center to in C such that there exists a solution u of the differential equation z’ = g(t, z), taking the value xo at the point t o , and D is analytic in L. Using the relation u’(t) = g(t, c ( t ) ) , and the definition of g and it is immediately verified by induction on n that all derivatives d”)(t,) belong to R”; hence (9.3.5.1) u ( t ) belongs to R” for t E L n R. This proves that the restriction u of u to L n R is a solution of (10.4.1) (see Section 8.4, Remarks), such that u(to) = xo . But by (10.5.2),any solution M.’ of (10.4.1 ) in a ball M of center t , such that iv(to) = x o coincides with u in L n M , hence is analytic at the point t o . Q.E.D. 21,

(10.5.4) Remark. When K = R, the proof of (10.5.1) shows that the inequality (10.5.1 .I ) is still valid when f i s lipschitziaii in 1 x H for a constant k 3 0, i.e. such that condition (a) of (10.4.6) is satisfied and that ll,f(t, xl) - f ( t , x2)ll d k . llxl - x,I/ for any t E I, x,,x 2 in H ; J can then be taken as an interval of origin (or extremity) t o , containing t o , u and u are primitivesof regulatedfunctions in J, and the relations Ilu’(t) - f ( t , u(t))11 < cl, llu’(t) - f ( t , v(t))II < E, are only supposed to hold in the complement of an at most denumerable subset of J. The uniqueness result (10.5.2) holds likewise (when K = R) under the only assumption thatfis locally lipschitzian (10.4.6) in I x H, when one takes for J an interval having to as origin or extremity, and one only requires that the relations u‘(t) = f ( t , u ( f ) ) ,u ’ ( t ) = g(t, u(t)) hold in the complement of an at most denumerable subset of J.

288

X

EXISTENCE THEOREMS

(10.5.5) Let f be continuously differentiable in I x H i f K = C, locally 11)schitzian in 1 x H i f K = R. Suppose v is a solution of (10.4.1) defined in an open ball J: If - t o / < r, suclz that j c I , that tl(J)c H,and that t + f ( t , 4 t ) ) is bounded in J . Then there exists a ball J‘ : It - toI < r’ contained in I , with r‘ > r, and a solution of (10.4.1) dejined in J’ and coinciding with v in J . (a) K = R. By assumption, we have 11 f ( t , v(t))ll < M for t E J, hence ilu’(t)ll < M in the complement of an at most denumerable subset of J. This implies I ~ D ( S ) - c(t)il < MIS - tI for s, t in J by the mean value theorem (8.5.2). From the Cauchy convergence criterion (3.14.6) we conclude that the limits c((t0 - r ) + ) and u((t, + r ) - ) exist and belong to v(J) c H. By (10.4.6), there exists a solution )ixl (resp. w,) of x’ = f ( t , x ) defined in an open ball U, (resp. U,) of center t , + r (resp. t , - r ) , contained in I , and taking the value v ( ( t o + r ) - ) (resp. u((t, - r ) + ) ) at this point; from (10.5.4)it follows in addition that (resp. coincides with 11 in U, n J (resp. U, n J), and the proof is therefore concluded in that case. (One may observe that it has not been necessary to check the existence of derivatives on the left or on the right for v (extended by continuity) nor for w1 and w,, at the points to - r and to + r.) (b) K = C. For any complex number 4‘ such that l i l = I , put t = t, + is, with s > 0, and P,(s)= ~ ( t+, is). Then the same argument as in (a) proves that u,(r-) exists and is in H ; hence there exists a solution H!, of x’ = f ( t , x ) defined in an open ball V, of center t , + i r , contained in J, such that w,(I, + i r ) = u,(r-). From (10.5.4) it follows that itsi and u coincide in the intersection of J n V, with the segment of extremities t , and t , + ( r ; as these functions are analytic in J n V,, they coincide in J n V, by (9.4.4). Now cover the compact set ( t - t o [= r with finitely many balls Vcc( 1 < i < m ) ; if V,{ n VCj# @, the functions w,, and coincide in VC4n VCj, for both coincide with 11 in the nonempty open set J n V,, n VCj,and we have only to apply (9.4.2) (to show that the preceding intersection is not empty, remark that the assumption implies r l i i - ijl < p i + p i , where p i , pi are the radii of VCi and V,,; hence there is 2 ~ 1 0 I,[ such that rAlii - iil < p i and and r(l - A ) l i i - ijl < p i ; it follows that the point t, + r((l - A)(, Aij) belongs to J n V,, n Vij). There is therefore a solution of x’ = f ( t , x) equal to v in J, to wi, in each of the V l i ,and there is an open ball of center t , and radius Y’ > r contained in the union of these sets (3.17.1 1), which ends the proof. ~

3

~

)

+

(10.5.5.1) It follows from (10.5.5) that if r , is the 1.u.b. of all numbers r such that j c I a n d F ) c H, either ro = co,or, if J, is the open ball It - tol < y o , one of the two relations jo$ I , $ H holds.

+

zi(J,)

5

COMPARISON OF SOLUTIONS OF DIFFERENTIAL EQUATIONS

289

(10.5.6) Let f, g be two continuously diflerentiable mappings of I x H into E, and suppose that, in I x H , 11 f ( t , x) - g(t, x)II 6 c( and Ij D,g(t, x)I/ 6 k. Let ( t o , xo) be a point qf’ I x H , p, p two numbers > 0, and q($ = ehc - 1 Lieht (a p) -for 2 0. Let u be an approximate solution of x’ = g(t, x),

+ +


c and taking the value y at t o , which contradicts the definition of c.

+

+

+

When K = R,one may, in the statement of (10.5.6), replace J by an open interval ]c, d [ containing the point t o . (10.5.6.1) We again remark that if K = R, we can relax in (10.5.6) the hypotheses on f and g, supposing merely that y is lipschitzian for the constant k, andflocally lipschitzian in I x H. PROBLEMS

1. Let f ( i ,x ) be a real valued continuous function defined in the set I t / < a, 1x1 < b in R2,such that f ( r , x ) < 0 for tx > 0, and f ( r , x ) z0 for t x < 0. Show that x = 0 is the unique solution of the differential equation x’ = f i t , x ) defined in a neighborhood

290

X

EXISTENCE THEOREMS

of 0 and such that x(0) = 0 (use contradiction, and consider, in a compact interval containing 0, the points where a solution reaches its maximum or minimum). 2. Let f ( t , x) be the real valued continuous function defined in R Z by the following conditicns: f ( t , x) = -2, for x > t 2 , f ( t , x) = -2x/t for 1x1 < t 2 , f ( t , x ) = 2 f for x < - t Z . Let (y,) be the sequence of functions defined by y o ( t ) = /’, y.(t)

=

J:

f ( u ,Y,-I(L~))L/u

for n > 1. Show that the sequence (y.(t)) is not convergent for any t # 0, although the differential equation x’ = f ( f , x ) has a unique solution such that x ( 0 ) 0 (Problem I). a, 3. For any pair of real numbers CL > 0, /3 > 0, the function equal to -(t - C L )f o~ r t -: to 0 for a < t < P, to (t - 8)’ for t > /3, is a solution of the differential equation x’ = 2 1 ~ I ”such ~ that x ( 0 ) = 0. Let uo be an arbitrary continuous function defined in a compact interval [a, / I ] , and define by induction

rr.(t) = 2

J’:

~ U - ~ ( . ds S ) for ~ ~ t~ E~ [a, 01.

Show that if y is the

largest number in [a, b] such that u O ( t )= 0 in [a, y ] , the sequence ( i d converges uniformly in [a, b] to the solution of x ’ = 2 1 ~ 1 ’ which /~ is equal to 0 for a < t < y , to ( t y ) 2 for y < r < h. (Consider first the case in which uo(t)= 0 for t < y , u o ( t ) k(t - y)2 for y < t < b. Next remark that, replacing if necessary uo by u l , one may suppose that uo is increasing in [a, b ] ; observe that for any number E > 0, there are two numbers k l > 0, k z > 0 such that in [a, b] -

4.

where vo(t)= 0 if t < 0, vo(t)= t 2 if t > 0.) The notations being those of Section 10.4, suppose K == R,fis continuous and bounded in I x H, and let M = sup llf(r, x)I/. Let x o be a point of H, S an open ball of ( ~ . x )xE HI

center xo and radius r, contained in H . (a) Suppose in additionfis uniformly continuous in I x S (a condition which is automatically satisfied if E is finite dimensional and I is contained in a compact interval I. such that f is continuous in l o x H). Prove that for any E > 0, and any compact interval [ t o ,to h] (resp.[to - h, t o ] ) contained in I and such that h < r/(M E ) , there exists in that interval an approximate solution of x’ =f ( t , x) with approximation E , taking the value xo for t = t o . (Suppose 6 > 0 is such that the relations Itl - f z I < 6, 11x1- xz I1 < 6 imply Ilf(t1, x l ) - ] ( I 2 , x2)l/ < E ; consider a subdivision of the interval [ t o , t o h] in intervals of length at most equal to inf(6, 6/M), and define the approximate solution on each successive subinterval, starting from to.) (b) Suppose E is finite dimensional and I = ]to - a, to a[. Prove that there exists a solution of x’ = f ( r , x ) , defined in the interval [ t o ,to c] (resp.[ro - c, t o ] )with c = inf(a, r/M), taking its values in S, and equal to xo for t = to . (“ Peano’s theorem ”: for each n, let u, be an approximate solution with approximation l/n, defined in J, = [ t o , t o c - ( l / n ) l ,whose existence is given by (a). Observe that for each m , the restrictions of the functions u. (for n 3 m ) to J,, form a relatively compact subset of the normed normed space KE(J,,,) (7.5.7), and use the “diagonal process” as in the proof of (9.13.2); finally apply (10.4.3)and (8.7.8).) 5. Letf’be the mapping of the space ( c o )of Banach (Section 5.3, Problem 5) into itself, such

+

+

+

+

+

+

that, for x

= ( x . ) , f ( x ) = (yJa

with y n =

+ -.n +1 l

Show that f is continuous in

(cd, but that there is no solution of the differential equation x’ - f ( x ) , defined in a

neighborhood of 0 in R, taking its values in (co), and equal to 0 for t = 0. (If there was

5

COMPARISON OF SOLUTIONS OF DIFFERENTIAL EQUATIONS

291

such a solution u ( t ) = ( u n ( f ) ) ,compute the value of each u,(t) by straightforward integration, and show that the sequence (u.(t)).to does not tend to 0 for t # 0.) (a) The notations being those of Section 10.4, let f be analytic in 1 x H if K = C, locally lipschitzian in 1 x H if K = R. Let lo be an open ball of center to and radius a, contained in I , and S an open ball of center xo and radius Y, contained in H. Let h(s, z ) be a continuous function defined in [0, a [ x [0, r [ c R2, such that h(s, z ) 2 0 and that, for every S E [0, a[, the function z+h(s, z ) is increasing in [0, Y[. Suppose that: ( I ) llf(t, x)l/ < h ( / t- t o / , /lx - xoll) in I. x S ; (2) there exists an interval [0, a ] with dc < a, and a function rp, which is a primitive of a regulated function T’ in [O, dc], and is such that ~ ( 0= ) 0, ~ ( s E) [0, r [ and ~ ’ ( s ) > h(s, ~ ( s ) ) in the interval [0, CL], with the exception of an at most denumerable set of values of s. Show that there is a solution I / of x’ = f ( t , x), defined in the open ball J of center to and radius a , taking its values in S and such that u(to)= xo ; furthermore, in J, llu(t) - xoIl < p(lt tot). (Use (10.5.5) to prove that there is a largest open ball Jo of center t o , contained in l o , and in which there is a solution u of x ’ = f ( t , x ) , taking its values in S and such that lIo(t) - X O / / < y(lt - t o l ) in J o , and furthermore that solution is unique; use then the mean value theorem to prove by contradiction that J c Jo .) (b) Suppose that H = E, and that there is a function h(z) > 0 defined, continuous and ~

increasing in [0, + a [ and , such that

Io+-

- = t m , and that

):

llf(t,

x)ll < h(llxll) in

l o x E. Show that every solution of x ’ = f ( t , x) defined in a neighborhood of t o , is defined in lo (use (a)). (c) If lif(t, x)ll < M in lo s S, then there exists a solution of x ‘ = f ( / , x) in the ball J of center to and radius inf(a, r/M), taking its values in S and such that u to)= x o (take h(s, z ) = M). Suppose K = E = C, and a 2 r / M ; show that, unlessfis a constant, there is an open ball J’ 3 in which u can be extended to a solution of x’ = f ( t , x) taking its values in S . (Observe that, due to the maximum principle ( 9 .5 .9 ) , Iic’(t)l < M for t E J; for any 5 such that 151 = I , consider the function I{&) = u(to 5s); arguing as in (10.5.5), prove that the assumptions of (10.5.5) are satisfied.) I t is not possible to take for the radius of J’ a number depending only on a, r and M , and not on fitself, as x)/2)”” (Section 9.5, Problem 8), with t = xo = 0, a = r = the examplef(t, x) = ((I M = I , shows (n arbitrary integer i I). Letfbe a real valued bounded continuous function in the open polydisk P: It - t o / < a, Ix- xoI < b in R2, and let M = sup I f ( f , x ) l ; let r = inf(a, b/M), and let I =

+

+

+

(1.X) E

P

]to- r , to i.[. Let (D be the set of all solutions I ( of x ’ = f ( t , x). defined in I, taking their values in the open interval ]xo- b, x o b[ and equal to xo for t = to ; the set CI, is not empty (Problem 4(b)). For each t E I , let u ( f , t o , xo) = inf u(t), w(t, t o ,xo) =

+

US@

sup u ( r ) ; show that u and w belong to CD (Section 7.5, Problem 1 I ) ; v(resp. w) iscalled the “ E O

minimal (resp. muximal) solutim of x’ = f ( t , x) in I , corresponding to the point ( t o ,XO). u ( t , t o , xo) in an interval For each T E I , let $. = U ( T , t o , xo). Show that u(t, T , of the form [ T , T h [ , if T 2 t o , of the form IT - h , T ] if T < to (with h > 0). Conclude that there is a largest open interval I t I ,f 2 [ contained in ]to- a , to a[ and containing t o , such that u(t, t o ,x,) can be extended to a continuous function g defined in ]/I, f 2 [ , taking its values in ] x o- b, xo -t /I[, and such that, for every t E PI, t 2 [ , g ( . s )= u(s, t , g ( t ) ) in an interval of the form [t,t h[ if t > t o , of the form It - h , t ] if t < to with h > 0). (Ifg, is another such extension of u ( t , t o , xo) in an interval It;, t i [ ,show that g andg, coincide in the intersection of ] I , , f 2 [ and It;, f ; [by considering the 1.u.b. (resp. g.1.b.) crf the points s in that intersection such that g and gt coincide in [ t o , s[ (resp. in Is, t o ] ) . Furthermore, either t l : to - a (resp. f 2 = to n ) , or g(tl +) = xo Jr b (resp.

6)

+

+

+

y(ti-)

= xo

i b).

+

292

X

EXISTENCE THEOREMS

(a) Generalize Gronwall’s lemma (10.5.1.3) to the inequalities

where v, # 1 , t+h2, 81, O2 are regulated functions 3 0. (Use induction on the number of integrals on the right-hand side.) (b) Let K(t, s) be a continuously differentiable function 3 0 defined in [0, c] x [0, c]. Suppose there are two regulated functions g, h, defined and 3 0 in [0, c ] , such that aK(t, s)/af< g(f)h(s).Show that the inequality w ( t ) < p(t) +lotK(f, s)w(s)ds

for a regulated function w

> 0 implies

w ( t ) G vl(t)

+ el(t)j)(s)w(s)

ds

where v1 and 81 are functions which one can explicitly compute when the functions

v,g and r ( t ) = K ( t , t) are known (consider the function y ( t ) = majorize its derivative). (c) Apply (a) or (b) to the inequality

w ( f )< t

+ h 2 f j : c A S w ( sds) +

fd

1:

K(t, s)w(s)ds and

w(s) ds,

where h > 0. Let w be a real function defined in an open interval I c R, and suppose wis the primitive of a regulated function w’ whose points of discontinuity are isolated in I, and such that in each of these points w’(t+) > w ’ ( t - ) ; suppose in addition that if E is the set of points of discontinuity of w’, the second derivative w” exists in I - E and w”(x) w(x) in I - E. (a) Show that if a, b are two points of I such that w(a) = w(b) = 0, then w(x) < 0 for a < x < b (use contradiction). Conclude that for any three points xl < x < x 2 in I one has w(xl) sinh(x2 - x) w(x2) sinh(x - xl) w(x) G sinh(x2 - xl)

+

(consider the difference w(x)- u(x), where u is a solution of the equation u”(x) - u(x) = 0 taking the same values as w at the points x, and x2).

6. LINEAR DIFFERENTIAL EQUATIONS

The existence theorem (10.4.5) can be improved in special cases: (1 0.6.1) Let I c K be an open ball of center to and radius r. Let j’be continuous in I x E ifK = R,continuously differentiable in I x E i f K = C,and such that

6 LINEAR DIFFERENTIAL EQUATIONS

293

Ilf(t, X I ) - A t , xJll < k(lt - t0I)llxl - x2II for t E 1, XI,x2 in E, where 5 --+ k(5) is a regulated function in [0,r[. Then for every xo E E , there exists a

unique solution u of (1 0.4.1), defined in I, and such that u(to) = xo .

We only have to prove that, if c is the 1.u.b. of the numbers p such that 0 < p < r and that there exists a solution of (10.4.1) defined in It - tot < p and taking the value xo at t o , then c = r (by (10.5.4)). Suppose the contrary; then, by (10.5.4), there is a solution v of (10.4.1) defined in J : It - tol < c and such that v(to) = xo . We are going to show that the conditions of (10.5.5) are satisfied; applying (1 0.5.5) then yields a contradiction and ends the proof. As here H = E, the condition v(J) c H is trivially verified, so we have only to check that t -+f ( t , v ( t ) ) is boundedin J . Now, in the compact interval [0, c], k is bounded and so is the continuous function t -+ 11 f ( t , xo)II in the compact x)ll < set 3; hence there exist two numbers m > 0, h > 0 such that ,?(fI[ mllxll h for t E J and x E E. This implies IIv’(t)ll < rnllv(t)ll h for t E J ; if we write w(5) = IIv(to + A()II with 111 = 1, the mean value theorem shows that

+

+

w(t) < llx0II + hc + rnJ; w(c) dc. We therefore can apply Gronwall’s lemma (10.5.1.3), which shows that Ilv(t)II < aemlr-tol+ b in J (a and b constants), hence u is bounded in J, and so is IIf ( t , v(t))ll < mllv(t)ll + h. (1 0.6.1. I ) Here again, when K = R,the condition of continuity on f can be relaxed to condition (a) of Remark (10.4.6).

A linear differential equation is an equation (10.4.1) of the special form

x’

(1 0.6.2)

= A(t) *

x

+ b(t)

(=f (t,x))

where A is a mapping of I into the Banach space 9 ( E ; E) of continuous linear mappings of E into itself (Section 5.7), and b a mapping of I into E. We have here H = E, and by (5.7.4)

Ilf(r,

x1) -

f(C

X2N

< IIA(t>ll

*

11x1

- x2II

for all t E I , x l , x 2 in E. Applying (10.6.1) and (10.6.1.1) we therefore get: (10.6.3) Let I c K be an open ball of center to . Suppose A and b are regulated in I if K = R, analytic in I if K = C. Then, for every xo E E , there exists a unique solution u of (10.6.2), defined in I and such that u(to) = xo .

Observe that if b = 0, and xo = 0, the solution u of (10.6.2) is equal to 0. From (10.6.3) we easily deduce the apparently more general result:

294

X

EXISTENCE THEOREMS

(10.6.4) The assumptions being the same as in (10.6.3), f o r every s E I and every xo E E, there is a unique solution u of (1 0.6.2) defined in 1 and such that u(s) = xo .

Replacing t by t - t o , we may assume that to = 0. Suppose I is a ball of radius r ; the mapping

t+r2-

t-s St - r2

is an analytic homeomorphism of 1 onto itself, mapping s on 0: indeed, one has t -s t'=r2-=it-1

hence, if It1

(

r_' I---- r2 - Is[') s r2 - St

< r, we have lr2 - it1 < r(r

+ Isl)

whence

our assertion follows from the fact that, conversely,

t=r2-

t' - s

St' - 1

.

Now, if

and

one sees at once that if v is the unique solution of the differential equation X' = A,(t)

defined in I and such that v(0)

.x + bl(t)

= xo, then

is the unique solution of (10.6.2), defined in 1 and such that u(s) = xo.

7 DEPENDENCE OF THE SOLUTION ON PARAMETERS

295

When E = K", A ( t ) = (aij(t))is an n x n matrix, b ( t ) = (b,(t))a vector, the a i j ( t ) and b,(t) being regulated in 1 if K = R , analytic if K = C; if

x = (xi)' 1 (10.6.6)

D"x - a,(t)D"-'x - . . . - a,-,(t)Dx - a,(t)x = b(t)

are equivalent to special systems of type (10.6.5); one has only to write < p < n, and (10.6.6) is equivalent to

x1 = x, x p = D P - ' x for 2

X ~ = X ~ +for ~

(10.6.7)

x;

= al(t)x,,

1 0 such that, for any pair ( u , u ) of approximate solutions, with approximation E , of any equation x' = f ( t , x) with f E a), such that u and u are defined in I and u(to) = u(to), the relation lIrr(r) - u ( t ) l / < ~ ( l t /E ), would hold for every t E I. (For any a E 10, 1 [, letf be the continuous function equal to x / t for 1x1 < t2/(a- t ) , 0 < t < a , and for t > a , and independent of x for the other values of ( t , x) such that t > 0; define f ( t , x) = f ( - t , x ) for t < 0. Take a : 0; let u ( t ) = E t for It/ < a , and take for u a solution of x' = f ( t ,x) E for the other values oft.) 7. The notations being those of Section 10.4, suppose E is finite dimensional and f is continuous in I x H; let ( t o ,x o ) be a point of I x H, J an open ball of center to contained in I, S an open ball of center xo such that S c H. Supposefis bounded in J x S, and the following conditions are verified: ( I ) There is af most one solution of x' = f(t, x) defined in an open interval contained in J and containing t o , and taking the value xo for t = t o . (2) There exists a sequence ( u , ) , ~of~continuous mappings of J into S such that

+

u.(t)

= xo

+Itr

f(s, r i n - l ( . s ) ) d.s

for n

>1

and

t E J.

0

+

(3) For every t E J , u " + ~ ( / )- u,(t) converges to 0 when n tends to m. Show that in every compact interval J' c J containing to the sequence(u,)converges uniformly to a solution of x' = f(/, x) equal to xo for t = t o . (Observe that the sequence (u.) is equicontinuous; use Ascoli's theorem (7.5.7), as well as (3.16.4) and (8.7.8).) 8. Suppose E is finite dimensional, w and f verify the conditions of Problem 5(a), and in addition, for every t E 10, a [ , the function z+ w(t, z ) is increasing in [0, +a[. There is then at most one solution of x' = f(t, x) defined in an interval [0, a [ c [0, a[ and taking the value xo for t = 0 (Problem 5(a)). Suppose in addition that there exists, in an interval J = [0, a ] C [0, a[, a sequence (u,Jnb0 of continuous mappings of J into S such that I&)

= xo

+s,'f(s,

U " - ~ ( S ) ) ds

(a) For every t E J, let y,(t)

=

for n > 1 and

l l ~ , , + ~( t u,,(t)/I, ) z.(t)

=

t E J.

supy,+,(t), and w ( t ) = k 2 0

inf z.(t). Show that the functions z, and w are continuous in J (use Problem 1 1 of XbO

Section 7.5). (b) Let t , t - h be two points of J (h > 0); show that, for every 6 > 0, there is an N such that, for n > N, lyn(t)- y.(t

- h)/
z,(t - h)). Hence

(by (8.7.8)).

t-h

(d) Conclude that w(t) = 0 in J (same argument as in Problems 4(b) and 5(a)), and using Problem 7, prove that the sequence (u,) converges uniformly in J to a solution of x = f ( t , x ) taking the value xo for t = 0. 9. The notations being those of Section 10.4, suppose E is finite dimensional, and f is continuous and bounded in I x H. Suppose in addition there is at most one solution of x' = f ( t , x) defined in any open interval J C I containing t o , and equal to xo E H for t = t o . Suppose that, for any integer n > 0, there exists an approximate solution it. of x' = f ( t , x), with approximation I/n, defined in 1 and taking its values in H, and such that u,(to) = xo . Show that in any compact interval contained inI, thesequence ( i t n ) is uniformly convergent to a solution u of x' = f ( t , x), taking its values in H and such that u(to)= x o . (Use the same argument as in Problem 7.)

8. DEPENDENCE OF THE SOLUTION O N INITIAL CONDITIONS

(10.8.1) Let f b e locally lipschitzian (10.5.4) in I x H if K I x H if K = C.Then,for any point (a, b) E I x H :

= R, analytic

in

(a) There is an open ball J cl of center a and an open ball V c H of center b such that,,for every point ( t o ,x,) E J x V, there exists a unique solution t u(t, t o , x,) of (10.4.1) defined in J, taking its values in H and such that d t , to xo) = xo * (b) The mapping ( t , t o , x,) + u(t, t o , x,) is uniformly continuous in JxJxV. (c) There is an open ball W c V of center b such that, for any point ( t , t o , x,) E J x J x W, the equation xo = u(tO,t , x ) has a unique solution x = u(t, t o , x,) in V. --f

9

9

(a) By assumption, there is a ball J, c I of center a and a ball B, c H of center b and radius r such that in J, x B, , 11 f ( t , x)ll Q M, and 11 f ( t , xl) f ( t , x2)l/ < k * I/xl - x211 for tE J,, x l , x 2 in B,, By (10.4.5) and (10.5.2) there is an open ball J, c J, of center I,, and a unique solution v of (10.4.1) defined in J,, taking its values in H and such that v(a) = b. We are going to see that the open ball V of center b and radius 4 2 , and the open ball J of center a and radius p , answer our specifications as soon as p is small enough. Apply

304

X

EXISTENCE THEOREMS

(10.5.6) to the case c1 = = 0; this shows that there exists a solution of (10.4.1) defined in J, with values in Bo , taking' the value xo E V at the point t o E J,

provided we have IIu(t) - 611

(1 0.8.1 .I)

for

every

IIv(t)

t~ J.

But

+ IIu(to) - xollekl'-'ol < r

by

the

mean

value

theorem,

we

have

< M It - a1 < M p forevery t E J ; as by assumption llxo - bll < r / 2 ,

- bll

the inequality (10.8.1 . I ) will be satisfied if p is such that (10.8.1.12)

(M p + -I)e 2 k p < r

Mp+

which certainly will be satisfied for small values of p > 0, since the left-hand side of (1 0.8.1.2) tends to 1-12when p tends to 0. (b) From the mean value theorem, we have (10.8.1.3)

for

to,t,,

(10.8.1.4)

- u(t2

IIu(t1, t o 9 xo)

t o 9 x0)II

9

dM

It2

- 1,

I

t , in J, xo in V. By (10.5.1), we have IIu(t, t o 9 XI)

- u(t, t o

9

x2)ll

< e2kp1x2 - x1 I

for t , t o in J, x , , x2 in V. Finally, (10.8.1.3) for Ilu(t1,

t2

9

xo) - xoll

1 , even if the right-hand sides ,f; are continuously differentiable functions. We say that an equation (10.9.1) is conipletely integrable in A x B if, for every point (xo,yo) E A x B, there is an open neighborhood S o f x o in A such that there is a unique solution u of (10.9.1), dejned in S, with values in B, and such that 4 x 0 ) = Yo. We will suppose in what follows that U is continuously chfferentiuble in A x B; for each (x,y ) E A x B, D, U ( x ,y ) (resp. D, U ( x ,y ) ) is an element of 9 ( E ; 9 ( E ; F)) (resp. 9 ( F ; 9 ( E : F))), which can be identified to the continuous bilinear mapping (sl, s,) -+ (D, U(x,y ) . sl) . s, of E x E into F, written (sl, s2) -+ D, U(x,y ) . (sl, s,) (resp. the continuous bilinear mapping ( t , s) -+ (D,U(x, y ) . t ) . s of F x E into F, written (1, s) -+ D, U ( x ,y ) . ( t , s)) (5.7.8); furthermore, the linear mapping s1 + ( D 1U ( x ,y ) . sl) s, of E into F, for each s, E E, is the derivative at the point (x, y ) of the mapping x + U(x,y ) . s2 of E into F, by (8.2.1) and (8.1.3); similarly, the linear mapping t -+ (D, U(x,y ) t ) * s of F into F, for each s E E, is the derivative at the point (x, y ) of the mapping y -+ U(x,y ) . s of F into F. 1

(10.9.4) (Frobenius’s theorem) Suppose U is continuously differentiable in A x B if K = R, t,tIice continuously riifferentiuble if K = C. In orrler that (10.9.1) be conipletely integrable in A x B, it is necessary and sufljcient that, for each (x,y ) E A x B, the relation (10.9.4.1) Di u(X,Y )

(Si, $2)

(u(x,J’) ’ S1, S2) + D, U(x, Y>. (U(X,Y ) . s 2 , $1)

-t D, U(x,Y ) *

= D, U.u,Y> . ( s 2

9

holds,for any pair (sir s,) in E x E.

s1)

9

THE THEOREM OF FROBENIUS

309

(a) Necessity. Suppose u is a solution of (10.9.1) in an open ball S c A of center xo such that zi(xo)= yo; then, from (10.9.2) and the assumption it follows that u‘(x) is differentiable in S ; moreover, for any s,EE, the derivative at the point xo of the mapping x -+ u’(x) . s, is sI -+ u”(xo) . (s,, s2)by (8.12.1). But by (10.9.2), that derivative is also (using (8.2.1), (8.1.3), and (8.9.1)) s1 -+

(D, U(x0 Y o ) * 3

Sl)

. s2

+ (D, U(x0 Yo) . (u’(x0)

*

7

s1))

*

s2.

Using the relation (10.9.2) again, and expressing that the second derivative of u at the point xo is a symmetric bilinear mapping (8.12.2), we obtain (10.9.4.1) at the point (xo, yo). But by assumption that point may be taken arbitrarily in A x B, hence the result. (b) Suficiency. Let So c A be an open ball of center xo a n d radius ct To c B an open ball of center yo and radius /3, such that U is bounded in So x To, let 11 U ( x ,y)II < M. We consider for a vector z E E the (ordinary) differential equation (where 5 E K) (10.9.4.2)

M” =

U(xo+ 52, w) * z

= f(5,

1.0,

z)

and observe that if u satisfies (10.9.2) in a neighborhood IIx - xoII < p of llzll < p is a solution of (10.9.4.2) in the ball < 1, in K , taking the value yo for 5 = 0 (which already proves uniqueness of u by (10.5.2)). Now the right-hand side of (10.9.4.2) is continuously differentiable for 151 < 2, IIM, - yo/l < /3 and llzll < a/2, and we have ,(fI[ M’, z)ll < Mllzll for such values. Applying (10.5.6) to f and to g = 0, we conclude that for any z E E such that ilzll < /3/2M, there is a unique solution 5 .+ v(5, z ) of (10.9.4.2) defined for 141 < 2, taking its values in H and such that a(0, z ) = y o . We are going to prove that thefimction u ( x ) = z*( I , x - xo) is a solution of (10.9.1) in the ball IIx - xoII < /3/2M. Now, for lizll < /3/2M and 151 < 2, we know from (10.7.3) that u is continuously differentiable, and that t -+ D, ~ ( 5z,) is, for 151 < 2, the solution of the linear differential equation

xo,5 -,u(xo + 5 2 ) for

v‘ = D, f ( 5 , u ( 5 , z), Z ) v + D, f(5, 4 5 , z), Z ) taking the value 0 for 5 = 0. For any sI E E, write g(5) = D, v(5, z) s,; we , z ) . g ( 5 ) + D, f(5, 4 5 , z), z ) . s1 and from the have g ’ ( 0 = D,,f(5, ~ ( 5z), definition 0f.L this can be written

d(5)= NO . M5L 4 + B(5). s1 + W(5) (s1,4 *

O 5z, ~ ( 5z>), , 4 5 ) = D2 U(xo + 52, 4 5 , z)), B(t) = ~ ( X + C(5) = D, U(xo+ 42, v(5, z)). We want to prove that g(5) = 5U(xo + 52, u(5, z)) * s1

with

310

X

EXISTENCE THEOREMS

and we therefore consider thedifferenceh(t)

-

g(5) - tB(5) sl.We have

= g ( ( ) - (U(x0

+ ~ z , v ( (z)), 's1=

h ' ( 0 = A(5) . Mt), z>f B(5) . s1 + W(5). (S1r

4 (K5) z , Sl)

- B(5) . s1 - t C ( 0 . ( z , $1 - lluPll, then 5 is regular for u. (Use (11.1.3), and from the convergence of the series m

"=O

5-"pu"p.conclude that the series

x ]a Hilbert basis of E. Let S be an arbitrary infinite compact subset of C, and let (p,,) be a denumerable set of points of S , which is dense in S (3.10.9). Show that there is a unique element L I E 9 ( E ) such that u(e,) = pne, for every n > 1 ; prove that the spectrum of u is equal to S , whereas the eigenvalues of u are the p,, . If 5 E S, 5 is not equal to any of the pn, and vc = u - 5 . l E, show that v : ( E ) is dense in E but not equal to E(use (6.5.3) to prove the first statement). Show that the Tpectrum of the operator u defined in (11.1.1) is the disk 151 S 1 in C; I I has no eigenvalue. If v c = u - 5 . I,, show that for 151 < 1, v,(E) is not dense in E, but for 151 1, vc(E) is dense in E and distinct from E (cf. (6.5.3)). Let E be a complex Banach space, Eo a dense subspace of E. Show that for any element I I f 9 ( E 0 ) , the spectrum of u contains the spectrum of its unique continuous extension ri to E (5.5.4). Give an example in which these spectra are distinct and an example of an operator u E Y(EO)and of a spectral value 5 of u such that, if u 5 = u - 5 . l E , v c is a bijective mapping of Eo onto itself (in Problem 3, consider the subspace Eo of E consisting of the (finite) linear combinations of the vectors en). Note that this is impossible if Eo is a Bunuch space, for every continuous bijective linear mapping of Eo onto itself is then a homeomorphism (12.16.8). Let E be a complex separable Hilbert space, (en),,>] a Hilbert basis of E; let u be a continuous operator in E such that, for every pair of indices h, k , one has (&,)I e x )> 0. (a) Show that the number p(u) (Problem I(b)) belongs to sp(u). (Note that for 151 > &I), if v c (u - 5 . I E ) - I , one has

+

:

5.

6.

:

m

(O;(eh)

I 4 = -"C (un(eh)I e.d5-"-' =O

and use Problem 7(b) of Section 9.15.) (b) Suppose in addition that for some integer n > I , there exist an integer k > 1 such that (u"(ek)I e b )= d > 0. Prove then that p(u) > d''" (observe that for every integer m > I , (unm(ek) I e r )> d"). (c) Suppose p(u) > 0 and that the point p(u) is a pole of the function 5 + u c . Prove

x 5" m

that there exists then an eigenvector x =

5" > 0 for every n. (Let N

en of u corresponding to p(u), such that

" = I

be the order of the pole p(u) of v c , and let

316

XI ELEMENTARY SPECTRAL THEORY

Show that (wN(eh)I ek)< 0 for every pair h, k , and w N # 0 by assumption, and use the fact that U W N = p(u)wN .) (d) Suppose that (u(el) I eA)> 0 for every pair of indices h, k (this, by (b), implies p(u) > 0), and in addition suppose that p(u) is a pole of v c . Show that p ( u ) is then a simple pole of v z . (Observe that ( d / d ( )- v c = v t and prove that, if one had N > I , then one would have wA = 0; this would imply (wN(ek)I eA)= 0 for every k ; using the relatidn uwN = p ( u ) w N , observe that if ( w N ( e k ) 1 eh) = 0 for one index h , then wN(eA) 0.) Prove that there exists an eigenvector z = = x(.en of N corresponding to p ( u ) and such that 1

=c~ “ e is, an eigenvector of the adj,oint u* corresponding to p ( u ) (cf. Section one has 7. > 0 for every n, or 7“ 0 for every (otherwise, one would get the contradictory inequality In I 7. I 0 for every n. Next show that if y

n

=Z

11.5),

II

n

majoration of each 7” derived from p(u)T. =C qr(u(ek)1 en)).Conclude finally that all k

eigenvectors of u corresponding to p(u) are scalar multiples of z (exchange u and (“Theorem of Frobenius-Perron.”)

id*).

2. COMPACT OPERATORS

Let E, F be two normed (real or complex) spaces; we say that a linear mapping u of E into F is conTpact if, for any bounded subset B of E, u(B) is relatively compact in F. An equivalent condition is that for any bounded sequence (x,) in E, there is a subsequence (x,J such that the sequence ( ~ ( x , ~ ~ ) ) converges in F. As a relatively compact set is bounded in F (3.17.1), it follows from (5.5.1) that a compact mapping is continuous.

Examples (11.2.1) If E or F is finite dimensional, every continuous linear mapping of E into F is compact (by (5.5.1), (3.17.6), (3.20.16), and (3.17.9)).

(1 1.2.2) If E is an infinite dimensional normed space, the identity operator in E is not compact, by F. Riesz’s theorem (5.9.4). (11.2.3) Let I = [a, b] be a compact interval in R, E = gC(l) the Banach space ofcontinuouscomplex-valued functions in I (Section 7.2), (s, t ) -+ K(s, t ) a continuous complex-valued function in I x I . For any function f ’ E,~ the mapping r --t K(s, t ) f ( s )ds is continuous in I by (8.11.1); denote this

Jab

2 COMPACT OPERATORS

317

function by U f . Then the mapping f + Uf of E into itself is linear; we prove that it is compact. Indeed, if g = U j ; we can write, for to E I , t E I,

s(r>- d t o ) =

(1 1.2.3.1)

(K(s,

r ) - K(s, t o ) ) f ( s )d s .

.lab

As K is uniformly continuous in I x I (3.16.5),for any E > 0 there is a 6 > 0 such that the relation It - tol < 6 implies IK(s, t ) - K(s, t o ) [< E for any s E I ; hence, for a n y f in E

(11.2.3.2)

Ig(t) - g(t0)l

d E(b - a>llfll

by the mean-value theorem. This shows that the image U(B) of any bounded set B i n E is equicontinuozrs at every point to of I (Section 7.5); on the other hand, for any t E I, we have similarly Ig(t)l d kllf I/ if IK(s, t ) l d k / ( b - a ) in I x I . By Ascoli’s theorem (7.5.7),U(B) is relatively compact in E.

(11.2.4) With the same notations and assumptions o n K as in (11.2.3),let now F be the space of complex-valued regulated functions in I (Section 7.6), which is again a Banach space, when considered as a subspace of the space g,-(l);Uf’ is then defined as in (11.2.3)for any f E F, and the inequality (1 1.2.3.2)still holds. The argument in (11.2.3)then proves that U is a coinpact mapping of F into E.

(11.2.5) If’u,

L’

are two compact mappings of E into F, u + u is compact.

Let (x,) be a bounded sequence in E ; by assumption, there is a subsequence (x;) of (s,)such that (u(x;)) converges in F. As the sequence (x:) is bounded in E, there is a subsequence ( x i ) of (x;) such that (c(xi)) converges in F. Then by (3.13.10)and (5.1.5), the sequence (u(x1)+ xi)) converges in F. Q.E.D.

(1 1.2.6) Let E, F, El, F, be normed spaces, f’ a continuous linear mapping into E, g a continuous linear mapping of F into F,. Then,f o r any compact mapping u of E into F, u1 = g u o f is a compact niapping of El into F,. of El

For if B, is bounded in E 1 , f ( B l ) is bounded in E by (5.5.1), u ( f ( B , ) ) is relatively compact in F by assumption, and g(u(,f(B,))) is relatively compact in F, by (3.17.9).

318

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ELEMENTARY SPECTRAL THEORY

(11.2.7) I f u is a compact mapping of E into F, the restriction of u to any rector subspace El of E is a compact mapping of El into u(E,).

For by (11.2.6), that restriction is a compact mapping of El into F. If B is a bounded subset of El,-) is then a compact subset of F, and as __ - u(B) c u(E,), u(B) is relatively compact in u(E,). Example (11.2.8) With the same notations and assumptions on the function K as in (11.2.3), let now G be the prehilbert space defined by the scalar product

( f ) g ) = ~ a b f ( t ) g ( r ) don t the set gC(I)(6.5.1); we write the norm f)”’ = ~ ~ , to f ~ distinguish ~ a it from the norm l l f l i = supIf(t)l, and we still

(fI

f E l

with the norm llfll; the identity mapping f + f o f denote by E the space gC(l) E into G is continuous, since i l f i l z < ( b - a)’/’ * llfll by the mean value theorem; but it is not bicontinuous, nor is G a Banach space. The CauchySchwarz inequality (6.2.1) is written here

With the same notations as in (1 1.2.3), we therefore deduce from (11.2.3.1) and (11.2.8.1) that It, - t,l < 6 implies

Is(t1) - g(t2)I < E(b .llflla and similarly Ig(t)l < k(b . 1l.f 11’ for any t E 1. Hence, by the same

(1 1.2.8.2)

9

argument as in (11.2.3), f - Uf is a compact mapping of G into E; and as the identity mapping of E into G is continuous, ,f -+ U ’ is also a compact mapping of G into G by (11.2.6). (11.2.9.) Let E, F be two Banach spaces, E, (resp. F,) a dense subspace of E (resp. F), u a compact mapping of E, into F, , ii its unique continuous extension as a mapping of E into F (5.5.4). Then u“(E) c F, , and u” is a coiiipact mapping ofE into F, .

I t is immediate that any ball llxil < r in E is contained in the closure of any ball of center 0 and radius > Y i n E, (3.13.13) hence any bounded set in E is contained in the closure of a bounded set B in E, . But ii( f3) is contained in the closure in F of the set C(B) = u(B) by (3.11.4); now, u(B)

2 COMPACT OPERATORS

319

is relatively compact in F,, i.e. its closure in F, is compact, hence closed in F, and therefore equal to its closure in F. This shows that G(B) is contained in F, and relatively compact in that space. Q.E.D. (11.2.10) Let E be a normed space, F a Banach space, (u,) a sequence of nzuppings in Y ( E ; F) (Section 5.7) w~hichconverges to u in Y(E; F). Then, if every u, is compact, u is compact. Let B be any bounded set in E ; as F is complete, all we have to do is to prove that u(B) is precompact (3.17.5). Now B is contained in a ball /IxJId a ; for any E > 0, there is no such that n 2 no implies /Iu - u,II 6 6 / 2 4 and therefore (by (5.7.4)) IIu(x)- u,(x)II 6 ~ / for 2 any x E B. But as u,,(B) is precompact, it can be covered by finitely many balls of centers y j (1 6 j < m) and radius ~ / 2For . any x E B, there is therefore a j such that IIu,,(x) - yjll Q 4 2 , hence IIu(x) - yj/l 6 E , and the balls of centers y j and radius E cover u(B). Q.E.D. In particular, any limit in Y(E; F) of a sequence of mappings of finite rank is compact by (11.2.1) and (11.2.10). Whether conversely any compact mapping is equal to such a limit is still an open problem (see Problem 4).

PROBLEMS

Let E be a Banach space, A a bounded open subset of E, F a finite dimensional vector space. Show that for any p 3 1, the identity mapping f-f of the Banach space 9LP'(A) (Section 8.12, Problem 8) into 9Pp-l)(A)(the latter being replaced by %$' (A) for p I ) is a compact operator. (Use the mean value theorem and Ascoli's theorem.) Let u be a compact mapping of an infinite dimensional Banach space E into a normed space F. Show that there is in E a sequence (x,) such that IIxJ = 1 for every n, and lim u(x,) = 0. (Observe that there is a number a > 0 and a sequence (y") in E such that "+a,

l / y l l = 1 for every n, and lly,, - y, I/ > a for m # n (Section 5.9, Problem 3, and (3.16.1)), and consider the sequence (~(y,,)).) Conclude that if the image by u of the sphere S : /lx/l= 1 is closed in F, it contains 0.

Let E be a separable Hilbert space, (e,) a Hilbert basis of E. If u is a compact mapping of E into a normed space F, show that the sequence (u(e.)) tends to 0. (Use contradiction, and show that it is impossible that the sequence (u(e,)) should have a limit 0 # 0 in F.) If, conversely, F is a Banach space and the series of general term /lu(e,)lj2 is convergent, show that u is compact (use the Cauchy-Schwarz inequality to prove that the image of the ball l/xl/< 1 by u is precompact). Let F be a normed space having the following property: there exists a constant c > 0 such that, for anyfinite subset ( a i ) 1 6 i bof n F, and any E > 0, there exists a decomposition E = M N of E into a direct sum of two closed subspaces, such that M is finite dimensional, d(u,, M) < E for I < i < I ? , and if for any x E F, x = p ( x ) q ( x ) , where

+

+

320

XI

ELEMENTARY SPECTRAL THEORY

< c . d(x, M). Show that, under that assumption, any compact linear mapping of a normed space E into F is a limit in Y(E; F) of a sequence of linear mappings of finite rank (use the definition of precompact spaces). Show that any Hilbert space satisfies the preceding condition, as well as the spaces ( c d (Section 5.3, Problem 5 ) and I' (Section 5.7, Problem 1). 5. Let I = [ a ,b ] be a compact interval in R, K(s, t ) a complex valued function defined in I x 1, and satisfying the assumptions of Section 8.11, Problem 4. Show that if U is defined as in (11.2.3), U is still a compact mapping of E = %',(I) into itself. p ( x ) E M and q(x) E N, then llq(x)ll

3. T H E T H E O R Y OF F. RlESZ

We will need repeatedly the following lemma: (11.3.1) Let u be a continuous operator in a normed space E, v = 1, - 11, L, M two closed vector subspaces of E such that M c L, M # L, and v(L) c M. Then there is a point a E L n M such that llall < 1 and that, for any x E M, lMa>- u(x>Il 2 t .

c

By assumption, there is b E L such that b 4 M, hence d(b, M) = CY > 0. Let M be such that \lb - yll < 2a, and take a = ( b - y)/lib - y / l ; we have llall = 1, and, for any z E M, a - z = ( b - y - Ilb - yllz)/llb - yll; but as y + 116 - yllz E M , we have 116 - y - Ilb - yllzll 2 a, hence ]la - zil 2 for any z E M. But, for x E M, we have u(a) - u(x) = a - (x v(a) - ~ ( x ) )and , by assumption, x u(a) - v(x) E M ; hence our conclusion. y

E

+

+

+

(11.3.2) Let u be a compact operator in a normed space E, and let v = I, - u. Then: (a) the kernel v-'(O) isfinite dimensional; (b) the image v(E) is closed in E; (c) ti(E) hasjinite codimension in E; (d) if ~ ~ ' ( = 0 {) 0 } , then v is a linear honieomorphism of E onto v(E) (cf. (11.3.4)).

(a) For any x E N = v-'(O), we have u(x) = x , hence the image of the ball B: llxll < 1 in N by u is B itself; by assumption u(B) is relatively compact in E, hence in N since N is closed in E. But this implies that N is finite dimensional by Riesz's theorem (5.9.4). there is then a sequence (x,) in E, such that (b) Suppose y = lim ~(x,,) (3.13.13). Suppose first that the sequence (d(x,,, N)) is n-+ m

unbounded; then, by extracting a subsequence, we may suppose that

3 THE THEORY OF F. RlESZ

lirn d(x,, N)

=

+

00.

Let z,

= x,/d(x,

321

, N); it is immediate that d(z, , N) = 1,

fl+W

and therefore there is t, E N such that I/z, - t,l/ < 2. Let s, = z, - t,, and observe that by definition we have u(s,) = u(z,) = o(x,)/d(x,, N), and d(s,, N) = 1. From the assumptions we deduce at once that lirn u(s,) = 0. n+ w

But the sequence (s,) is bounded in E ; as u is compact, there is a subsequence (s,J such that (u(snk)) converges to a point a E E. As lirn (s, - u(s,)) = 0, we also have lirn s,, n-m

v(a) = lim u(s,,)

= a, = 0,

n+

w

hence, as x -+ d ( x , N ) is continuous, d(a, N)

=

1. But

and this contradicts the definition of N.

k-+ w

We therefore can suppose that the sequence (d(x,, N)) is bounded by a number M - I ; there is then a sequence x; such that x, - x; E N and Ilx;l\ 6 M ; as c ( x 3 = u(x,), we may suppose that IIx,II 6 M. Then as u is compact there is a subsequence (xnk) such that (u(xnk))converges to a point b E E ; as xnk- u(x,,) = ~(x,,) tends to y , (x,,,)tends to b + y and by continuity we have u(b + y ) = y , which proves that y E v(E), hence u(E) is closed. (c) T o say that u(E) has an infinite codimension in E means that there exists an infinite sequence (a,) of points of E such that a, does not belong to the subspace Vn-l generated by o(E) and by a,, . . . , a,-, for every n. Now each V, is closed since u(E) is closed (using (5.9.2)). By (11.3.1)we can define by induction a sequence (b,) such that 6, E V,, 6, $ V n - l , llb,/l 6 1, and IIu(b,) - u(bj)I/2 3 for any j 6 n - I . This implies that the sequence (u(b,)) has no cluster point, contradicting the assumption that u is compact. (d) In order to prove that 21 is a homeomorphism of E onto v(E) when V ' ( 0 ) = {0}, it is only necessary to show that for any closed set A c E, u(A) is closed in E (hence in o(E)) (3.11.4).But this is proved by exactly the same argument as in (b), replacing throughout E by A (and N by (0)).

(11.3.3) Under the same assumptions as in (11.3.2),define inductively N, = v - ' ( o ) , N , = U - l ( N k - l )for k > 1, F, = @), F, = V ( F k - 1 ) f o r k > 1. Then : (a) The N , form an increasing sequence of finite dimensional subspaces, the F, a decreasing sequence of finite codiniensional closed subspaces. (b) There is a smallest integer n such that N k + l = N , for k 2 n ; then Fk+l= F, for k Z n, E is the topological direct sum (Section 5.4)of F,and N,, and the restriction of 1: to F, is a linear homeomorphism of F, onto itself. (a) Define by induction

zl1

= 11, ti, = i I k - l

0

t'; I

claim that

ilk

= 1, - u,,

322

XI

ELEMENTARY SPECTRAL THEORY

induction on k , for U , and the result follows at once from the inductive hypothesis and from (11.2.6) and (11.2.5). Then by definition Nk = a ~ ' ( 0 and ) F, = zik(E), and our assertion follows from (11.3.2).

where Zlk =

(IE

uk

is

- Uk-1)

compact:

O

this

is

shown

by

( 1 -~ U ) = 1, - uk-1 - U -k uk-1

O

(b) Suppose Nk # Nk+1 for every k . We have U(Nk+,) C Nk for k 3 1 ; by (11.3.1), there would exist an infinite sequence (xk) of points of E such that xk E Nk, xk $ Nk-1, llxkll < 1 for k > 1 and IIu(xk) - U(Xj)II > f for any j < k . This implies that the sequence (u(xk))has no cluster point, contradicting the assumption that u is compact. Similarly, suppose Fk+l# Fk for every k . We have v(Fk) c F,+, for k > 1 ; by (11.3.1), there would exist an infinite sequence (x,) of points of E such that xk E Fk, xk $ Fk+l, llxkll < 1 for k 2 1, and IIu(xk) - U(Xj)II > for a n y j > k . This again implies contradiction, hence there exists a smallest integer m such that Fk+,= Fk for k 2 m . Next we prove that N, n F, = (0): if y E F, n N , , then there is x E E such that y = v,(x), and on the other hand v,(y) = 0; but this implies that tiZn(x)= 0, hence x E N,, = N, , and y = v,(x) = 0. By definition, we have F, c F, and v(F,) = F,; let us prove that F, = F,. Otherwise, we would have nz > n ; let z be such that z E F,-, c F,, and z $ F,; as U ( Z ) E F, = v(F,), there is a t E F, such that u(z) = v ( t ) , i.e. z - t E N, c N,; but as z - t E F,, we conclude that z = t , and our initial assumption has led to a contradiction. For each x E E we have v,(x) E F, = F,, and as u,(F,) = F, by definition of nz, there is y E F, such that v,(x) = u,(y), hence x - y E N, , and therefore E = F, + N , . This last sum is direct since F, n N, = ( 0 ) ; F, is closed and N, is finite dimensional, therefore (5.9.3) E is the topological direct sum of F, and N,. Finally, the restriction of I' to F, is surjective and its kernel is F, n N, c F, n N, = {0}, hence it is also injective. By (11.3.2(d)) that restriction is a linear homeomorphism of F, onto itself, and this ends the proof.

(11.3.4) Under the same assumptions as in (11.3.2), 11 is injectiile (i.e. v-'(O) = {0)), then v is surjective, hence a linear homeomorphism of E onto itself. For the assumptions imply that Nk = (0) for every k , hence n = 1 and N, is reduced to 0, therefore F, = E by (11.3.3) and the result follows from (11.3.3).

4

SPECTRUM OF A COMPACT OPERATOR

323

PROBLEMS

1. Let E, F be two Banach spaces, f a continuous linear mapping of E into F such that f(E) = F; then, there exists a number m > 0 such that for any y E F, there is an x E E for whichf(x) = y and Ilxll < mllyll (12.16.12). (a) If (y.) is a sequence of points of F which converges to a point b, show that there exists a subsequence ( y n k ) and , a sequence (x,) of points of E, which converges to a point a and is such that f(xk) = y,,, for every k . (Take (y,J such that the series of general term IIy.,+ - ynk/lis convergent.) (b) Let u be a compact mapping of E into F, and let u =f- u. Show that u(E) is closed in F and has finite codimension in F. (Follow the same pattern as in the proof of (11.3.2), using (a).) (c) Define inductively F1 = u(E), Fk+l = u(f-'(Fk)) for k > 1 ; show that there is an integer n such that = F, for k > n (same method). (d) Take E = F to be a separable Hilbert space, and let (en)nS1 be a Hilbert basis of E. Definefand u such that f(e,) = e n - 3for n > 4,f(e,) = 0 for n < 3, u(e,) = e.-z/n for n > 6, u(el)= u(e3)= 0, u(e2)= - e 2 , u(e4)= e l , u(e5) = e2 (e3/5). Define inductively NI = u-'(O), N,+,= u-'(f(Nk)) f o r k > I ; show that the N kare all distinct and finite dimensional. 2. Let E, F be two normed spaces,fa linear homeomorphism of E onto a closed subspace f(E) of F, u a compact mapping of E into F, and let u =f- u. (a) Show that u-'(O) is finite dimensional and u(E) is closed in F; furthermore, if u-'(O) = {O}, u is a linear homeomorphism of E onto u(E). (Follow the same method as in (11.3.2).) (b) Define inductively NI= u-I(O), N,+, = u-'(f(N,)) for k > 1 ; show that there is an integer n such that N k + = N,for k 2 n. (c) Give an example in which, when F1 = u(E), and F,,, = u(f-'(F,)) for k > 1 , the F, are all distinct (take for E = F a separable Hilbert space, and for f a n d u the adjoints (Section 11.5) of the mappings notedfand u in Problem l(d)). 3. Let E be a Banach space, g a continuous linear mapping of E onto itself such that Ilg11 ( 1 - 211g11)/2.) 4. In the space E = I' (Section 5.7, Problem 1; we keep the notations of that problem), ) e 2 k f 2 (k > O ) , f ( e , )= eo, f(e2n+l)= let f be the automorphism of E such that f ( e Z k = e2k-1 for k I , and let u be the compact mapping such that u(e,) = 0 for n # 1 , and u(el)= e o . If u =f- u, and the Fkand N,are defined as in (11.3.3), show that Na+l # Nkand F, + # F, for every k .

+

4. SPECTRUM OF A C O M P A C T OPERATOR

(11.4.1 ) Let u be a compact operator in a complex normed space E. Then: (a) The spectrum S of u is an at most denumerable compact subset of C , each point of which, with the possible exception of 0, is isolated; 0 belongs to S if E i s injinite dimensional.

324

XI

ELEMENTARY SPECTRAL THEORY

(b) Each number A # 0 in the spectrum is an eigenvalue of u. (c) For each A # 0 in S, there is a unique decomposition of E into a topological direct sum of two subspaces F(A), N(A) (also written F(A; u), N(A; u)) such that: (i) F(A) is closed, N(A) i s j n i t e dimensional; (ii) u(F(2)) c F(A), and the restriction of u - A . 1, to F(A) is a linear homeomorphism of that space onto itself; (iii) u(N(A)) c N(A) and there is a smallest integer k = k(A), called the order oj'A (also written k(A; u)),such rhat the restriction to N(A) of(u - A . I , ) k is 0. (d) The eigenspace E(A) of u corresponding to the eigenvalue A # 0 is contained in N(A) (hence finite dimensional). (e) I f A, p are two difeerent points of S , distinct from 0 , then N(p) c F(A). (f) I f E is a Banach space, the function (-+ ( u - ( I,)-', which is dejned and analytic in C - S , has a pole of order k(A) at each point A # 0 of S. Let A # 0 be any complex number; as A-'u is compact, we can apply the Riesz theory (Section 11.3). By (11.3.4), if A is not an eigenvalue of u, I - A- ' u is a linear homeomorphism of E onto itself, and the same is true of course of u - A . 1, = -A(l, - A - l u ) , i.e. A is regular for u, which proves b). Suppose on the contrary A is an eigenvalue of u ; then the existence of the decomposition F(A) + N(A) of E with properties (i), (ii), (iii), follows from (11.3.3), as well as (d) (E(A) is the kernel noted N, in (11.3.3)). To end the proof of (c), we need only show the uniqueness of F(A) and N(A). Suppose there is a second decomposition E = F' N' having the same properties, and write v = u - A . 1., Then, any x E N' can be written x = y + z where y E F(A), z E N(A); by assumption there is h > 0 such that vh(x)= 0, hence vh(y) = 0; as the restriction t o F(A) of vh is a homeomorphism by assumption, y = 0 and x E N(1). This proves that N' c N(A), and a similar argument proves N(1) c N'. Next, if x = y + z E F' with y E F(n), z E N(A), we have vk(x) = vk(y), hence vk(F')c F(A); but as v(F') = F', this implies F' c F(A), Denote by u l , u2 the restrictions of u t o F(A) and N(A), respectively. From the relation (u2 - A . l N ( n ) ) k = 0, it follows by linear algebra (A.6.10 and A.6.12) that there is a basis of N(1) such that the matrix of u2 - I . IN(,, with respect to that basis is triangular with diagonal 0; if d = dim (N(A)), the determinant of u2 - ( lN(l)is therefore equal to ( A - ( ) d and this proves that u2 - ( . lN(n) is invertible if 5 # A. Let us prove on the other hand that u, - ( . lF(A)is invertible for ( - A small enough: we can write u1 - [ . 1 F(i) = v1 + (A - 5) . IF(,, with u1 = u1 - A . We know by (c) that V , is invertible; by (5.7.4), we therefore have I ~ V ; ~ ( X ) I I < IIv;' II * /Ixll in F(A), which can also be written IIu,(x)I/ 2 c . l/xil with c = llv;'/l-l. Now if i # 0 and

+

4 SPECTRUM OF A COMPACT OPERATOR

325

u1 - ( . IF(,) is not invertible, this implies, by (b) (applied to F(A) and u l , using (11.27)) that there would exist an x ;f; 0 in F ( I ) such that u l ( x ) = ('x, hence li- A1 . llxll = ilol(x)j/ 3 c . IIxll, which is impossible if - A1 < c. This shows that for [ # 0, i# I , and - I )< c, u - i .I, is invertible (since its restrictions to F(A) and N(A) are), i.e. ( is not in S; therefore all points A # 0 in S are isolated, and S is at most denumerable. By (b), for each A # 0 in S, there is x # 0 in E such that u(x) = Ax, hence 121 . ~~x~~ < 1 ~ . ~~x~~ ~ ~ by1 (5.7.4), and 111 < IIull, which proves S is compact. To end the proof of (a), suppose E is infinite dimensional; if u were a homeomorphism of E onto itself, the image u(B) of the ball B: ~1~~~ < 1 would be a neighborhood of 0 in E, and as it is relatively compact in E, this violates Riesz's theorem (5.9.4). If p is a point of S distinct from 0 and A, and x E N(p), we can write x =y z with y E F ( I ) , z E N(I). We have seen above that the restriction of w = u - /i . I, to N(A) is a homeomorphism; as wh(x)= 0 for h large enough, and wh(y)E F(I), wh(z)E N(A), we must have wh(y)= wh(z) = 0, which proves statement (e). in C - S follows If E is a Banach space, the analyticity of ( u - i. I,)-' from (11.1.2). With the same notations as above, A is not in the spectrum of u l , hence (by (11.2.7)) (ul - . IF(,,)-' is analytic in a neighborhood of A ; in particular, there are numbers p > 0 and M > 0 such that

+

for x E F(A) and l i - AI < p. On the other hand, we can write u2 -5 . I,(,, = (A - i). I,,,)+ 1i2 with u2 = u2 - A . I,(,, , and we know that for 5 # A, u2 -( . I,(,, is invertible; moreover, we can write

since 11; = 0. From this it follows that there is a number M' > 0 such that l i - 21k . l l ( u ~- i . IN(~.J-'(x)Il< M'llxll for l i - A1 < p, I # A and for any x E N(1). Now any x E E can be written x = y z with y E F(A), z E N(A), and there is a constant a > 0 such that llyll < a/lxl/and llzll < allxll (5.9.3); therefore we see that, for li - I1 < p, i# A, and any ~ E E we , have

+

l i - I l k ll(u - i. 1E)-'(X)Il

< 4 M p k + M')IIxII. < a ( M p k + M') for if A

In other words, l i - Alk . ll(u - i .l,)-'Il and - A1 6 p ; by (9.1.5.2), this implies that A is a pole of order < k for ( u - i* I,)-'. But by definition there is an x E N ( I ) such that z~:-'(x) # 0, hence (i-A)"-'((u - i I,)-'(X)) is not bounded when i# I tends to A, and this proves that A is a pole of order k , and ends the proof of (1 1.4.1). 9

326

XI

ELEMENTARY SPECTRAL THEORY

We say that the dimension of N(1) is the algebraic multiplicity of the eigenvalue A. of u, the dimension of the eigenspace E(1) its geometric n d tiplicity; they are equal if and only if k(1) = 1 ; when E is a Banach space, this is equivalent to saying that 1 is a simpre pole of ( u - 5 . l&'. (11.4.2) Let E be a Banach space, E, a dense subspace of E, u a compact operator in E,, ii its unique continuous extension to E. Then /he spectra of u and ii are the same, and for each eigenoalue A # 0 of u, N(A, u) = N(1, u"), E(1, u) =: E(1, ii) and k(1, u) = k(1, ii).

We know that u" is compact and maps E into E, , by (11.2.9); if A # 0 is an eigenvalue of u", any eigenvector x corresponding to 1 is such that x = 1-'C(x) E E,, hence A is an eigenvalue of u, and E(1, u") c E(1, u ) ; the converse being obvious, we have sp(ii) =sp(u) and E(A, u) = E(A, u") c E, for each eigenvalue A # 0. Considering similarly the kernels of ( u - A . IEJk and of its extension (ii - A . I E)k we see that they are equal, hence k(1, u) = k(A, u") and N(A, u) = N(1, u") c E, .

PROBLEMS

1. Let E be a coniplex Banacli space, u a compact operator in E; we keep the notations of (11.4.1), and in addition, we write pi (or p i , J and qA=- l E - p i the projections of E onto N(h) and F(h) in the decomposition of E as direct sum F(h) i N(h). (a) Show that -pi is the residue of the meromorphic function (u - 5 . I E ) - l at the pole h, for every A E sp(u) such that h .?; 0. (b) If h1, . .. , h, are distinct points of the spectrum sp(u), show that the projections p i , ( I < j < r ) commute, and that p i , I ' . . -+-pi, is the projection of E onto N(h,) -I . . . -t N(hJ in the decomposition of E as direct sum of that subspace and of F(hl) n F&) n . . . n F(h,). 2. Let E be an infinite dimensional complex Banach space, u a compact operator in E, (uJngl a sequence of compact operators in E, which converges to I I in the Banach space Y(E). (a) Prove that for any bounded subset B of E, the union u.(B) is relatively compact

u"

in E. (Show that it is precompact.) (b) If A E C does not belong to sp(u), show that there is an open disk D of center h and an integer no such that, for n > n o , the intersection sp(u,,) n D = 0(use (8.3.2.1)) and ( [ i n- 5 . IE)-' converges uniformly to (u - 5 . I E ) - I for 5 E D. (c) Let (p.) be a sequence of complex numbers such that p.~sp(u,) for every n ; such a sequence is always bounded. If h is a cluster point of ( p J , show that h E SP(N). (One can assume that h = lim pn 0; there is then x. E E such that llxn/l= 1 and u.(xn) = h.x,; use then (a).)

"'m

+

4

SPECTRUM OF A COMPACT OPERATOR

327

(d) Conversely, let h # 0 be in sp(u). Show that for each n there is (at least) a number E sp(u,) such that h = lim p.. (Otherwise, one can assume that there is an open

p.

"+ m

disk D of center h and radius Y, such that D n sp(u) = {A} and D n sp(u.) = @ (extract from (u,) a suitable subsequence). Let then ^J be the road t + A Yeirdefined in [0,277];consider the integral

+

sy(lln 5 -

* IE)-l(5

-

d[

=0

for k

>0

and use (b) to obtain a contradiction.) (e) Let h # 0 be in sp(u), and let D be an open disk of center h and radius r such that D n sp(u) = {A}; there exists no such that, for n > no the intersection of sp(un) and of the circle 15 hl = r is empty (use (c)). Let pI,... , p, be the points of D n sp(u,),

c k(pj;

~

and write k , =

uJ. Show that there exists nl such that, for n

j=l

> n l , k, > k(h;u).

(Use the same method as in (d), multiplying (u, - 5 . by a suitable polynomial in 5 of degree k , .) Give an example in which k, > k(h; u) for every n. (f) With the notations of (e), let p =pi.", pn=

c pPJ,""; show that

j =

1

limp, = p

n- m

in the Banach space 9 ( E ) (use (b), and Problem I). Deduce from that result that there exists n2 such that, for n 2 n 2 , N, = N(p, ; u,) . . . N(p, ; u.) is a supplement to F(h; u) in E. (Suppose n is such that IIp --p.li < 1/2; if there was a point x, E F(h) n N. such that /lx, /I = I , then the relations p(x.) = 0, p,,(x,) = x, would contradict the preceding inequality. Prove similarly that the intersection of N(h; u ) and of the subspace F(pI ; u,) n . . . n F(p,;u,) is reduced to 0.) 3. Let u be a compact operator in an infinite dimensional complex Banach space E, and let P(5) be a polynomial without constant term; put v = P(u). Show that the spectrum sp(u) is identical to the set of numbers P(h), where h E sp(u); furthermore, for every p E sp(v), N(p; u ) is the (direct) sum of the subspaces N(h,; u ) such that P(&) = p, and F(p; u ) the intersection of the corresponding subspaces F(&; u). (Let V be any closed subspace of E, such that u(V) C V, and let uv be the restriction of u to V. Show that there is a constant M independent of V and n, such that ll(P(uv))'ll < M" lIuqI1. Apply that remark and Problem 1 of Section 11 .I, taking for V a suitable intersection of a finite number of subspaces of the form F(h; u).) 4. Let E be a separable Hilbert space, (en)n3oa Hilbert basis of E. Show that the operator u defined by u(e,) = e.+l/(n 1 ) for n > 0 is compact and that sp(u) is reduced to 0 (more precisely, u has no eigenualue). 5. Let u be a continuous operator in a complex Banach space E. A Riesz point for u is a point h in the spectrum sp(u) such that: (I) h is isolated in sp(u); (2) E is the direct sum of a closed subspace F(h) and of a finite dimensional subspace N ( h ) such that u(F(h)) c F(h), u(N(h))c N(h), the restriction of u - h . 1, to F(h) is a linear homeomorphism and the restriction of u - h . 1, to N ( h ) is nilpotent. (a) If h and p are two distinct Riesz points in sp(u), show that N(p) C F(h), and F(h) is the direct sum of N(p) and F(h) n F(p). (b) A Riesz operator u is defined as a continuous operator such that all points f 0 in the spectrum sp(u) are Riesz points. For any E > 0, the set of points h E sp(u) such that 1x1 > E is then a finite set { p i ,. . ., p , } ; let pz be the projection of E onto N(pJ in the decomposition of E into the direct sum N(pJ -t F(pJ ( I < i < Y), and let

+ +

+

u =u -

2u

i=l

0

p i . Show that sp(v) is contained in the disk

Problem I ) that lim I1u"Il"" n-m

< E.

151 < E , hence (Section 11 .I,

328

XI

ELEMENTARY SPECTRAL THEORY

(c) In the Banach space Y(E), let X be the closed (11.2.10) subspace of all compact operators. Show that, in order that u E Y(E) be a Riesz operator, it is necessary and sufficient that lim (d(u", Y ) ) ' '= " 0. (To prove that the condition is necessary, use n-rm

(b), observing that u" = u" i- w,, where w, is an operator offinirerank, hence compact. To prove that the condition is sufficient, use the result of Problem 3 of Section 11.3, which can be interpreted in the following way: if ilgIl< 4, then either h = 1 does not belong to sp(y u ) or is a Riesz point for y -t u.)

+

5. C O M P A C T OPERATORS IN H I L B E R T SPACES

Let E be a prehilbert space, u an operator in E. We say u has an adjoint if there exists an operator u* in E such that (11.5.1) for any pair of points x,y in E. It is immediate that the adjoint u* is unique (when it exists), and (by Section 6.l(V)) that then (u*)* exists and is equal to u. It is similarly verified that when the operators u and v have adjoints, then u + v, AM, and uv have adjoints respectively equal to u* + v*, Xu*, and u*u*. (11.5.2) If u is continuous and has an adjoint, then u* is continuous and lJu*JJ = JIuJ)in Y(E). Zf E is a Hilbert space, every continuous operator in E has an adjoint. From (11.5.1) and the Cauchy-Schwarz inequality (6.2.4) we deduce for any pair x, y ; taking x = u*(y), we get Ilu*(y)II d llull . llyll for any y E E, which proves the continuity of u* and the inequality JIu*JJ d JJuJ1; the converse inequality is proved by interchanging u and u* in the argument. If E is a Hilbert space and u is continuous, then, for any Y E E , the linear form x-+ (u(x)Iy) is continuous, and by (6.3.2) there exists a unique vector u*(y) such that (11.5.1) holds. From the uniqueness of u*(y), we conclude that u* is linear, hence the adjoint of u. The second statemento f (11.5.2) does not extend to prehilbert spaces. An operator u in a prehilbert space E is called self-adjoint (or hermitian) if it has an adjoint and if u* = u ; the mapping (x, y ) -+(u(x) I y ) = ( u ( y ) I x ) is

then a hermitian form on E; the self-adjoint operator u is called positioe (resp. nondegenerate) if the corresponding hermitian form is positive (resp. nondegenerate); one writes then u > 0. For any operator u having an adjoint, u + u* and i(u - u*) are self-adjoint operators.

5

COMPACT OPERATORS IN HILBERT SPACES

329

(1 1.5.3) (i) If a continuous operator u in a prehilbert space E has an adjoint, = then u*u and uu* are selfadjoint posiiive operators, and j(u*uI( = (Juu*(( llu\lz = 11u*11’. In particular, i f u is self-adjoint, I I U ’ I / = 1 1 ~ 1 1 ~ . (ii) If P is the orthogonal projection of E on a complete vector subspace F (Section 6.3), P is apositive hermitian operator. Conversely, i f E is a Hilbert space, every continuous operator P in E which is hermitian and idempotent (i.e. P2 = P) is the orthogonal projection of E on the closed subspace P(E) (these operators are called the orthogonal projectors in LZ(E)).

(i) The fact that u*u and uu* are self-adjoint follows from the relations (u*)* = u and (uv)* = v*u*; moreover (u*u(x) I x) = (u(x) I u(x)) 2 0 for any x E E, and it is proved similarly that uu* is positive. Further this last relation shows that IIu(x)11’ < IIu*u(x)II . llxll by Cauchy-Schwarz, hence (by (5.7.4)) llul12 < IIu*uII. On the other hand, /Iu*uII < IIu*Il . llull = l\ull’ by (5.7.5) and (1 1.5.2), and this concludes the proof of (i). (ii) If P is the orthogonal projection of E on a complete subspace F, then (P . x l y - P y ) = 0 for x E E, y E E, hence (P . x Iy) = ( P . x IP - y ) = (x I P .y ) , which proves that P is hermitian, and it is positive since ( P x I x) = ( P x I P x) 2 0.Conversely, suppose E is a Hilbert space, and P2 = P = P*; as the then, for all x,y in E, ( P . x I y - P . y ) = ( x I P . y - P ’ . y ) = O ; relation y = P x implies P * y = P2 . x = P * x = y , P(E) is the kernel of 1, - P, hence is a closed vector subspace; furthermore, for any y E E, y - P y is orthogonal to every P . x,in other words to P(E), which proves (ii). (1 1.5.4) If E is a Hilbert space, the adjoint of any compact operator u in E is a compact operator. As E is complete, it will be enough to prove that the image u*(B) of the -

ball B: llyll < 1 isprecompact. Let F = u(B), which is a compact subspace of E, and consider, in the space %,-(F) (Section 7.2) the set H of the restrictions to F of the linear continuous mappings x -+ (x 1 y ) of E into C,where y E B; we prove that H is relatively compact in V,(F). Indeed, we have I(x - x’I v ) ~ < IIx - x’/I by the Cauchy-Schwarz inequality, since llyll < !, which shows that H is equicontinuous; on the other hand F is contained in the ball llxll < IIuII, hence I(x Iy)l < llull for anyy E B and a n y x E F ; Ascoli’s theorem (7.5.7) then proves our contention. Therefore, for any E > 0, there exist a finite number of points y j (1 < j < m) in B such that for any y E B, there is an indexj such that I(u(x) I y - yj)l < E for any x E B. But by (1 1.5.1 ) this last inequality is written I(x I u*(y) - u*(yj))I < E , and either u*(y) = u*(yj) or we can take x = z/llzll, where z = u*(y) - u*(yj);we therefore conclude that I/ u*(y) - u*(yj)II < E , and this ends the proof.

330

XI

ELEMENTARY SPECTRAL THEORY

Note that the proof that u*(B) is precompact still holds when E is not complete; but it can happen that in a prehilbert space E, a compact operator has an adjoint which is not compact.

(11.5.5) Let u be a compact operator in a complex preliilbert space E , having an adjoint u* which is compact. Then : (a) The spectrum sp(u*) is the image of sp(u) by the mapping 5 + [. (b) For each A # 0 in sp(u), k(A; u) = k(X; u*). (c) I f u = u - 1 * 1 , , then u*(E) is the orthogonal supplement (Section 6.3) of v-'(O) = E(A; u), and the dimensions of the eigenspaces E(A; u) and E(X; u*) are equal. (d) The subspace F(X; u*) is the orthogonal supplement of N(A; u), and the dimensions of N(A; u) and N(X; u*) are equal.

We have u* = u* - X l,, hence (u(x)I y ) = (x I u*(y)) from (11.5.1), and therefore the relation u(x) = 0 implies that x is orthogonal to the subspace v*(E). Now by (11.4.1) applied to u*, u*(E) is the topological direct sum of F(X; u*) and of the subspace u*(N(X; u*)) of N(X; u*), and from linear algebra (A.4.17) it follows that the codimension of u*(E) is equal to the dimension of u*-'(O) = E(X; u*); hence we have dim E(A; u) < dim E(X; u*). But u = (u*)*, hence we have dim E(A; u) = dim E(X; u*); furthermore, the orthogonal supplement of E(A; u) contains u*(E) and has the same codimension as v*(E), hence both are equal, which proves (c). This also shows that for any eigenvalue A # 0 of u, X is an eigenvalue of u*, and as the converse follows from the relation u = (u*)*, we have also proved (a). The same argument may be applied to the successive iterates vh of u, and shows that the image of E by u * ~= (oh)* is the orthogonal supplement of the kernel of vh. Using (11.3.2), (11.4.1), and the relation u = (u*)*, this immediately proves (b) and (d). Theorems (1 1.4.1) and (11.5.5) can be translated into a criterion for the solutions of the equation u ( x ) - Ax = y : (1 1.5.6)

Under tlze assumptions of (11.5.5) :

(a) If2 is riot in the spectrunz of u, the equation u(x) - Ax = y has a unique solution in E for every y E E. (b) IfA # 0 is in the spectrum of u, a necessary and suficient condition for y E E to be such that the equation u(x) - Ax = y have a solution in E is that y be orthogonal to the solutions of the equation u*(x) - Ax = 0.

5 COMPACT OPERATORS IN HILBERT SPACES

331

For a finite dimensional space, this reduces to the classical criterion for existence of a solution of a system of scalar linear equations.

(11.5.7) Let u be a compact self-adjoint operator in a complex Hilbert space E. Then: ( a ) Every element of the spectrum sp(u) is real and k(A) = 1 for every eigenvalue A # 0 of u. (b) If A, p are two distinct eigenvalues of u, the eigenspaces E(A) and E(p) are orthogonal. (c) Let (p,) be the strictly decreasing (finite or infinite) sequence of eigenvalues > 0 , (v,) the strictly increasing (finite or infinite) sequence of eigenvalues p , whence it (11.1.2) the mapping [ -+ (u - [ lE)-' is analytic for follows at once that the mapping 5 + ( I E - tu)-' is analytic for < l/p. Now, for 5 in a sufficiently small neighborhood of 0, the power series m

n=O

runconverges to ( I E - @)-I

in 9 ( E ) (8.3.2.1); by (9.9.4) that power

< l/p. Furthermore, for each r such series converges for every { such that that 0 < r < l/p, if M is the maximum of Ij(lE - tu)-' 11 for 151 = r, the Cauchy inequalities (9.9.5) yield IIu"lI < M/r" < Mp". In particular, if we use (11.5.3),

332

XI

ELEMENTARY SPECTRAL T H E O R Y

we get here 1 1 ~ 1 1 ~ 0 , and any x E E, llu"(x)112 < l l ~ " - ~ ( x )I!u"+l(x)/l Schwarz). 0

.

5 COMPACT OPERATORS IN HILBERT SPACES

337

+

(b) Suppose E is a Hilbert space, and u is a compact self-adjoint operator. If u(x) 0, show that u"(x) # 0 for any integer n > 0, and that the sequence of positive numbers m, = llu"+'(x)ll/llu"(x)l~ is increasing and tends to a limit, which is equal to the absolute value of an eigenvalue of u. Characterize that eigenvalue in terms of the canonical decomposition of x ; when does the sequence of vectors u"(x)/ llu"(x)11 have a limit in E ? (Use (11.5.7).) 11. Let u be a compact self-adjoint operator in a complex Hilbert space E, and let f b e a complex valued function defined and continuous in the spectrum sp(u). Show that there is a unique continuous operator u such that (with the notations of (11.5.7)), the restriction of u to E(pk) (resp. E(vk), E(0)) is the homothetic mapping y -f(pk)y (resp. y + f ( v , ) y , y + 0). This operator is written f ( u ) ; one has ( f ( u ) ) *=.f(u). If g is a second function continuous in sp(u), and h = f + g (resp. h = f g ) , then h(u) = f ( u ) -t- g(u) (resp. h(u) = f ( u ) g ( u ) ) . In order thatf(u) be self-adjoint (resp. positive and self-adjoint), it is necessary and sufficient that f ( 6 ) be real in sp(u) (resp. f(5) 0 in sp(u)); in order that f ( u ) be compact, it is necessary and sufficient that f ( 0 ) = 0. 12. Let u be a compact positive hermitian operator in a complex Hilbert space E. Show that there exists a unique compact positive hermitian operator u in E such that v2 = u ; v is called the square root of u.

Let E be a separable complex Hilbert space, (e,Jnbl a Hilbert basis of E. Let u be the compact operator in E defined by u ( e l )= 0, u(e.) = e n W l / nfor n > 1 . Show that there exists no continuous operator u in E such that u2 = u. (Observe first that H = u*(E) -is a closed hyperplane orthogonal to el, and that it is contained in H' = u*(E); as H' is orthogonal to x l= u(el), conclude that necessarily x1 = 0; next consider x2 = u(e2), and observe that u(v(e2))= 0, hence necessarily x 2 = hel, where is a scalar; but this implies x2 = 0, hence u(e2)= 0, a contradiction.) 14. Let E be a separable complex Hilbert space, ( e J n b o a Hilbert basis, u the compact positive hermitian operator in E defined by u(eo)= 0, u(e.) = e./n for n > 1 .

13.

The point a =

x (e,/n) does not belong to u(E). Let Eo be the dense subspace of E m

+

n=1

which is the direct sum of u(E) and of the one-dimensional subspace C(eo a ) . Show that the restriction u of u to Eo is a compact positive hermitian operator which is nondegenerate, although its continuous extension d = u to E is degenerate; furthermore, in the canonical decomposition (11.5.8) of the vector eo a E Eo , the summands do not all belong to Eo .

+

Let U be a compact operator in a complex Hilbert space E, and denote by R and L the respective square roots (Problem 12) of the compact positive hermitian operators U * U and UU*,respectively. Show that there exists a unique - continuous operator V in E, whose restriction to F = R(E) is an isometry onto U(E), whose restriction to the orthogonal supplement F' to F is 0, and which is such that U = VR (observe that llUxll= l l R x l l f o r e a c h x ~ E ) . P r o v ethat R = V * U = R V * V , a n d L = VRV*. (b) Let (m,) the full sequence of strictly positive eigenvalues of R, and (a,) a corresponding orthonormal system (Problem 8). If h, = Va,, show that (b.) is a n orthonormal system, and that, for any x E E, Ux - = x m . ( x I a&, where the series on the

15. (a)

right-hand side is convergent (if R,x (11.5.7), that lim IlR n-r m

-

"

= k= 1

"

mA(xIak)ak, show, using the proof of

R,I/ = 0, and apply (a)). Deduce from that result that (cL.)

is also the full sequence of strictly positive eigenvalues of L , and that (b,) is a corresponding orthonormal system. The sequence (m,) is also called the full sequence of singular values of U.

338

XI

ELEMENTARY SPECTRAL THEORY

(c) Let (p.) be the sequence of distinct eigenvalues # 0 of U,arranged in such an order that 1p.l > (pn+llfor every n for which is defined; let d, be the dimension of N(pn), and let (h.) be a sequence such that h, = pI,Ih,( > lh,,+ll for every n for which is defined, and for each k for which px is defined, the indices n for which h, = pk form an interval of N having dx elements. Show that, for each index n such that h,, and a, are defined, N(pk)for 1

< k < r , and

n (h,I < n a , . (Let V be the (direct) sum of the subspaces n

I= I

i=1

let Uv be the restriction of U to V; show that there is in V a Hilbert basis (eJ)lsJs, such that (U(eJ)Ie x )= 0 for k >j ; for n < m, if W, is the subspace of V having e l , ..., en as a basis, let U. be the restriction of U to W,, and let P,,be the orthogonal projection of E on W,. Show that

n lhjI2 is

j= 1

equal to the determinant of U ~ U=, P, U *UP,, and apply Problem 9(a).) (d) Let T be an arbitrary continuous operator in E, and let (y,,) (resp. (6,)) be the full sequence of singular values of UT (resp. T U ) . Show that y. < a. IITll (resp. 6, < a. IITll) for all values of n for which a,, y. and 6, are defined (if S = TU, observe that S * S < IITIIZU*Uand use Problem 8(d)). (e) Suppose T is also a compact operator, and let (fin) be the full sequence of its singular values. Show that n" y J < J=l

(n" a,)( fi6,) J=1

J=I

for all values of n for which

a,, fin, and ynare defined (apply Problem 9(b)). 16. Let E be a complex Hilbert space, (an)a sequence of points of E, (A,) a sequence of real numbers. Show that if the series u(x) = h,(x I a&. is convergent in E for n

every x E E, u is a hermitian operator in E. The convergence condition is always satisfied if the series of general term h,l~anllzis absolutely convergent. If in addition (a,) is a n orthonormal system, the convergence condition is satisfied if the sequence (A,) is bounded. When the h. are 2 0 and the convergence condition is satisfied, u is a positive hermitian operator. 17. Let E be a complex Hilbert space, Eo a dense vector subspace of E; ( x l y ) and //XI/ denote the scalar product and the norm in E. Suppose a second norm //xllois given on E,, for which E, is a Banach space, and is such that, for x E E,, one has llxll < a .Ilxllo, where a is a constant (in other words, the identity mapping lE0 of Eo with the norm /lxllo,into Eo with the norm IlxlI, is continuous). (a) Let U be a hermitian operator in Eo (which a priori is not assumed to be continuous); show that if U is continuous for the norm llxllo, it is also continuous for the norm llxll. (If I/U//o is the norm in 2'(EO) when Eo is given the norm Ilxll,, show that, for every integer n , one has, for x E Eo ,

Use the inequality IIUkxllz< llxll. IIU2k~II(Problem lO(a)).) (b) Let 0 be the continuous extension of U to E, which is a hermitian operator in E. Show that the spectrum of 0 is contained inthespectrumof U(when U isconsideredas an endomorphism of the Banach space Eo for the norm llxllo). (If 5 is a regular value for U,observe that ( U - 5 . l E 0 ) - I can be extended by continuity to E, using (a).) (c) Every eigenvalue h of U is real; if V = U - A . lE0, and if E(h)= V - ' ( O ) is finite dimensional in Eo, V(E,) is supplementary and orthogonal to E(A) in the prehilbert space Eo, and in addition it is closed for the norm llxll,; it follows from (12.16.8) that the restriction of V to V(E,) = F(h) is a linear homeomorphism of

5

COMPACT OPERATORS

IN

339

HILBERT SPACES

F(h) onto itself for the norm I/x/Io.Show that if p = 0 - h . l,, then P-l(O) = E(h) and P(E) = in E (apply (a) to the inverse of the restriction of V to F(h)). (d) Deduce from (c) that if U (or a power of U ) is a compact operator in Eo (for the norm Ilxllo),then 0 is a compact operator. (If (A,) is the sequence of eigenvaluesof U, deduce from (b) and (c) that the subspace of E orthogonal to all the subspaces E(h,) is the kernel of 0.) 18. Let E be an infinite dimensional complex Hilbert space. For a positive hermitian operator T i n E, the following conditions are equivalent: (1) T(E) is dense in E; (2) T-'(O) = {O}; (3) (Tx I x) > 0 for any x # 0 (use the Cauchy-Schwarz inequality applied to (TxJy)); (4) Tis nondegenerate. We say that a continuous operator Uin E is quasi-hermitian if there exists a nondegenerate positive hermitian operator T such that TU= U*T. (a) Show that every eigenvalue of U is real; if V = U - h . l E and if V-'(O) is finite dimensional, then E is a direct topological sum of V(E) and V-'(O), and h is a simple pole of (U - 5 . lE)-'. (Consider the Hilbert space obtained by completing E for the scalar product (Tx I y ) , and apply Problem 17.) a: = 1. (b) Let (01") be an infinite sequence of distinct real numbers such that

"

Let E be the Hilbert sum of a sequence of finite dimensional Hilbert spaces En such that dim(E,) = n (n 2 1). In En, let ( e i J l be a Hilbert basis, U, an operator in En such that Uneln= cciein ei+,,"for i < n - 1 and U,e,, = tL,e,, . Prove that llUnl.il< 2, but for any complex number 5 such that 151 = 1, ll(U, 5 . lEn)-'ll 2 44;. There is a unique continuous operator U on E whose restriction to each En is Un; show that U is quasi-hermitian and that the CL, are its eigenvalues, but that its spectrum contains the circle 151 = 1. (c) Let U be a compact quasi-hermitian operator. Prove that if U # 0, the spectrum of U cannot be reduced to the point 0, the eigenvalues hn # 0 of U are simple poles of (U- 5 . I,)-' and N(h,; U )= E(&; U ) and T(E(h,; U ) )= E(h.; U *). Furthermore, the intersection of the subspaces F(h,; U ) is equal to U -'(O), and the intersection of U-'(O) and U(E) is reduced to 0 (method of (a)). (d) Suppose E is separable, and let U be a compact continuous operator in E having the following property: there is a sequence (h,) of real eigenvalues of U * such that the sum of the E(X,; U * ) is dense in E. Prove that U is quasi-hermitian. (Show that (for a suitable k(n)), and there is in E a total sequence (b.) such that U*b, = hkCn)bn a,(x I b,)b, with suitable a. > 0; cf. Problem 16.) define T such that Tx

+

+

=I n

(e) Suppose E is separable, and let U be a compact operator in E satisfying the following conditions: (1) all the eigenvalues An of U are real and k@.; U )= I for every n ; (2) the intersection of the subspaces F(hn;U)is equal to U-'(O); (3) the intersection of U -I(O) and U(E) is (0). Prove that U is quasi-hermitian (use (11.5.5) and (d)). (f) Suppose E is separable, and let (eJnbobe a Hilbert basis for E. Prove that the operator U defined by Ue2.

= 0,

Ue2,+

=

nfl

+

for all n 2 0, is compact and quasi-hermitian, but the sum U -I(O) U(E) (which is an algebraic direct sum) is not a topological direct sum (cf. Section 6.5, Problem 2). m

(g) With the same notations as in (f), let U be the operator defined by Ueo =

e,/n,

"=I

Ue, = e,/nZ for n 2 1 ; using (e), show that U is compact and quasi-hermitian, but the sum U-'(O) U(E) is not dense in E; conclude that U * is not quasi-hermitian.

+

340

XI ELEMENTARY SPECTRAL THEORY

19. Let E be a Hilbert space, U a continuous operator in E. Suppose there is an element n f 0 in E such that: ( I ) the elements a, = U"a for n > 0 (with Uon = a ) form a total sequence in E; (2) the image of E by the hermitian operator V = U U * is the onedimensional subspace D = K a . Let Eo be a closed vector subspace of E not reduced to 0 and such that U(Eo) c Eo; let Uo be the restriction of U to Eo , and Po the orthogonal projection of E onto Eo . (a) Show that Eo cannot be orthogonal to D . (Observe that for any y orthogonal to D, U y = - U * y , and conclude that if Eo was orthogonal to D, one would have U p = (- I)"U*"yfor every y E Eo; show that this contradicts the assumption that the U"n form a total sequence in E.) (b) Prove that for any x E Eo , U:x = Po U,*x = Po U * X . (c) Prove that the image of Eo by Vo = Uo U z is not reduced to 0, hence is the subspace Do = Po(D) of dimension 1 (use the fact that Poa # 0 and the result of (b)). (d) Prove that the elements U;(Pon)constitute a total sequence in Eo. (Let Fo be the closed vector subspace of Eo generated by that sequence, and F6 the orthogonal supplement of Fo in Eo. Prove that FL is orthogonal to D and that Uo(Fb) = U(Fb) = Fb; conclude as in (a).) 20. Let E be a separable Hilbert space, U a compact operator in E whose spectrum consists of 0 and of an infinite sequence (AJof distinct eigenvalues # 0 such that k(h,) = 1 and that E(hJ is one dimensional for every n. For each n , let a. be an Figenvector of U corresponding to h,, and b, an eigenvalue of U * corresponding to h, (see (11.5.5)); a. and b, are chosen in such a way that (a. I b,,) = I . (a) Let A and B be the closed vector subspaces of E respectively generated by the a. and the b,, B' and A' the orthogonal supplements of B and A, respectively. Show that B' is stable for U , contains U - l ( O ) and that the restriction of U to B' has a spectrum reduced to 0. (b) Suppose the series of general term lh,, . lla,ll~ IIb.11 is convergent. Show then h,(x I b.)a,, where the series is convergent in E; U - l ( O ) that for any x E A, U X

+

+

=c "

then contains A n B'. (c) Give an example in which A n B' is infinite dimensional and U ( A n B') = A n B'. The following method may be used: let E be a Hilbert sum of three Hilbert spaces of infinite dimension F, R, S, having respective Hilbert bases (f"),(r,,), and (.yn); define a sequence (an)in F -1 R such that f. is the projection of a, in F and the closed vector subspace generated by the a, is F t R. To do this, observe that in a Hilbert , closed vector subspace generated by the space H with a Hilbert basis ( e n ) n 3 othe eo en for n > I is equal to H, and take for F R the Hilbert sum of a sequence o f Hilbert spaces all equal to H. Define U such that Unn= h,,a,,, Urn= p,,r,,-l for I? 2 I , Url = 0, Us,, = p , , ~ . + ~ where , the sequences (ti,) and (pn)converge rapidly enough to 0 (cf. Section 11.2, Problem 3). One thus gets B = F, A = F I R, and A n B'= R.

+

+

6. T H E FREDHOLM INTEGRAL E Q U A T I O N

We now apply the preceding theory to the example (11.2.8). We consider here the prehilbert space G of continuous complex-valued functions in I = [a, h], with ( f i g ) =Jabf(t)z) rlt, and the operator U such that U f is the function

6 THE FREDHOLM INTEGRAL EQUATION

341

t -+ @ K ( s , t)f(s) ds.

We say that the operator U is defined by the kernel function K. (11.6.1 ) The compact operator U in G has a compact adjoint which is defined by the kernel function K* such that K*(s, r ) = K ( t , s).

We prove for a d x

< b the identity

which, for x = 6,will yield the result by definition. Both sides of (11.6.1.1) are differentiable functions of x in [a,b ] , by (8.7.3) and Leibniz's rule (8.11.2); they vanish for x = a, and their derivatives are equal at each x E [a, b ] by (8.7.3) and (8.1 1.2), hence they are equal everywhere in [a, b ] (8.6.1 ). We leave to the reader the expression of the criterion (1 1.5.6) for that particular case (the " Fredholm alternative ").

If K ( t , s) = K(s, t ) (in which case the kernel K is called hermitian), the compact operator U is self-adjoint. As the prehilbert space G is separable ((7.4.3) or (7.4.4)), it can be considered as a dense subspace of a Hilbert space (6.6.2), and therefore we can apply to the operator U the results of (11.5.8). We shall denote by (An) the sequence of the (positive or negative) eigenvalues of U, each being repeated a number of times equal to its mu/tiplicity, and ordered in such a way that lAnl b / A n + , [ ; and we will denote by (q,) an orthonormal system in G such that, if the values of n for which An = pk (resp. A,,= v k ) are ni, m + I , . . . , m + r, then q m ,v",+~, . ..,qm+r constitute a basis for the eigenspace E(pk) (resp. E(vk)); we therefore have U ( q , ) = Anqnfor each t i . The q, are called eigenfunctions of the kernel K.

e

(11.6.2)

If K

is a hermitian kernel, the series

c A; d [ d t n

1 A:

is conwrgent and

n

jabIK(&t)12 d s .

a

Indeed, if we apply the Bessel inequality (6.5.2) to the function s -+ K(s, t ) and to the orthonormal system ((p"), we obtain, for any N

n= 1

342

XI

ELEMENTARY SPECTRAL THEORY

i.e.

c 2: N

(11.6.2.1)

fl=

1

lcpn(t)I2

G [IK(s9 t>I2 ds

for every t E I . Integrating both sides in I and using the relations (cp, and (11.6.1 .I) yields the result. The canonical decomposition in be written f

=

c c,

cpn

n

+f o ,

e of any function f

where c,

=

( f I cp,)

I cp,)

1

G (11.5.7) can here f(t)cp,(t) d t ; but, as

=la b

=

E

already observed,,f, may fail to be in G; on the other hand, the series

e,

1 cncpn

converges in the Hilbert space but not in general in the Banach space E = %?,-(I) (the series cncpn(t)will not necessarily converge for every t E I).

1 n

However: (11.6.3)

f (t)

If K is a hermitian kernel, and .f = Ug f o r a function g E G (i.e. K(s, t ) g ( s )ds), then rhe series c, cp,(t) converges absolutely and

=lab

uniformly

tof(t)

We have in

1 n

in 1.

c the canonical decomposition g = 1dncpn + g o ; as

V is a

n

continuous linear mapping of G into E = %?,-(I) (11.2.8), U can be extended to a continuous linear mapping of into E, and Ug, = 0, hence we have -f = Ug = Andncpn,where now the convergence is in E; i.e. the series

1

e

A,, d, cp,(t) converges uniformly to f ( t ) in E; as c, = (fI cp,)

= (Ug I cp,)

=

n

( g I Ucp,) = 1,(g I cp,) = And, , we have proved (11.6.3) except for the statement on absolute convergence. But for any integer N, we have, by Cauchy-Schwarz (for finite dimensional spaces)

and the right-hand side is bounded by a number independent of N, by Bessel's inequality (6.5.2) and (11.6.2.1). (11.6.4) I f K is hermitian, and 1 # 0 is not in the spectrum of U, the unique solution f'of the equation Uf - Af = g , f o r any g E G, is such that

where the series is absolutely and uniformly convergent in I, and dn = ( g I cp,).

6 THE FREDHOLM INTEGRAL EQUATION

343

e

We know that the unique solution of U f - Af = g in belongs to G since G (11.5.6), and by (11.5.11) we have c, = (fl cp,) = l/(A,, - 2). As g /If= Uh we can apply (1 1.6.3), and this proves the result. g

E

+

(11.6.5) Under the same assurnptions us in (11.6.4), the unique solution of U f - /If = g can be written

with

where the series is absolutely and unifornily coniIergerit for (s, t ) E I x I. By the proof of (11.6.3), we have

An dncp,,(t) = Q ( t ) ,the series convergn

ing absolutely and uniformly in I. As 1

A(An - A)

+ -1= A2

An

A(Ai -A)

the formula in (11.6.4) gives

The theorem will follow when we have proved the uniform convergence of the series A: Icpn(s)12: for there is a 6 > 0 such that IA, - A1 2 6 for each n, fl

hence

by Cauchy-Schwarz, and this will prove that the series

is absolutely and uniformly convergent in I x I ; the conclusion then results from (8.7.8).

344

XI ELEMENTARY SPECTRAL THEORY

Now consider the function H(s, t )

=I

b

K ( u , s ) K ( t , u ) du; for each fixed

I~qn(s)qn(t)

t E I, we can apply to it (11.6.3), and we see that H(s, t ) = n

where the series is convergent for any pair (s, t ) E I x I. In particular H(s, s ) = il: Iqn(s)12 for all s E I, and H(s, s) is continuous; by Dini’s n

theorem (7.2.2), the convergence is uniform in I. Q.E.D.

(11.6.6)

If K is hermitian, then

uniformly,for s E I.

With the notations of the proof of (11.6.5), we have (11.6.6.1) n-tm

\

k= 1

uniformly for s E I ; if we evaluate the integral in the statement of (1 1.6.6), using the fact that the qkare eigenvectors of U , and that they are orthogonal, we obtain the expression in the left-hand side of (11.6.6.1), whence the result. -

In general, the series

d,cp,(s)q,(t)

wiI/ not be contlergent for all

n

(s, t ) E 1 x I ; but we have the special result:

(11.6.7) (Mercer’s theorem) Suppose the coinpact operator Udejiied 61) the hertnitian kernel K(s, t ) is posititre. Then we have K ( s , t ) = 1, q,(s)q,,(t), n

where the series is absolutely and uniformly coniiergent in I x I.

We recall that we have here I n> 0 for every n (11.5.9). We first prove that for each s E I , the series I nlqn(s)I2 is convergent. For any s E I, we have K ( s , s) > 0. Otherwise, there would exist a neighborhood V of s in I such that B(K(s‘, t ) ) Q - 6 < 0 for (s’, t ) E V x V. Let q be a continuous mapping of I into [0, 11, equal to 1 at the points, to 0 in I - V (4.5.2). Then we have

6 THE FREDHOLM INTEGRAL EQUATION

345

by (8.5.3). But the left-hand side is (Ucp I cp), and this violates the assumption that U is a positive operator. Remark now that for any finite number of eigenvalues Ak (1 < k d n ) ~ , ( s , t ) = ~ ( s t, ) -

&(Pk(S)fpk(t)is the kernel function of a positive operk= 1

ator U,,, for we have

(un

f

f 1 f) = (uf I f) -k = 1 I(f I ‘Pk)12;

but the right-hand side of that equation can be written (Ug Ig) with fl

g=f-

k= 1

(fI (Pk)(Pk,as is readily verified, hence is positive by assumption.

Therefore, by (5.3.1) it follows from K,(s, s) 2 0 that the series is convergent, and we have

c A,,

Icpn(s)12

1 A,, lcpn(s)12

,< K(s, s) for all s E I. By Cauchy-

n

Schwarz, we conclude that

for all (s, t ) E I x I. Hence, as K(t, t ) is bounded in I, f o r f i x e d s E I, the series A,,cp,(s)cp,,(t)is uiiifornily conrergent for t E I. By (11.6.6), (8.7.8), and

1

-

fl

An cp,I(s)cp,,(t) = K(s, t ) for all (s, t ) E I x I since

(8.5.3), we conclude that n

t -+ IK(s, t ) -

2

1A , l a ) c p n ( t ) l fl

particular, we have K(s, s) = series

1 A,, lq,(~)1~is n

proves that the series

is continuous in I and its integral in I is 0. In

c An

Icp,,(~)1~; by Dini’s theorem (7.2.2) the

n

therefore uniformly contiergent in I, and (11.6.7.1) -

1 A,, cp,(s)cp,(t)

is absolutely and uniformly convergent

n

in I x I, which ends the proof.

Remarks (11.6.8) The result (11.6.7) is still true when we only suppose that U has a finite number of eigenvalues vk < 0 (1 < k < m).For (11.5.7(c)) shows then that in the space FL+,,orthogonal supplement of E(v,) -.. E(v,) in G, the restriction of the operator U is positive, and we apply (11.6.7) to that

+ +

346

XI

ELEMENTARY SPECTRAL THEORY

operator, which, as is readily verified, corresponds to the kernel function K(s, t ) - A,,q h ( ~ ) q h (where t), h runs through all the indices (in finite num-

1 h

ber) such that Ah < 0. The conclusion is then immediate. (11.6.9) We can consider the operator U in a larger prehilbert space, namely the space F, of regulated functions (Section 7.6) which are continuous on the right (i.e. such thatf(t+) = f ( t ) for a < t < b) and such that f ( b ) = 0; for such a function the relation Jab If(t)12 dt = 0 implies f ( t ) = 0 everywhere in 1 = [a, b], for it implies f(t) = 0 except at the points of a denumerable subset D (by (8.5.3)), and every t such that a < t < b is limit of a decreasing sequence of points of I - D. The space G may be identified to a subspace of F, , by changing eventually the value of a continuous function f E G at the point 6; it is easily proved (using (7.6.1)) that G is dense in F, . The argument of (11.2.8) then shows that U is a compact mapping of F, into the Banach space E = gC(I) (and a fortiori a compact mapping of the prehilbert space F, into itself). All the results proved for the operator U in G are still valid (with their proofs) when G is replaced by F, .

PROBLEMS 1. Extend the results of Section 11.6 (with the exception of (1 1.6.7)) to the case in which K(s, t ) satisfies the assumptions of Section 8.1 1, Problem 4 (use that problem, as well as Section 11.2, Problem 5). 2. In the prehilbert space G of Section 11.6, let (J,) be a total orthonormal system (Section 6.5); let

KAs, t )

=

kh(slz)

k= 1

and

HAS) = /ablK.(s, t)l df

(the “nth Lebesgue function” of the orthonormal system (fn)).For any function g E G,

let s,(g)

n

=

(g I f x ) f x , so that s,(g)(x) =

k= 1

1.”

K,(x,

t ) g ( r ) dt

for any x

E I.

(a) Prove that if, for an x o E I, the sequence (H,(xo)) is unbounded, then there exists a function g E G such that the sequence (s,,(g)(xo))is unbounded. (Use contradiction, and show that under the contrary assumption it is possible to define a strictly increasing sequence of integers (n,), and a sequence (g,) of functions of G, with the following properties: (1) let ch = sup IJ:K.(xo,

“

assumption), let dk = c1 -1 c2 sup(ml,. . ., mk-,); then m,

t)gh(t) dt

1

(a number which is finite by

+ . . . + ck-1, let mk ~Kn,(xo,t ) dt, and let qr 2,+l(9, + l)(d, + k ) ; (2) let vkbe a continuous func=

nl

=

Jnu

I

, dt tion such that vk(a)= Tt(b) = 0, Ivk(t)l< I in I and J o b ~ . , ( x o t)Tk(f)

I

mk/2

(see Section 8.7, Problem 8); then g k = ~ , / ( 2 ~-t ( 9I)).~ Then show that the function m

g=

k=l

g k is continuous in 1 and contradicts the assumption: to evaluate the integral

6 THE FREDHOLM INTEGRAL EQUATION

gi

Knk(xO,t ) g ( t ) dt, split g into ih

majorize the two other ones (“method of the gliding hump”).) (b) Show that for the trigonometric system (Section 6.5) in I = [-1, 11, the nth Lebesgue function is a constant h, , and that lim h, = w (observe that

I I

n- m

/A/‘“

(h-l)ln

for 2 < k

+

sin n v t 2 dt 3 sin vt kn

< n). Conclude that, for any xo E I, there exists a continuous function g in I,

such that g(- 1 ) = g(l) = 0, for which the partial sums

.

5 (1’g ( t ) e - i K ” r

h=-n

dt)etkff”/2

-1

of the “Fourier series” of g are unbounded for x = xo (Cf. Section 13.17, Problem 2.) 3. Let g be a continuous complex valued function defined in I = [- 1 , I ] and such that g(- 1) = g(1) = 0; g is extended to a continuous function of period 2 in R. Let K(s, t ) be the restriction ofg(s - t ) to I x I ; ifg(-t) =g(t), the compact operator U defined by the kernel function K(s, I ) is self-adjoint. Show that the functions ~ “ ( t= ) enntt/2/i are eigenvectors of U , the corresponding eigenvalue being the “Fourier coefficient ” a.

= J : l

g(t)e-“”“ dt of g.

Using that result and Problem 2, give examples of a hermitian kernel function K for which the series of general term h , a v . ( t ) has unbounded partial sums for certain values of s and t , and of a positive hermitian kernel function K for which there is a function f e G such that the series

rn

C (fI vn)vn(t) has unbounded partial sums for

fl=l

4.

certain values o f t . Let I = [-2v, 271, and define K(s, t ) in I x I to be equal to the absolutely convergent -sin ns . sin nt for 0 C s < 2n, 0 C t < 2 7 , and to 0 for other values of (s, t ) .=inz in I x I. Give an example of a functionfg G such that in the canonical decomposition off, fo does not belong to G. (The eigenfunctions of K are the functions pnsuch that v,(t)= 0 for -2v < t < 0, y,(t) = n-1’2sin nt for 0 C t < 2n. Take forfa continuous

series

function in I equal to 2 n - t in [0,27r], and show that the series verges everywhere in I, but has a discontinuous sum.)

m

C (f1 ~.)v,(t) con-

n=1

5. With the general notations of Section 11.6, let K be a hermitian kernel defined in

I x I, and let U be the corresponding self-adjoint compact operator in G. Show that for every h > 0, Uhcorrespondsto the hermitian kernel Kh,which is defined inductively by K 1 = K, and Kh(s, t )

=jab w

u)K(u, t ) du.

-

Prove that for h 3 2, Kh(s,t ) = C ht v n ( s ) v n ( t ) ,the series being absolutely and

”= 1

uniformly convergent in I x I. Show in addition that

and that the sequence (Ahfl/Ah)is increasing, and has a limit equal to lh1)’, where h, is an eigenvalue of K of maximum absolute value (use Cauchy-Schwarz).

348

XI ELEMENTARY SPECTRAL T H E O R Y

K be an arbitrary continuous kernel function in I x I, and let U be the corresponding compact operator in G . Let M be a finite dimensional subspace of G such that U(M) c M ; let ( + h ) l d h < n be an orthonormal

6. With the notations of Section 11.6, let

basis of the space M, and write

u+h

=

C

k=l

ahk

4 k . Show that

(For each f E I, apply Bessel's inequality (6.5.2) to the function s +K(s, t ) and the orthonormal system ( # h ) in G.) Let (A,) be the sequence defined (for the operator U ) in Section 11 -5,Problem 15(c). Prove that the series

m

"=l

Ih,12 is convergent, and

(Apply the preceding result to any sum of subspaces N(pk), with the notations of Section 11.5, Problem 15(c).) 7. Give an example of an hermitian kernel K(s, t ) , such that, if U is the corresponding compact operator in G , and V the square root of U 2 (Section 11.5, Problem 12), there is no hermitian kernel to which corresponds the compact operator V . (If there existed such a kernel, Mercer's theorem (11.6.7) could be applied to i t ; take then for K the first example in Problem 3.) 8. In order that the compact operator U defined by an hermitian kernel K(s, t ) be positive, show that a necessary and sufficient condition is that K be (in I x I ) a function of positive type (Section 6.3, Problem 4; to prove that the condition is necessary, write the inequality

JabfT dt K(s, t ) f (s) ds 3 0 Jab

for a function f which is 0 outside arbitrary small neighborhoods of a finite number of points x i of 1 (1 ,< i < n). To prove that the condition is sufficient, use the same method as in Section 8.7, Problem I). Conclude from that property and from Section 6.3, Problem 8 and Section 6.6, Problem 5 , a new proof of Mercer's theorem. 9. (a) A kernel function K(s, t ) defined in 1 x I (with I = [a, b ] ) and satisfying the assumptions of Section 8.1 1, Problem 4, is called a Volterra kernel if K(s, t ) = 0 for s> t . Let M = sup IK(s, t ) l . If U is the compact operator in G corresponding (S,

¶)El

Y

I

to K (Problem l), show that U" corresponds to a Volterra kernel K, such that (Kn(s,f ) l < M"(t - s)"-'/(n - l ) ! for n > 1 and s < t (use induction on n). Deduce from that result that the spectrum of U is reduced to 0, and that for any 4 E C , ll(lE - [ U ) - ' - lE1I < M1 1. Show that for that kernel, the function R(s, 1 ; A) in (11.6.5) is equal to exp((t - s)/A) for s < t , and to 0 for s > t. (Use (8.14.2) to compute U".) 10. Let F, be the prehilbert space defined in (11.6.9) for the interval I = [0, I ] , and let U be the operator defined in Problem 9(b), so that for every function x E F,, y = Ux is the function t - t

Jot

x(s)ds. The space

F+ is a dense subspace in a Hilbert space E

(6.6.2); U is extended by continuity to a compact operator in E, again written U (11.2.9). The closed subspaces Eo of E, distinct from 0 and E, and such that U(E,) c Eo

will be determined in this problem.

7 THE STURM-LIOUVILLE PROBLEM

349

(a) Show (using (7.4.1)) that the operator U satisfies the condition of Section 11.5, Problem 19, with a equal to the constant function of value I in 1; let a. E E, be such that Po a = so a. with so > 0 and I/ao11 =- 1 , so that 0 < so < 1. For every x E Eo , V O X= sb(x1 ao)ao(notations of Section 11.5, Problem 19). (b) For every 5 E C not 0, let f(5) = 1 - si((Uo 5f)-'ao I a,). Prove that for ( # 0, -

f(- n An

the series being absolutely and uniformly convergent in I x I (it is supposed, as we may, that 0 is not one of the A,). We observe that (d) follows from (11.6.3) and (11.7.8) when the additional assumption is made on u’ that it!’ is continuous in 1. To prove (d) in general, let ti (1 d i d wz) be the points of f where M)’ has a discontinuity, and let cli = w’(ti+) - w ’ ( t i - ) . Then the function u = w

m

+ i=

ctiK,, satisfies all the conditions of (d) and in addition 1

has a continuous derivative, by (11.7.6). Using (1 1.7.10.1 ) we conclude the proof of (d). From the fact that the identity mapping of E = +?,-(I) into G is continuous, it follows that for the functions w satisfying the conditions of (d), we can also write MI = c,cp,, the sequence being convergent in the n

prelzilbert space G. To prove (c) it will then be enough to show that the set P of these functions lli is dense in G. Now, for any function u E G, consider the continuous function M’, equal to u in [~+.!-,b-~], m m

to a linear function x + M X + fi satisfying the first (resp. second) bounduy condition (11.7.2) in

and to a linear function in each of the intervals

7 THE STURM-LIOUVILLE PROBLEM

355

We can in addition suppose that at the points a, b, the value of w, is 0 or 1 ; it is then clear that lu(x) - w,,,(x)I < llull

+ 1 in each of the intervals

and therefore IIu - iv,,J2 is arbitrarily small by the mean value theorem; as w, satisfies all conditions in (d), this proves our assertion. Once (c) is thus proved, it is clear that the total sequence (cp,,) must be infinite, and (applying (11.6.2)), (a) is also completely proved. Finally, (e) and (f) follow at once from (11.5.11).

Remark. It is possible to obtain much more precise information on thecp,and A,,, and to prove in particular that AJn2 tends to a finite limit (see Problems 3 and 4).

PROBLEMS

1. Let 1 = [ a , b] be a compact interval in R , and let Ho be the real vector space of all real-valued continuously differentiable functions in I; Ho is made into a real prehilbert space by the scalar product (x 1 y) =Job (x’y‘

+ xy) df .

(a) Show that Ho is separable (approximate the derivative of a function x E Ho by polynomials (7.4.1)); Ho is therefore a dense subspace of a separable Hilbert space H (6.6.2).

(b) If (x.) is a Cauchy sequence in the prehilbert space Ho , show that the sequence (x.) is uniformly convergent to a continuous function u in I, and that if (y,,)is a second Cauchy sequence in Ho having the same limit in H, then (yn)converges uniformly in I to the same function v ; the elements of H can thus be identified to some continuous functions in I, which however need not be differentiable at every point of I. (Observe that for every function x for any function z z’(u) = z’(b)

=

E

E

H o , Ix(t) - x(u)l

< (t - a ) 1 / 2 ( j o b x dt)’” ’2

in I.) Show that,

Ho which is twice continuously differentiable in I and such that

0,(v I z) = -

\

uz“ dt

. a

+

fab

uz dt.

(c) Let a , ,8 be two real numbers, q a continuous function in I. Show that in N o , the function x

+ @(x) =

s.“

(x” -tqx2)df

-

~ ( x ( u )-) P(x(b))’ ~ is continuous. Let A be the

subset of H consisting of the functions x such that

Jab

x 2 dt = 1 (observe that this is not

a bounded set in the Hilbert space H). Show that in A n H o , the g.1.b. of @ ( x ) isfinite. (One need only consider the case a > 0, ,8 > 0. Assume there is a sequence (x,) in

356

XI

ELEMENTARY SPECTRAL THEORY

A n H, such that lim rD(x,) = - m, and, if y,, = n-

dt)

liZ

, lim yn = -1 cu ; consider n-m

m

the sequence of the functions y ,

= x,/y,,

and derive a contradiction from the fact that,

y', dt 0, and on the other hand, there is an interval [a,c] jab a number p > 0 such that Iy.(t)l > p for every n and every point t [a, c].)

on one hand lim

=

n-m

C

I and

E

(d) Let p1 be the g.1.b. of @(x) in A n H o . Show that if (x.) is a sequence in A n Ho such that lim @(x,) = p,,(x,) is bounded in H (same method as in (c)). Deduce from n-t m

that result that, by extracting a convenient subsequence, one may assume that the sequence (x,) is uniformly convergent in I to a function u (which, however, need not a priuri belong to H) (use Ascoli's theorem (7.5.7)). (e) @(x)is a quadratic form in Ho , i.e. one has @(x t y ) = @(x) @ ( y ) 2 'Ux, y ) , where Y is bilinear; for any function z which is twice continuously differentiable in 1

+

and such that z'(n) = z'(b) = z(a) = z(h) = 0,one has 'Y(x, z ) =

-jab

XZ"

-'

dt

+lab

qxz dt ;

z ) can be defined by the same formula for any function v continuous in 1. Show that for any such function z and any real number [, one has '€'(u,

and deduce from that result that one must have

lab+ (uz" - quz

p I u z )dt = 0.

Hence, if w is a twice continuously differentiable function such that w" = qrc - p l u , one has [ab (u - w)z" dt = 0 by integration by parts;conclude that u - w is a polynomial

of degree < 1 (observe that by substracting from u - w a suitable polynomialp of degree 1, there exists a function z such that z" = u - w - p, z(a) = z(b) = z'(a) = z'(b) = 0). Hence u is twice continuously differentiable, satisfies the differential equation us - qu

+ plu

u2 dt = 1 ; furthermore, u'(a)

last statement, express that for any z 2.

E

= 0,

= - au(a),u'(h) = pu(6).

H,, cD(u

+ [z) > p1 J

" (u a

(To prove the

+ [ z ) dt,foranyreal ~

number 6.) (a) With the notations of (11.7.10), suppose first that k l k 2 # 0, and let a = h,/k,, p = - h 2 / k 2 . Show that the q ncan be defined (up to sign) by the following conditions: (1) p, is such that, o n the sphere A : (yl y ) = 1 in G , the function (I) (defined in Problem l(c)) reaches its minimum for y = pl,and that minimum is equal to A, ; (2) for n > 1 , let A, be the intersection of A and of the hyperplanes ( y I pr)= 0 for 1 < k c IZ - 1 ; then F~is such that on A,, CD reaches its minimum for y = rp", and that minimum is equal to A,. (The characterization of y1 follows at once from the results of Problem 1; use the same kind of argument to characterize vn.) (b) If k l = 0, k2 # 0,prove similar results, replacing a by 0 in @, but replacing the sphere A by its intersection with the hyperplane in G defined by y ( a ) = 0. Proceed similarly when k , # 0 and kZ = 0, or when k , = k 2 = 0.

7 THE STURM-LIOUVILLE PROBLEM

357

(c) Under the assumptions of (a), let z , , . . . , z , - ~ be n - 1 arbitrary twice continuously differentiable functions in I, and let B(zl, . . . , z " - ~ )be the intersection of A and of the n - 1 hyperplanes ( y I z x )= 0 ( I < k < n - 1). Show that in B(zl, . . .,zn-,), the function CU reaches a minimum p(zl, .. ., z,- at a point of B(zl, . , . , zn-,),and that h, is the 1.u.b. of p ( z l , . . . , z . - ~ ) when the z , vary over the set of twice continuously differentiable functions in I (the "maximinimal" principle; same method as in (a) to prove the existence of the minimum; the inequality is proved by the same method as in Section 11.5, Problem 8). Extend the result to the cases k , k 2 = 0. 3. (a) One considers in the same interval I two linear differential equations of the second order y" - y I y hy = 0, y" - q2y hy = 0, with the same boundary conditions (11.7.2); let (hi')),(hi2))be the two strictly increasing sequences of eigenvalues of these two Sturm--Liouville problems. Show that if 41 < q 2 ,then hi') < hiz) for every - hb2)l < M for every n (use the maximinimal n,and if I q l ( t ) - q2(t)l < M in I, then principle). (b) Conclude from (a) that there is a constant c such that

+

+

for every 11, with I : h - a . (Study the Sturm-Liouville problem for the particular case in which y is a constant.) 4. (a) Let y be any solution of (11.7.3) in I = [a,h ] for h > 0. Show that there are two constants A, w such that y is a solution o f the integral equation

(*I

y ( t ) = A sin J h ( t

1.

+ w ) -1- I

dh

t

q(s)y(s)sin Jh(t

- s) ds.

Show that there exists a constant B independent of h, such that A' < B(y I y ) (use Cauchy-Schwarz in order to majorize the integral on the right-hand side of (*)). (b) Deduce from (a) that if, in the Sturm-Liouville problem, k 1 k 2# 0 or k l = k2 = 0, then there are two constants C o , C,, such that, for every n, and every t E I lp,,(t)- J2// sin Jhn t 1

< co/n

and

Ipi(t) - 4 211JXn cos JGtl

< c,

with I = b - a

(use (a), and the result of Problem 3(b)). What is the corresponding result when only one of the constants k , , k 2 is O ?

APPENDIX

ELEMENTS OF LINEAR ALGEBRA

Except for boolean algebra (Section 1.2) there is no theory more universally employed in mathematics than linear algebra; and there is hardly any theory which is more elementary, in spite of the fact that generations of professors and textbook writers have obscured its simplicity by preposterous calculations with matrices. We shall give a brief survey of the concepts and results of linear algebra which are used in this book. For a more complete account we refer the reader to the works of Halmos [I I], Jacobson [13], or Bourbaki [4].

1. VECTOR SPACES

(A.l.1) Throughout this appendix, K is a fixed (commutative) field, whose elements are called scalars. A vector space orer K (or simply a vector space, if there is no ambiguity about K) is a set E endowed with the structure defined by two mappings: (x, y ) --f

x +y

(A,x) + Ax

of E x E into E

(called addition)

of K x E into E

(called scalar multiplication)

having the following properties:

+y + z)

(1.1)

x + ( y + z ) = (x + y ) + z

(1.2)

x+y=y+x

(1.3)

there exists an element 0 E E such that x = 0

(written x

+ x for all x E E l

t Experience shows that, contrary to the opinion of some authors, there is no risk of confusion in using the same symbol 0 to denote the zero elements of all vector spaces (and of K in particular). 358

1 VECTOR SPACES

(I.4) (II.I) (II.2) (11.3)

for each x EE there exists an element -x EE such that x (J.. + µ)x = J..x + µx J..(x + y) = J..x + J..y

359

+ (-x) = 0

= (J..µ)x (II.4) I · x = x J..(µx)

(in these conditions, x, y, z are arbitrary elements of E and J.., µ are arbitrary elements of K). Conditions (I. I) to (1.4) express that Eis a commutative group with respect to addition. It follows that, if (x;); e H is any finite family of elements of E, is unambiguously defined. (We recall the convention that the sum

IX;

ieH

L X; =0.)

ie 0

From (1.3) and (11.2) we deduce that J..x = J..(x + 0) = J..x + J..O, and therefore J..O = 0 for all J.. EK. From (II.I) it follows that 2x = (). + O)x = J..x + Ox, so that Ox= 0 for all x EE. Finally, from these two relations and (I.4), (II. I), and (II.2) we deduce that h + 2( - x) = ).0 = 0, and 2x + ( -1)x = Ox = 0, so that (-J..)x = J..( -x) = -(J..x). The elements of E are often called vectors. (A.1.2) An additive group consisting of the element O alone is a vector space: the scalar multiplication is the unique mapping of K x {O} into {O}. The field K is a vector space over itself, the scalar multiplication being multiplication in K. If K' is a subfield of K and if E is a vector space over K, then E is also a vector space over K' if we define the scalar multiplication to be the restriction to K' x E of the mapping (1, x) ~ J..x. (A.1.3) If E is a vector space, a vector subspace (or simply subspace) of E is defined to be any subset F of E such that the relations x E F, y E F imply h + µy E F, for all scalars 1, µ. The restriction to F x F (resp. K x F) of the addition (resp. scalar multiplication) in E is a mapping into F, and therefore F is a vector space with respect to these mappings. This justifies the terminology. If (x;) 1 ,. ;,;;" is any finite family of vectors in F, and if U;) 1 ,. ;~" is a family

" of scalars, then by induction on n we see that

I

form

I

i= 1

A;X; E

F. A vector of the

i= 1

A;X;

(where

X;

EE and A; EK) is called a linear combination of the

x;. (A linear combination is always a finite sum, even if E is, for example, a normed space (Section 5.1) in which certain infinite sums are defined (Section 5.3).)

360

APPENDIX:

ELEMENTS OF LINEAR ALGEBRA

In any vector space E the subsets ( 0 ) and E are vector subspaces (called the tricial subspaces). If (M,),e, is any family of vector subspaces of E, the intersection M, is again a vector subspace. If A is any subset of E, the

n

as1

intersection M of the subspaces which contain A (such subspaces exist, for example E itself) is therefore the smallest subspace containing A. We say that M is generated by A, or that A is a systeni of generators of M. (A.1.4) The i m i o r subspace M generated b y A is the set of all linear conibitiations o j j n i t e families of elenietits of A.

It follows from above that such linear combinations belong to every vector subspace containing A, and therefore belong to M. Conversely, if x = & l x r and y = d J y j are two linear combinations of elements of A, i

then so is 1,x

+ py = i

+ 1(pdj)yJ. Hence the set of all such linear

J

(l.yl)xi

J

combinations is a vector subspace of E, which contains A (by (11.4)) and therefore coincides with M .

In particular, take A to be the unioti of a family (NP)PeJof vector subspaces of E. Every linear combination of elements of A i s then of the form x, x, . . . xP,, where (fl,), jdn, is any finite family of elements of J, and xPJE N P Jfor each j . The set of all these sums is therefore the smallest vector subspace containing all the N,. It is called the sum of the N, (not to be confused with their union!) and is denoted by N, . When J = { I , 2},

+

+

+

+

1

P E J

the notation N, N, is also used, and similarly for any finite set of indices, It is clear that M N = N M and M ( N + P) = (M + N ) + P for any three subspaces M, N,Pi n E. The relation M c N is equivalent to M + N = N. For every x E E, the subspace of E generated by x is written Kx (and is sometimes called a " ray '' when x # 0). If (x,), A is any family of vectors in E, the subspace generated by the family is therefore Kx,,

+

+

+

asA

(A.1.5) Let ( E J U E lbe a/iy family of vector spaces. We shall construct a new vector space E, called the direct sun? (or external direct sum) of the family (E,) and denoted by @ E,. The elements of E are the families ael

x = (x,), E , , where x, E E, for all ct and x, = 0 f o r all except a j n i t e nuniber of indices. The vector x u is called the coniponent of index M in x. Addition and

scalar multiplication in E are defined by the formulas

2 LINEAR MAPPINGS

361

((Ax,) belongs to E because 10 = 0). The conditions of (A.l .I) are immediately verified (the element 0 of E is the family (x,) for which x, = 0 for all e). When J = (1, 2}, the notation El 0E, is used, and similarly for any finite set of indices. When J is finite, the set E is equal to the product set E,.

fl

aeJ

If J is arbitrary and H is a subset of J (other than @ or J), the vector space @ E, can be identified in an obvious way with

ael

@ E, (as"

0 @ En). (aeJ-H

When all the E, are equal to K, their direct sum is denoted by K'". It is the set of all mappings e -+ ).(A of I into K such that A(e) = 0 for all but a finite number of indices e E I.

2. L I N E A R MAPPINGS

(A.2.1) Let E, F be two vector spaces over the same field K. A mapping u : E F is said to be linear if it satisfies the condition --f

(A.2.1 .I)

u(Ax + P A = Au(x) + P ( Y >

for all scalars A, p and all vectors x, y in E. In particular, we have u(0) = 0. By induction on n it follows from (A.2.1 .I) that (A.2.1.2) i Q n is a family where (xi)l is any finite family of vectors in E, and of scalars. A linear mapping of E into E is called an endomorphism of E. A linear mapping of E into K is called a linear form on E.

(A.2.2) Let u : E -+ F be a linear mapping. If M is any subspace of E, it is immediately verified that its image u(M) is a subspace of F. If N is any subspace of F, then its inverse image u-'(N) is a subspace of E. If ( M a ) is any

family of subspaces of E, then u ( T Ma)=

a

u(M,).

In particular, u(E) (which is called the image of u and is written im(u)) is a subspace of F, and u-'(O) (which is called the kernel of u and written ker(u)) is a subspace of E. The mapping u is injective if and only if u-'(O) = {O}, because the relation u(x) = u(x') is equivalent to u(x - x') = 0. If u is bijective, it is called an ison?orphism of E onto F. (If also F = E, then u is an autoniorphism of E). If u is bijective, it is clear that the inverse mapping u-' : F -+ E is linear, and therefore an isomorphism of F onto E.

362

APPENDIX: ELEMENTS OF LINEAR ALGEBRA

Two vector spaces E, F are said to be isomorphic if there exists an isomorphism of E onto F. In that case, any theorem proved for E, involving vectors and subspaces of E, immediately gives a corresponding theorem for F, involving the images of the vectors and subspaces in question. If u : E -,F is an injective linear mapping, then u can be considered as an isomorphism of E onto its image u(E). Examples of linear mappings (A.2.3) The identity mapping of a vector space E onto itself (often denoted by 1 or IE, or simply 1 or I ) is linear. So is the unique mapping of E into a vector space consisting of 0 alone. For every I E K, the mapping h, : x -+ Ax is an endoinorphism of E, called the homothetic mapping with ratio I . If I = 0, its image is the zero subspace ( 0 ) of E. If I # 0, then hl is bijective and its inverse automorphism is the homothetic mapping with ratio A- because we have x = I - ’ ( I x ) by (11.3). For each x, E E, the mapping 5 -+ >

. . . ,xJ.

Since fo is not zero, the formula (A.6.6.1) now follows from the definition of det(u t i ) . 0

MULTILINEAR MAPPINGS. DETERMINANTS

6

(A.6.7)

375

det(u) # 0 ifand only if u is bijective.

If u is bijective, it has an inverse u-l such that zi u-' = 1., Hence, by (A.6.6) and (A.6.5.2), we have det(u) det(i4-') = I , so that certainly det(u) # 0. If u is not bijective, then it is not injective (A.4.19), hence there exists b, # 0 such that u(b,) = 0. There exists a basis (b;), of E containing 6 , (A.4.5), and we have fo(b,, . . . , 6,) # 0, whereas fo(u(b,), . . . , u(b,)) = 0. Hence det(u) = 0. (A.6.8) Let ( b , ) l s be a basis of E, and let M ( u ) = ( z j i ) be the matrix of u with respect to the two bases (b;)and (b;)of E (or, as is usually said, tllenzatrix of I I witA respect to the basis (b;)). Since ,fo(h,, . . . , b,) # 0, the formulas (A.6.1 .I)and (A.6.5.1) give (A.6.8.1)

det(4

=

c

EU

%(l)lMU(2)2

..

*

%("),

3

U

where CJ runs through the symmetric group 6,of all permutations of (1, 2, ..., n}. The determinant of the matrix M(u) is by definition the determinant of u. This provides the link between our theory and the classical theory of determinants in its original form. We shall not need to use this latter theory, and we leave the task of transcribing our results into the old-fashioned notation to those readers who are interested in this type of calculation. In applications it is always much simpler to go back to the definition (A.6.5), as we shall illustrate by considering the eigenvalues of an endomorphism. (A.6.9) The definition of the eigenvalues of an endomorphism u is that given in ( l l . l . l ) , except that the field C is now replaced by an arbitrary field K. It follows immediately from (A.6.7) that these eigenvalues are the roots of the equation (called the characteristic equation of u)

det(u - 1 lE) = 0.

(A.6.9.1)

The formula (A.6.8.1) shows immediately that the left-hand side of this equation is a polynomial of degree n in A, with leading coefficient (- 1)". In what follows we shall assume that the field K is algebraically closed, so that det(u - A IE) factorizes into linear factors (A, - A)(& - A) * . (A, - A). (A.6.10) (A.6.10.1)

There exists a basis (b,, . . . , 6,) of E such that

+

u(bJ = A j b j ai.;+,bi+,

+ + ainh, 1 . -

(I

< i < n).

376

APPENDIX: ELEMENTS OF LINEAR ALGEBRA

Conuersely, i f (bi)l

is a basis with this property, then

det(u - A lE) = (A, - A)(A2 - A)

* . . (A, - A).

The proof is by induction on n. By hypothesis, there exists a vector b, # 0 in E which is an eigenvector for the eigenvalue A,; in other words, u(b,) = A, b, . Let us split E into a direct sum Kb, V, and let p : E + V be the corresponding projection (A.3.4). The mapping x -p(u(x)) is an endomorphism of V, and hence there exists a basis (bl, . . . , bnwl)of V such that

+

P(u(bi))=pibi

+ Ui,i+lbi+l +

+ ~ ( ~ , , - ~ b , - , (1 < i < n - 1)

and consequently

u(bJ

= pibi

+~

+ +

( ~ , ~ + ~ bai,n-.lb,,-l ~ + ~

+ ai,,b,

(1

< i < n - 1)

for suitable scalars m i , . We now have

fo(~(b1) - Ab1, . . ., u(bn) - Abn) =fo((/ll

- A)bi

* * . > (Pn-1

+ . + alnb, , *

-1)bn-i

A)b2

( ~ 2

+. + * *

~ 2 b, ,

,.. .,

+an-1,nbn,(An-A)btJ.

If we expand the right-hand side by means of (A.6.1 . I ) and use the definition of an alternating multilinear form, we see easily that the only term which does not vanish is (PI - m

2

- A)

. . . (Pn- 1 -

A W L ,

-

A)fO(b,, . . . , b,),

(P,,-~ - A)(A, - A). This and therefore det(u - A . lE) = ( p l - A)(p2 - A) proves that the p i are (except possibly for their order) the scalars A,, . . . , An-l, and the calculation above also establishes the second assertion of (A.6.10). The matrix of u with respect to a basis satisfying the conditions of (A.6.10) is said to be lower triangular.

(A.6.11) (A.6.11 . l )

For each integer k > 0, we have det(uk - I * 1E) = (A: - A)(Ak, - A)

*

(A: - A).

For it follows from the formulas (A.6.10.1) that U k ( b i ) = A!bi

+

bi+,+ . *

*

+ apb,

and the result therefore follows from (A.6.10).

(1 < i < n )

7 MINORS OF A DETERMINANT

377

(A.6.12) The endomorphism u is a nilpotent element of the ring End(E) and on1.y if all its eigenvalues are zero.

If u is nilpotent, it follows from (A.6.11) that all the eigenvalues of u are zero. Conversely, if all the ,Ii are zero, the formula (A.6.10.1) shows, by induction on k , that uk(E) c Kbk+ + K b , + , + * * + Kb, if k < n, and finally that u"(E) = {0}, that is to say, u" = 0.

-

7. M I N O R S OF A D E T E R M I N A N T

(A.7.1) Let E be a vector space of dimension n over K, and let ( b i ) l d i 6 n be a basis of E. For each subset I of the index set A = { 1, 2, . . . , n } , let E(I) be the subspace of E generated by the 6 , with i E I . Then E is the direct sum of E(I) and E(A - I). If I = { i l , i , , . . . , ir}, where i , < iz < ... < i,, let j , be the bijection of K' onto E(I) such that j , ( e k )= b,, (1 < k < r ) , where (ek)l4 k G r is the canonical basis of K' (A.4.4). Also letp, be the linear mapping of E onto K' such that p,(bi,) = ek for 1 < k < r, and p,(bj) = 0 if j $ I. The kernel ofp, is therefore E(A - I), and the restriction ofp, to E(1) is a bijection of E(1) onto K'. (A.7.2) Let u be an endomorphism of E and let M ( u ) = ( a j i ) be its matrix with respect to the basis (bi) (A.6.8). If I , J are two subsets of the index set A, having the same number of elements r, consider the endomorphism uJ, = p u 0 j , of K'. Its matrix with respect to the canonical basis (e,) of K' consists of those a j i for which i E I and j~ J. The determinant of this matrix (that is to say, det(u J l ) ) is called the r x r minor of det(u), corresponding to the basis ( b J l i Q n of E and the subsets I, J of the index set A. 0

(A.7.3) An endomorphism u of E is of rank r i f and only i f all the s x s minors (where s > r ) in det(u) relatiue to (bi)are zero and at least one of the r x r minors is nonzero.

Let p be the rank of u. With the notation of (A.7.2), we have ujI(K') = pj(u(E(I))), hence (A.4.18)rank(uJ I ) = dim(uJ1(K')) < dim(u(E(1))) < dim u(E) = rank(u) = p . If r > p , we therefore have det(u,,) = 0, by (A.4.19) and (A.6.7). On the other hand, there exists a subset I, of A, containing p elements, such that E(1,) is supplementary to ker(u), and a subset J, of A containing p elements, such that E(A - J,) is sumlementarv to u(E) (A.4.5).

378

APPENDIX: ELEMENTS OF LINEAR ALGEBRA

It follows that u, restricted to E(To), is a bijection of E(Io) onto u(E) (A.4.19), and that p J orestricted to u(E) is a bijection of u(E) onto KP (A.3.5). Hence u~~~~ is bijective and therefore det(uJo,,) # 0 (A.6.7). The proposition follows immediately from these remarks. (A.7.4) With the preceding notations let us now take 1 = J = { 1, 2, . . . , m}, hence A - I = A - J = {m + 1, . . . , n } , and let us suppose in addition that uA-,,I = 0, in other words the matrix M(u) has the form

where X = M(u,,) is an m x m matrix, Y = M(u,,A - ,) an m x (n - m) matrix and Z = M(u,-,, A-I) an (n - m) x (n - m ) matrix (0 standing for the zero ( n - m) x m matrix). Then we have det(M(u)) = det(X) det(Z).

(A.7.4.1)

For if u is the endomorphism of E(1) having X as matrix with respect to (bi)l i 6 m , we have, with the notations of (A.6.10), f(u(b,),* . . > u(bm), u(bm+l), . * . 4bn)) = f ( v ( b l ) , * * * v(b,n), u(b,n+1), . . . 4bn)). 7

9

But the mapping ( ~ 91 *

* * 9

xm)

-tf(xl, * . ., x m

9

u(bm+1),

*

., u(bn))

is an alternating m-linear form on (E(I))m, hence, by (A.6.5), f(v(bl),

*

. ., u ( b m ) ,

. . u(bn)) X ) f ( b i , . . . , b,, , u ( b m + 1 ) , . . ., u(bn))*

u(bm+l)?

= (det

'9

For each j 2 rn + 1, let us write u(bj) = CJ + cy, with c j E E(I), c; By definition of an alternating multilinear form, we have

E

E(A-I).

. ., b m 4 b m + I), . . ., u(bn)) =f(b1, . . ., b m c;+1, . c:)* Let )Y then be the endomorphism of E(A - 1) having 2 as matrix with respect f(b1, *

9

* *

9

9

to ( b j ) m + l Q j Qby n ;definition w(bj) = cj" f o r j e A - I. The mapping ( x m + 1,

* *

9

xn) +f(b1,

*

. ., brn

is an alternating ( n - m)-linear form on (E(A f(b,,

. . . ,b,,

9

Xm + 1,

.. ., xn)

- I))"-",

~ ( b , + ~.). ., , ~ ( b , ) = ) (det Z)f(bl,

hence we get similarly

. . . , b,)

= det 2

which proves (A.7.4.1). By induction on r, we conclude that for any " triangular

7 MINORS OF A DETERMINANT

matrix of matrices ”

where X i j is an m i x mj matrix, we have (A .7.4.2)

det 0 = (det XII)(det X 2 * )

(“ Computation of a determinant by blocks ”).

(det XrV)

379

REFERENCES

[ I ] Ahlfors, L., “Complex Analysis.” McGraw-Hill, New York, 1953. [2] Bachmann, H., “Transfinite Zahlen” (Ergebnisse der Math., Neue Folge, Heft I). Springer, Berlin, 1955. [3] Bourbaki, N., “ Elements de Mathkmatique,” Livre I, “Theorie des Ensembles ” (Actual. Scient. Ind., Chaps. I, 11, No. 1212; Chap. 111, No. 1243). Hermann, Paris, 1954-1956. [4] Bourbaki, N., “ Elements de Mathkmatique,” Livre 11, “Algebre,” Chap. 11 (Actual. Scient. Ind., Nos. 1032, 1236, 2nd ed.). Herrnann, Paris, 1955. [5] Bourbaki, N., “ Elkments de Mathkmatique,” Livre 111, “Topologie gknkrale” (Actual. Scient. Ind., Chaps. I, 11, Nos. 858, 1142, 4th ed.; Chap. IX, No. 1045, 2nd ed.; Chap. X, No. 1084, 2nd ed.). Hermann, Paris, 1949-1958. 161 Bourbaki, N . , “Elements de Mathkmatique,” Livre V, “ Espaces vectoriels topologiques” (Actual. Scient. Ind., Chap. 1, 11, No. 1189, 2nd ed.; Chaps. 111-V, No. 1229). Hermann, Paris, 1953-1955. [7] Cartan, H., “Seminaire de I’Ecole Normale Superieure, 1951-1952: Fonctions analytiques et faisceaux analytiques.” [8] Cartan, H., “Theorie elementake des fonctions analytiques.” Hermann, Paris, 1961. [9] Coddington, E., and Levinson, N., “Theory of Ordinary Differential Equations.” McGraw-Hill, New York, 1955. [lo] Courant, R., and Hilbert, D., “Methoden der mathematischen Physik,” Vol. I, 2nd ed. Springer, Berlin, 1931. [ I I] Halmos, P., “Finite Dimensional Vector Spaces,” 2nd ed. Van Nostrand, Princeton, New Jersey, 1958. [I21 Ince, E., “Ordinary Differential Equations.” Dover Publications, New York, 1949. [I31 Jacobson, N., “Lectures in Abstract Algebra,” Vol. 11, “Linear Algebra.” Van Nostrand, Princeton, New Jersey, 1953. [I41 Kamke, E., “Differentialgleichungen reeller Funktionen.” Akad. Verlag, Leipzig, 1930. [I51 Kelley, J., “General Topology.” Van Nostrand, Princeton, New Jersey, 1955. [I61 Landau, E., “Foundations of Analysis.” Chelsea, New York, 1951. [ 171 Springer, G . , “Introduction to Riemann Surfaces.” Addison-Wesley, Reading, Massachusetts, 1957. [I81 Weil, A,, “Introduction a I’ktude des varietes kahleriennes” (Actual. Scient. Ind., No. 1267). Hermann, Paris, 1958. [I91 Weyl, H., “Die Idee der Riemannschen Flache,” 3rd ed. Teubner, Stuttgart, 1955. 380

INDEX

In the following index the first reference number refers to the number of the chapter in which the subject may be found and the second to the section within the chapter. A

Abel’s lemma: 9. I Abel’s theorem: 9.3, prob. 1 Absolute value of a real number: 2.2 Absolute value of a complex number: 4.4 Absolutely convergent series: 5 . 3 Absolutely summable family, absolutely sumniable subset: 5.3 Adjoint of an operator: 11.5 Algebraic multiplicity of an eigenvalue: I I .4 Amplitude of a complex number: 9.5, prob. 8 Analytic mapping: 9.3 Approximate solution of a differential equation: 10.5 Ascoli’s theorem: 7.5 At most denumerable set, at most denumerable family: I .9 Axiom of Archimedes: 2.1 Axiom of choice: 1.4 Axiom of nested intervals: 2.1

B Banach space: 5.1 Basis for the open sets of a metric space: 3.9 Belonging to a set: 1 . 1 Bergman’s kernel: 9.13, prob. Bessel’s inequality: 6.5 Bicontinuous mapping: 3.12

Bijective mapping, bijection: 1.6 Bloch’s constant: 10.3, prob. 5 Bolzano’s theorem: 3.19 Borel’s theorem: 8.14, prob. 4 Borel-Lebesgue axiom: 3.16 Borel-Lebesgue theorem : 3. I7 Boundary conditions for a differential equation: 11.7 Bounded from above, from below (subset of R): 2.3 Bounded subset of R : 2.3 Bounded real function: 2.3 Bounded set in a metric space: 3.4 Broken line: 5.1, prob. 4 Brouwer’s theorem for the plane: 10.2, prob. 3

C Canonical decomposition of a vector relatively to a hermitian compact operator: 11.5 Cantor’s triadic set: 4.2, prob. 2 &-Capacityof a set: 3.16, prob. 4 Cartesian product of sets: 1.3 Cauchy’s conditions for analytic functions: 9.10 Cauchy criterion for sequences: 3.14 Cauchy criterion for series: 5.2 Cauchy’s existence theorem for differential equations: 10.4 MI

382

INDEX

Cauchy’s formula : 9.9 Cross section of a set: 1.3 Cauchy’s inequalities: 9.9 Cut of the plane: 9.Ap.3 Cauchy-Schwarz inequality: 6.2 Cauchy sequence: 3.14 D Cauchy’s theorem on analytic functions: 9.6 Center of a ball: 3.4 Decreasing function: 4.2 Center of a polydisk: 9.1 Degenerate hermitian form: 6.1 Change of variables in an integral: 8.7 Dense set in a space, dense set with respect Circuit: 9.6 t o another set: 3.9 Closed ball: 3.4 Denumerable set, denumerable family: 1.9 Closed interval: 2.1 Derivative of a mapping at a point: 8.1 Closed polydisk: 9.1 Derivative in an open set: 8.1 Closed set: 3.8 Derivative of a function of one variable: 8.4 Closure of a set: 3.8 Derivative with respect to a subset of R:8.4 Cluster point of a set: 3.8 Derivative on the left, on the right: 8.4 Cluster value of a sequence: 3.13 Derivative (second, pth): 8.12 Codimension of a linear variety: 5.1, Derivative (pth) with respect to an interval: prob. 5 8.12 Coefficient (nth) with respect to an ortho- Diagonal : 1.4 normal system: 6.5 Diagonal process : 9.13 Commutatively convergent series: 5.3, Diameter of a set: 3.4 Difference of two sets: 1.2 prob. 4 Differentiable mapping at a point, in a set: Compact operator: 11.2 8.1 Compact set: 3.17 Differentiable with respect to the first, Compact space: 3.16 second, . . . , variable: 8.9 Complement of a set: 1.2 Differentiable (twice, p times): 8.12 Complete space: 3.14 Differential equation: 10.4 Complex number: 4.4 Dimension of a linear variety: 5.1, prob. 5 Complex vector space: 5.1 Dini’s theorem: 7.2 Composed mapping: 1.7 Direct image: 1.5 Condensation point: 3.9, prob. 4 Conformal mapping theorem: 10.3, prob. 4 Dirichlet’s function: 3.1 1 Disk: 4.4 Conjugate of a complex number: 4.4 Connected component of a set, of a point Discrete metric space: 3.2 and 3.12 Distance of two points: 3.1 in a space: 3.19 Distance of two sets: 3.4 Connected set, connected space: 3.19 Constant mapping: 1.4 Contained in a set, containing a set: 1.1 E Continuity of the roots as function of parameters: 9.17 Eigenfunction of a kernel function: 11.6 Continuous, continuous at a point: 3.11 Eigenspace corresponding to an eigenvalue: Continuously differentiable mapping: 8.9 11.1 Convergence radius of a power series: 9.1, Eigenvalue of an operator: 1 1.1 prob. 1 Eigenvalue of a Sturm-Liouville problem: Convergent sequence: 3.13 11.7 Convergent series: 5.2 Eigenvector of an operator: 1 I . 1 Convex set, convex function: 8.5, prob. 8 Eilenberg’s criterion: 9.Ap.3 Coordinate (nth) with respect to an ortho- Element: 1.1 normal system: 6.5 Elementary solution for a Sturm-Liouville problem: 11.7 Covering of a set: 1.8

INDEX

Empty set: 1.1 Endless road: 9.12, prob. 3 Entire function: 9.3 &-Entropyof a set: 3.16, prob. 4 Equation of a hyperplane: 5.8 Equicontinuous at a point, equicontinuous : 7.5 Equipotent sets: 1.9 Equivalence class, equivalence relation: 1.8 Equivalent norms: 5.6 Equivalent roads: 9.6 Essential mapping: 9.Ap.2 Essential singular point, essential singularity: 9.15 Euclidean distance: 3.2 Everywhere dense set: 3.9 Exponential function: 4.3 and 9.5 Extended real line: 3.3 Extension of a mapping: 1.4 Exterior point of a set, exterior of a set: 3.7 Extremity of an interval: 2.1 Extremity of a path: 9.6 F

Family of elements: 1.8 Finer distance, finer topology: 3.12 Finite number: 3.3 Fixed point theorem: 10.1 Fourier coefficient (nth): 6.5 Fredholm equation, Fredholm alternative: 11.6 Frobenius’s theorem: 10.9 Frobenius-Perron’s theorem : 11.l, prob. 6 Frontier point of a set, frontier of a set: 3.8 Full sequence of positive eigenvalues: 11.5, prob. 8 Function: 1.4 Function of bounded variation: 7.6, prob. 3 Function of positive type: 6.3, prob. 4 Functional graph, functional relation: 1.4 Functions coinciding in a subset: 1.4 Fundamental system of neighborhoods: 3.6 Fundamental theorem of algebra: 9.1 1

G Geometric multiplicity of a n eigenvalue: 11.4 Goursat’s theorem: 9.10, prob. 1

383

Gram determinant: 6.6, prob. 3 Graph of a relation: 1.3 Graph of a mapping: 1.4 Greatest lower bound: 2.3 Green function of a Sturm-Liouville problem: 11.7 Gronwall’s lemma: 10.5

H Haar orthonormal system: 8.7, prob. 7 Hadamard’s three circles theorem: 9.5, prob. 10 Hadamard’s gap theorem: 9.15, prob. 7 Hausdorff distance of two sets: 3.16, prob. 3 Hermitian form: 6.1 Hermitian kernel: 1I .6 Hermitian norm: 9.5, prob. 7 Hermitian operator: 11.5 Hilbert basis: 6.5 Hilbert space: 6.2 Hilbert sum of Hilbert spaces: 6.4 Homeomorphic metric spaces, homeomorphism: 3.12 Homogeneous linear differential equation: 10.8 Homogeneous hyperplane: 5.8, prob. 3 Homotopic paths, homotopic loops, homotopy of a path into a path: 9.6 and 10.2, prob. 6 Hyperplane: 5.8 and 5.8, prob. 3 Hyperplane of support: 5.8, prob. 3 I

Identity mapping: 1.4 Image of a set by a mapping: 1.5 Imaginary part of a complex number: 4.4 Implicit function theorem: 10.2 Improperly integrable function along an endless road, improper integral: 9.12, prob. 3 Increasing function: 4.2 Increasing on the right: 8.5, prob. 1 Indefinitely differentiable mapping : 8.12 Index of a point with respect to a circuit, of a circuit with respect to a point: 9.8 Index of a point with respect to a loop: 9.Ap.l Induced distance: 3.10 Inessential mapping: 9.Ap.2

384

INDEX

Infimum of a set, of a function: 2.3 Infinite product of metric spaces: 3.20, prob. 7 Injection, injective mapping: 1.6 Integer (positive or negative): 2.2 Integral: 8.7 Integral along a road: 9.6 Integration by parts: 8.7 Interior point of a set, interior of a set: 3.7 Intersection of two sets: 1.2 Intersection of a family of sets: 1.8 Inverse image: 1.5 Inverse mapping : 1.6 Isolated point of a set: 3.10 Isolated singular point: 9.15 Isometric spaces, isometry: 3.3 Isomorphism of prehilbert spaces: 6.2 Isotropic vector: 6.1 1

Jacobian matrix, jacobian: 8.10 Janiszewski’s theorem: 9.Ap.3 Jordan curve theorem: 9.Ap.4 Juxtaposition of two paths: 9.6 K

Kernel function: 11.6 L Lagrange’s inversion formula: 10.2, prob. 10 Laurent series: 9.14 Least upper bound: 2.3 Lebesgue function (nth): 11.6, prob. 2 Lebesgue’s property: 3.16 Legendre polynomials: 6.6 and 8.14, prob. 1 Leibniz’s formula: 8.13 Leibniz’s rule: 8.11 Length of an interval: 2.2 Limitof a function, limit of a sequence: 3.13 Limit on the left, limit on the right: 7.6 Linear differential equation: 10.6 Linear differential equation of order n : 10.6 Linear differential operator: 8.13 Linear form: 5.8 Linear variety: 5.1, prob. 5

Linkedbyabroken line(points): 5.1, prob. 4 Liouville’s theorem: 9.1 1 Lipschitzian function: 7.5, prob. 12, and 10.5 Locally closed set: 3.10, prob. 3 Locally compact space: 3.18 Locally connected space: 3.19 Locally lipschitzian function: 10.4 Logarithm: 4.3 and 9.5, prob. 8 Loop: 9.6 and 10.2, prob. 6 Loop homotopy: 9.6 and 10.2, prob. 6 M

Majorant: 2.3 Majorized set, majorized function: 2.3 Mapping: 1.4 Maximal solution of a differential equation: 10.7, prob. 4 Maximinimal principle: 11.5, prob. 8, and 11.7, prob. 2 Mean value theorem: 8.5 Mercer’s theorem : 1 1.6 Meromorphic function: 9.17 Method of the gliding hump: I I .5, prob. 4, and 11.6, prob. 2 Metric space: 3.1 Minimal solution of a differential equation: 10.7, prob. 4 Minorant: 2.3 Minorized set, minorized function: 2.3 Minkowski’s inequality: 6.2 Monotone function: 4.2 Morera’s theorem: 9.10, prob. 2 N

Natural boundary: 9.15, prob. 7 Natural injection: 1.6 Natural mapping of X into X/R: 1.8 Natural ordering: 2.2 Negative number: 2.2 Negative real half-line: 9.5, prob. 8 Neighborhood: 3.6 Newton’s approximation method: 10.2, prob. 5 Nondegenerate hermitian operator: 11.5 Norm: 5.1 Normally convergent series, normally summable family: 7.1 Normed space: 5.1

INDEX

0 One-to-one mapping: 1.6 Onto mapping: 1.6 Open ball: 3.4 Open covering: 3.16 Open interval: 2.1 Open neighborhood: 3.6 Open polydisk: 9.1 Open set: 3.5 Operator: 11.1 Opposite path: 9.6 Order of an analytic function at a point: 9.15 Order of a linear differential operator: 8.13 Ordered pair: 1.3 Origin of an interval: 2.1 Origin of a path: 9.6 Orthogonal projection: 6.3 Orthogonal supplement: 6.3 Orthogonal system: 6.5 Orthogonal to a set (vector): 6.1 Orthogonal vectors: 6.1 Orthonormal system: 6.5 Orthonormalization: 6.6 Oscillation of a function: 3.14

385

Precompact set: 3.17 Precompact space: 3.16 Prehilbert space: 6.2 Primary factor: 9.12, prob. 1 Primitive: 8.7 Principle of analytic continuation: 9.4 Principle of extension of identities: 3.15 Principle of extension of inequalities: 3.1 5 Principle of isolated zeros: 9.1 Principle of maximum: 9.5 Product of a family of sets: 1.8 Product of metric spaces: 3.20 Product of norrned spaces: 5.4 Projection (first, second, ith): 1.3 Projections in a direct sum: 5.4 Purely imaginary number: 4.4 Pythagoras’s theorem: 6.2

Q Quasi-derivative, quasi-differentiable function: 8.4, prob. 4 Quasi-hermitian operator: 11.5, prob. 18 Quotient set: 1.8 R

P p-adic distance: 3.2 Parallel hyperplane: 5.8, prob. 3 Parseval’s identities: 6.5 Partial derivative: 8.9 Partial mapping: 1.5 Partial sum (nth) of a series: 5.2 Partition of a set: 1.8 Path: 9.6 and 10.2, prob. 6 Path reduced to a point: 9.6 Peano curve: 4.2, prob. 5, and 9.12, prob. 5 Peano’s existence theorem: 10.5, prob. 4 PhragmCn-Lindelof’s principle: 9.5, prob. 16 Picard’s theorem: 10.3, prob. 8 Piecewise linear function: 8.7 Point: 3.4 Pole of an analytic function: 9.15 Positive definite hermitian form: 6.2 Positive hermitian form: 6.2 Positive hermitian operator: 11.5 Positive number: 2.2 Power series: 9.1

Radii of a polydisk: 9.1 Radius of a ball: 3.4 Rational number: 2.2 Rank theorem: 10.3 Real line: 3.2 Real number: 2.1 Real part of a complex number: 4.4 Real vector space: 5.1 Reflexivity of a relation: 1.8 Regular frontier point for an analytic function: 9.15, prob. 7 Regular value for an operator: 11.1 Regularization: 8.12, prob. 2 Regulated function: 7.6 Relative maximum: 3.9, prob. 6 Relatively compact set: 3.17 Remainder (nth) of a series: 5.2 Reproducing kernel: 6.3, prob. 4 Residue : 9.15 Resolvent of a linear differential equation: 10.8 Restriction of a mapping: 1.4 Riemann sums: 8.7, prob. 1

386

INDEX

Riesz (F.)’s theorem: 5.9 Road: 9.6 Rolle’s theorem: 8.2, prob. 4 RouchB’s theorem: 9.17

S Scalar : 9.1 Scalar product: 6.2 Schoenflies’s theorem: 9.Ap., prob. 9 Schottky’s theorem: 10.3, prob. 6 Schwarz’s lemma: 9.5, prob. 6 Second mean value theorem: 8.7, prob. 2 Segment: 5.1, prob. 4, and 8.5 Self-adjoint operator: 1 1.5 Semi-open interval : 2.1 Separable metric space: 3.10 Separating points (set of functions): 7.3 Separating two points (subset of the plane): 9.Ap.3 Sequence: 1.8 Series: 5.2 Set: 1.1 Set of mappings : 1.4 Set of uniqueness for analytic functions: 9.4 Simple arc, simple closed curve, simple loop, simple path: 9.Ap.4 Simply connected domain: 9.7 and 10.2, prob. 6 Simply convergent sequence, simply convergent series: 7.1 Simpson’s formula: 8.14, prob. 10 Singular frontier point for an analytic function: 9.15, prob. 7 Singular part of an analytic function at a point: 9.15 Singular values of a compact operator: 11.5, prob. 15 Solution of a differential equation: 10.4 and 11.7 Spectral value, spectrum of an operator: 11.1 Sphere: 3.4 Square root of a positive hermitian compact operator: 11.5, prob. 12 Star-shaped domain: 9.7 Step function: 7.6 Stone-Weierstrass theorem: 7.3 Strict relative maximum: 3.9, prob. 6 Strictly convex function: 8.5, prob. 8

Strictly decreasing, strictly increasing, strictly monotone: 4.2 Strictly negative, strictly positive number: 2.2 Sturm-Liouville problem: 11.7 Subfamily: 1.8 Subsequence: 3.1 3 Subset: 1.4 Subspace: 3.10 Subspace of a normed space: 5.4 Substitution of power series in power series: 9.2 Sum of a family of sets: 1.8 Sum of a series: 5.2 Sum of an absolutely summable family: 5.3 Supremum of a set, of a function: 2.3 Surjection, surjective mapping: 1.6 Symmetric bilinear form: 6.1 Symmetry of a relation: I .8 System of scalar linear differential equations: 10.6

T Tangent mappings a t a point: 8.1 Tauber’s theorem: 9.3, prob. 2 Taylor’s formula: 8.14 Term (nth) of a series: 5.2 Theorem of residues: 9.16 Tietze-Urysohn extension theorem : 4.5 Titchmarsh’s theorem: 11.6, prob. 1 1 Topological direct sum, topological direct summand, topological supplement: 5.4 Topological notion: 3.12 Topologically equivalent distances: 3.12 Topology: 3.12 Total derivative: 8.1 Total subset: 5.4 Totally disconnected set: 3.19 Transcendental entire function: 9.15,prob. 3 Transitivity of a relation: 1.8 Transported distance: 3.3 Triangle inequality: 3.1 and 5.1 Trigonometric polynomials : 7.4 Trigonometric system: 6.5

U Ultrametric inequality: 3.8, prob. 4 Underlying real vector space: 5.1

INDEX

Uniformly continuous function: 3.1 1 Uniformly convergent sequence, uniformly convergent series: 7.1 Uniformly equicontinuous set: 7.5, prob. 5 Uniformly equivalent distances: 3.14 Union of two sets: 1.2 Union of a family of sets: 1.8 Unit circle: 9.5 Unit circle taken n times: 9.8

Vector space: 5.1 Volterra kernel : 11.6, prob. 8 W

Weierstrass’s approximation theorem: 7.4 Weierstrass’s decomposition: 10.2, prob. 8 Weierstrass’s preparation theorem: 9.17, prob. 4 Weierstrass’s theorem on essential singularities: 9.15, prob. 2

V

Value of a mapping: 1.4 Vector basis: 5.9, prob. 2

387

Z Zero of an analytic function: 9.15

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ISBN 0-12-215502-5 PRINTED IN THE UNITED STATES O F AMERICA

“Treatise on Analysis,” Volume I1 Enlarged and Corrected Printing First published in the French language under the title “Eldments d’Analyse,” tome 2, 2“ edition, revue et augment& and copyrighted in 1974 by Gauthier-Villars, Gditeur, Paris, France.

SCHEMATIC PLAN OF THE WORK I. Elements of lhe theory of sets

IV.

VI I. Spoces of continuous funct i ons

XXIV. E lementary di fferential topology XXV. Nonlinear problems

This Page Intentionally Left Blank

CONTENTS

Notation

............................

Chapter XI1

TOPOLOGY AND TOPOLOGICAL ALGEBRA

. . . . . . . . . .

1 . Topological spaces. 2. Topological concepts. 3. Hausdorff spaces. 4. Uniformizable spaces. 5 . Products of uniformizable spaces. 6. Locally finite coverings and partitions of unity. 7. Semicontinuous functions. 8. Topological groups. 9. Metrizable groups. 10. Spaces with operators. Orbit spaces. 1 1 . Homogeneous spaces. 12. Quotient groups. 13. Topological vector spaces. 14. Locally convex spaces. 15. Weak topologies. 16. Bake's theorem and its consequences.

Chapter Xlll

INTEGRATION

.......................

1. Definition of a measure. 2. Real measures. 3. Positive measures. The absolute value of a measure. 4. The vague topology. 5. Upper and lower integrals with respect to a positive measure. 6. Negligible functions and sets. 7. Integrable functions and sets. 8. Lebesgue's convergence theorems. 9. Measurable functions. 10. Integrals of vector-valued functions. 1 1 . The spaces L' and Lz. 12. The space L". 13. Measures with base p. 14. Integration with respect to a positive measure with basep. 15. TheLebesgue-Nikodym theorem and the order relation on MR(X). 16. Applications: I. Integration with respect to a complex measure. 17. Applications: 11. Dual of L'. 18. Canonical decompositions of a measure. 19. Support of a measure. Measureswith compact support. 20. Bounded measures. 21. Product of measures.

Chapter XIV

INTEGRATION IN LOCALLY COMPACT GROUPS

ix

1

98

. . . . . . . . 242

1. Existence and uniqueness of Haar measure. 2. Particular cases and examples. 3. The modulus function on a group. The modulus of an automorphism. 4. Haar measure on a quotient group. 5. Convolution of measures on a locally compact

vii

CONTENTS

viii

group. 6. Examples and partkular cases of convolution of measures. 7. Algebraic properties of convolution. 8. Convolution of a measure and a function. 9. Examples of convolutions of measures and functions. 10. Convolution of two functions. 11. Regularization. Chapter X V

NORMED ALGEBRAS A N D SPECTRAL THEORY

. . . . . . . . 304

1. Normed algebras. 2. Spectrum of an element of a normed algebra. 3. Characters and spectrum of a commutative Banach algebra. The Gelfand transformation. 4. Banach algebras with involution. Star algebras. 5. Representations of algebras with involution. 6. Positive linear forms, positive Hilbert forms, and representations. 7. Traces, bitraces, and Hilbert algebras. 8. Complete Hilbert algebras. 9. The Plancherel-Godement theorem. 10. Representations of algebras of continuous functions. I I. The spectral theory of Hilbert. 12. Unbounded normal operators. 13. Extensions of hermitian operators.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

References Index

NOTATION

In the following definitions, the first number is the number of the chapter and the second number is the number of the section within that chapter.

A-'

AB

support of a function: 12.6 characteristic function of a subset A of a set: 12.7 upper and lower envelopes of a family of real-valued functions: 12.7 upper and lower limit of a sequence of real-valued functions: 12.7 opposite of a group G: 12.8 group of automorphisms of a topological vector space E: 12.8 set of elements x-' where x E A (where A is a subset of a group in which the group law is written multiplicatively): 12.8 set of elements xy with x E A and y E B, where A and B are subsets of a group in which the group law is written multiplicatively: 12.8 normalizer and centralizer of a subset H of a group: 12.8 set of finite products xlxz . . . x, (with n arbitrary) of elements of a subset V of a group: 12.8 ix

x

NOTATION

ZL.

G*A

set of p-adic integers: 12.9, Problem 4 p-adic solenoid: 12.9, Problem 4 homogeneous space of left (right) cosets of a subgroup H in a group G : 12.10 and 12.11 union of the orbits of the points of a subset A of E, with respect to an action of G on E: 12.10 space of orbits with respect to G: 12.10 space of continuous complex-valued functions on X: 12.14 quotient normed space of a normed space E by a vector subspace F: 12.14 x'(x) for a vector x E E and a continuous linear form x' E E' (the dual of E): 12.15 transpose of a continuous linear mapping: 12.15 unitary group of a Hilbert space E: 12.15, Problem 8 space of continuous complex-valued functions with support contained in the compact subset K of X : 13.1 space of continuous complex-valued functions on X with compact support: 13.1 Dirac measure at the point x: 13.1 measure with density g with respect to p: 13.1 and 13.13 image of a measure p under a proper continuous mapping n: 13.1 and 13.1, Problem 8 measure induced on an open set U by a measure p (or restriction of p to U): 13.1 space of complex measures on X : 13.1 integral of f € X ( X ) with respect to p : 13.1 space of continuous real-valued functions with support contained in the compact subset K of X: 13.2

NOTATION

IPI

$59"

PY

xi

space of continuous real-valued functions on X with compact support: 13.2 conjugate of the measure D: 13.2 space of real measures on X: 13.2 real and imaginary parts of a complex measure: 13.2 set of positive measures on X: 13.3 positive and negative parts of a realvalued function J': 13.3 order relation between real measures : 13.3 absolute value of a complex measure: 13.3 set of lower semicontinuous functions on X which are bounded below by a function belonging to XR(X):13.5 upper integral off: 13.5 sum of a sequence (t,) of elements 20 of 8: 13.5 set of upper semicontinuous functions on X which are bounded above by a function belonging to X.(X): 13.5 lower integral off: 13.5 outer and inner measures of A c X: 13.5 equivalence class off (with respect to p): 13.6 order relation between equivalence classes: 13.6 sum and product of equivalence classes: 13.6 integral of a p-integrable function: 13.7 space of (finite) real-valued p-integrable functions: 13.7 measure of a p-integrable set: 13.7 integral of an equivalence class: 13.7 integral off over A: 13.9 upper integral off over A: 13.9 measure induced by p on a closed subspace Y: 13.9

xii

NOTATION

entropy of a finite partition a : 13.9, Problem 27 entropy of a finite partition a relative to a finite partition P : 13.9, Problem 27 least upper bound of a finite sequence of finite partitions: 13.9, Problem 27 entropy of a mapping u relative to a finite partition a : 13.9, Problem 28 entropy of a mapping u: 13.9, Problem 28 integral of a vector-valued function : 13.10 space of complex-valued p-integrable functions : 13.10 s* If I dp: 13.11 (Y If l2 dp)"': 13.1 1 space of (finite) real-valued squareintegrable functions : 13.11 space of p-negligible functions : 13.1 1 space of classes of pth power integrable functions: 13.11 and 13.11, Problem 12 N,U) for anyfin the classf: 13.11 space of complex-valued square-integrable functions: 13.11 space of classes of complex-valued square-integrable functions: 13.11 maximum and minimum in measure off: 13.12 space of real-valued p-measurable functions bounded in measure: 13.12 space of complex-valued p-measurable functions bounded in measure : 13.12 space of classes of real-valued p-measurable functions bounded in measure : 13.12 space of classes of complex-valued pmeasurable functions bounded in measure : 13.12 space of (finite) real-valued p-measurable functions: 13.12, Problem 2 space of classes of (finite) real-valued pmeasurable functions: 13.12, Problem 2

NOTATION

xiii

space of real-valued locally p-integrable functions : 13.13 space of complex-valued locally pintegrable functions : 13.13 space of classes of real (complex) valued locally p-integrable functions : 13.13 integral with respect to a complex measure : 13.16 support of a measure: 13.19 norm of a measure: 13-20 space of bounded real (complex) measures: 13.20 space of real (complex) valued continuous functions which ' tend to 0 at infinity: 13.20 integral with respect to the product measure A Q p : 13.21 product measure: 13.21 upper integral with respect to a product measure: 13.21 lower integral with respect to a product measure: 13.21 the function ( x , y ) H f ( x ) g ( y ) : 13.21 product of n measures: 13.21

m

0 n= 1

integral with respect to the product of the n measures: 13.21 product of an infinite sequence of measures: 13.21, Problem 9 translates of a function: 14.1 translates of a measure: 14.1 transforms of a function or measure under SHS-': 14.1

xiv

NOTATION

T AG(S),

mod,(u), mod(u)

one-dimensional torus: 14.2 modulus function on a group: 14.3 modulus of an automorphism of G: 14.3

field of p-adic numbers: 14.3, Problem 6 average off over a fibre: 14.4, Problem 2 convolution of n measures: 14.5 space of measures with compact support: 14.7

centralization of a measure: 14.7, Problem 6 convolution of a measure and a function: 14.8

convolution of two functions: 14.10 spectrum of an element x of an algebra A: 15.2 spectral radius of x : 15.2 spectrum of an algebra A: 15.3 Gelfand transformation: 15.3 Hardy spaces: 15.3, Problem 15 adjoint (in an algebra with involution): 15.4

Hilbert-Schmidt norm : 15.4 involution in MS(G): 15.4 algebra of Hilbert-Schmidt operators : 15.4

Problem 18 regular representation of a complete Hilbert algebra: 15.8 space of hermitian characters: 15.9 multiplication by the class of u : 15.10 space of bounded universally measurable complex-valued functions on K: 15.10 ( H e x J x )2 0 for all x : 15.11 function of a normal operator: 15.1 1 square root of a positive hermitian operator: 15.11 absolute value of an operator: 15.11, Problem 6 trace of a nuclear operator: 15.11, Problem 7 ( ~ ( X * X ) ) ” ~ : 15.4,

HZO

f (N) abs(T)

NOTATION

dom(T)

T*

xv

nuclear norm: 15.11, Problem 7 space of nuclear operators: 15.11, Problem 7 approximative point-spectrum of an operator: 15.1 1, Problem 9 domain of an unbounded operator: 15.12 adjoint of an unbounded operator with dense domain: 15.12 spectrum of an unbounded operator: 15.12 function of an unbounded normal operator: 15.12

CHAPTER XI1

TOPOLOGY AND TOPOLOGICAL ALGEBRA

As we said in the Introduction, we have sought to limit the material in this chapter to the minimum that the reader will require. We shall not stay to explore the refinements of general topology (filters, uniformities, separation axioms); instead, we shall pass as soon as possible to the category of uniformizable spaces, which are the only ones we shall meet in later chapters. Usually these spaces occur merely as “ ambient spaces ” in which it is convenient to operate, and we shall be mainly concerned-in conformity with the spirit of this book-with their separable metrizable subspaces. $or this reason we have included many criteria of metrizability and separability (12.3.6,12.4.6,12.4.7,12.5.8,12.9.1,12.10.10,12.11.3, 12.14.6.2,12.15.7,12.15.9, 12.15.10). These, together with Bake’s theorem and its consequences, are the only results in the chapter whose proofs are not straightforward. We have also included an account of various purely topological techniques which were not needed in the first volume (partitions of unity (12.6), semi-continuous functions (12,7)), and more than half the chapter is devoted to elementary concepts of topological algebra (topological groups, spaces with operators, topological vector spaces). All of this will be used constantly in the succeeding chapters.

1. TOPOLOGICAL SPACES

A topology on a set E is a set 0 of subsets of E (in other words, a subset of

p(E))satisfying the following two conditions :

(0,) The union of any family (AJA of sets belonging to D belongs to 0; (OJ The set E belongs to D,and the intersection of any two sets belonging to D belongs to D. 1

2

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

A topologicalspace is a set E endowed with a topology 0.The subsets of E which belong to 0 are called the open sets of the topological space E. Taking L = @ in (O,), it follows that the empty subset @ of E is always an open set, and from (0,Jwe see that E itself is an open set. Examples (12.1.1) The set D of open sets in a metric space E satisfies (0,)and (Oil) (3.5.2 and 3.5.3). This topology D is called the topology of the metric space E (or is said to be dejined by the distance given on E). Two topologically eguiualent distances (3.12) define the same topology. A topological space is said to be metrizable if its topology can be defined by a distance (and then this topology is also said to be metrizable). On any set E, the set 0 = E} is a topology, called the chaotic topology. It is not metrizable if E has at least two elements, because otherwise there would exist an open ball containing one of these elements and not the other, and this is impossible because E is the only nonempty open set. On a set E = {a, b } consisting of two elements, the set 0 = (0, { a } , E} is a nonmetrizable topology. If D,,0, are two topologies on the same set E, we say that 0, isfiner than O1 (or that 0, is coarser than D2)if 0, c D2. Two topologies on E are said to be comparable if one is finer than the other. The chaotic topology is coarser than all others. The discrete topology (i.e., the topology defined by the metric (3.2.51, for which D = V(E)) is finer than all others. Two topologies on E are not necessarily comparable; for example, let E = { a , b } be a set with two elements, and consider the two topologies 0, = (0, { a } , E} and 0, = {@, (61, El.

{a,

2. TOPOLOGICAL CONCEPTS

We have already remarked (3.12), in the context of metric spaces, that the notions of closed set, cluster point of a set, interior point of a set, frontier point of a set, neighborhood of apoinr (or of a set), dense set, continuous.function and homeomorphism are defined entirely in terms of the notion of an open sel in a metric space. Their definitions can therefore be transferred, without any change, to the context of arbitrary topological spaces. Moreover, all the properties involving these notions which were proved for metric spaces, and whose statements do not involve the distance (cf. (3.5) to (3.12)) remain valid for arbitrary topological spaces (and we shall therefore make use of them in the general case) with the following exceptions:

2 TOPOLOGICAL CONCEPTS

3

(1) The properties (3.8.11) and (3.8.12) are not true for arbitrary topological spaces, as is shown by the example of the space E = {a, b } with the topology (0, { a } ,E}. The existence of a distance is essential to the proof of these two propositions, although it does not feature in the enunciation. (2) We may define as in (3.9) the notion of a basis of open sets (or of the topology) of a topological space E. The criterion (3.9.3) is valid in general. We can also prove as in (3.9.4) that if there exists a denumerable basis for the topology of a topological space E, then there exists an at most denumerable set which is dense in E; but the converse is not valid for arbitrary topological spaces (Section 12.4, Problem 6).

By definition, a basis following property:

B of the topology of a topological space E has the

The intersection of any two sets of B is a union of sets of 8. Conversely, let B be a set of subsets of a set E. If B has the above property and if E E 8,then the set SJ of (arbitrary) unions of sets of B is a topology for which B is a basis. For it is immediately seen that 0 satisfies (0,)and (O,,). If E is a topological space and if F is any subset of E, then the set of intersections U n F, where U runs through the set of open subsets of E, satisfies the axioms (0,) and (0") and is therefore a topology on F. This topology on F is said to be induced by the topology on E. The set F, endowed with this topology, is called a subspace of E. With these definitions, all the propositions of (3.10) are valid for arbitrary topological spaces, with the exception of (3.10.9), which has to be stated in the following form: if the topology of E has a denumerable basis, then so does the topology induced on any subset of E. We have already remarked that the properties of compactness and local compactness for a metric space depend only on the topology, and not on the distance which defines the topology. We may therefore speak unambiguously of compact and locally compact rnetrizable spaces. We observe also that all the definitions and results of (3.19) relating to connectedness are valid for arbitrary topological spaces. (12.2.1) Let SJ1, SJ2 be two topologies on a set E. Then the followingproperties are equivalent:

(a) D2 is$ner than SJl; (b) if Ei denotes the topological space obtained by giving E the topology Di (i = 1, 2), then the identity mapping of E, onto E, is continuous; (c) for all x E E, every neighborhood of x in the topology Dlis a neighborhood of x in SJ2.

4

XI1 TOPOLOGY A N D TOPOLOGICAL ALGEBRA

The equivalence of (a) and (b) follows from the criterion of continuity (3.11.4(b)), and the equivalence of (b) and (c) from the definition of a continuous function.

Remark (12.2.2) Let (UJaE I be an open covering (3.16) of a topological space E. For a set G c E to be open in E it is necessary and sufficient that each of the sets G n U, should be open in the subspace U, . This follows immediately from the and (O1Jand the relation G = (G n U,). Taking complements, axioms (0,)

u

as1

it follows that a set F c E is closed in E if and only if each of the sets F n U, is closed in the subspace U, . (12.2.3) Let L be a subset of a topological space E. Then the followingproperties are equivalent: (a) L nV (b) (c)

For each x E L there exists a neighborhood V of x in E such that is closed in V ; L is an open subset of the subspace L (the closure of L in E); L is the intersection of an open subset cind a closed subset of E.

It is clear that (b) implies (c), L being the intersection of L with an open subset of E. Equally clearly, (c) implies (a). Let us show that (a) implies (b). For each x E L, we have V n L = V n L, because V n L is closed in V; this shows that in the subspace L the point x is an interior point of L, and therefore L is open in L. When L satisfies the equivalent conditions of (12.2.3), it is said to be a locally closed subset of E. (12.2.4) A procedure for constructing topological spaces which is frequently used is that of “patching together” a family of topological spaces in such a way that, in the topological space E so obtained, the Ed are identified with open sets of E. Since pairs of these sets may very well intersect, this identification requires that, for each pair of indices (A, p), we are given a homeomorphism of an open subset of Ed onto an open subset of E, . To be precise, suppose that for each pair (A, p) E L x L we are given:

(I) an open subset A,, of E,; (2) a homeomorphism AMa: Alp-+ A,, , satisfying the following conditions :

2 TOPOLOGICAL CONCEPTS

5

(I) A,, = E,, and h,, is the identity mapping l E A ; (11) for each triple of indices (A, p, v) and each x E A,, n A,, , we have h,,W E A,, and (12.2.4.1 )

hv,(x) = hv,(h,,(X))

(patching condition). Let F be the sum of the E,, which are therefore pairwise disjoint subsets of F (1.8). In F, consider the relation

R(x, y ) : " there exist A, p such that x y = h,,(x)."

E

A,, , y E A,, , and

This is an equivalence relation. It is reflexive by virtue of condition (I); it is symmetric because h,, and h,, are inverses of each other (by applying (11) with v = A, and then (I)); finally, it is transitive, because if x

E A,,

Y

y

= h,,(x) E A,, n A,,

9

z = h,,Ot),

then we have x = h,,(y) and therefore x E A,, n A,, by (11), and it follows now from (12.2.4.1) that z = h,,(x). We remark also that, by condition (I), the intersection of an E, and an equivalence class of R consists of at most one point. If E = F/R is the set of equivalence classes of R and if

n : F -+F/R = E is the canonical mapping, then each of the restrictions n, = n I E, : E, -,E is injective. Moreover, the sets n,(E,) form a covering of E. Now consider the set 0 of subsets X of E with the following property: for each A E L, the set n;'(X n n,(E,)) is open in E, . It is clear that 0 satisfies axioms (0,) and (01,), and is therefore a topology on E. We shall show that in this topology the sets n,(E,) are open in E and that n, is a homeomorphism of E, onto the subspace n,(E,) of E, for each A E L. In view of the definition of D and of the fact that n, is a bijection of E, onto n,(E,), it is enough to establish the equivalence of the following two properties of a subset X , of E,: (a) X, is an open set in the topological space El; (b) for each p E L, the set n;'(n,(X,) n n,(E,)) is open in E,. Taking p = A, we see that (b) implies (a). Conversely, if (a) is satisfied, then n n,(E,)) = A,, by definition, the condition (b) since we have n;'(n,(E,) signifies that, for each p E L, the set h,,(X, n A,,) is open in E,. Now X, n A,, is open in A,, , and h,, is a homeomorphism of A,, onto an open subset of E, . Hence (a) implies (b). The topological space E so defined is said to be obtained by patching together the E, along the A,, by means of the h,, .

6

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

3. HAUSDORFF SPACES

The notion of a limit is defined as in (3.13) for arbitrary topological spaces, and the criterion (3.13.1)is merely another formulation of the definition. However, the conclusion of (3.13.3)is no longer necessarily true: for example, if E is a set endowed with the chaotic topology (12.1.1),every sequence o f points of E has every point of E as a limit. A topological space E is said to be Hausdorff, and its topology is said to be a Hausdorf topology, if it satisfies the following " Hausdorff axiom":

Given any two distinct points a, b in E, there exists a neighborhood U of a and a neighborhood V of b which do not intersect. Every metrizable space is Hausdorff

(12.3.1) Let E be a topological space, A a subset of E, and let a be a cluster point of A. Then a mapping f of A into a Hausdorff space E ' has at most one limit at the point a with respect to A. For if a', b' were two distinct limits off at the point a, there would exist neighborhoods U', V' of a', b', respectively, having no point in common. But by hypothesis there would be a neighborhood W of a in E such that f ( W n A) c U' and f (W n A) c V', and this is absurd because W nA # 0. Hence we may continue to use the notation

lim

x E A ,x

f ( x ) to denote the

+

a

unique limit o f f a t the point a. The propositions of (3.13) which do not refer to distances will remain valid for mappings (in particular for sequences) from an arbitrary topological space to a Hausdorf space, with the exception of (3.13.13)and (3.13.14).

(12.3.2) Every subspace of a Hausdorfspace is Hausdorf. (12.3.3) Every topology which isfiner than a Hausdor# topol5gy is Hausdorff. These are immediate consequences of the definitions.

(12.3.4) In a Hausdorff space E, every finite subset is closed. For if ( a i ) l s i s nis a finite sequence of points in E, a point b distinct from all the a, cannot be a cluster point of the set of the a i , because for each i there

3 HAUSDORFF SPACES

exists a neighborhood V i of b which does not contain a i , and V = then a neighborhood of b which does not contain any of the ai .

7

n Vi is n

i= 1

(12.3.5) LetS, g be two continuous mappings of a topological space E into a Hausdorffspace E’. Then the set A ofpoints x E E such that f ( x )= g(x) is closed in E.

The proof is similar to that of (3.15.1) (which is a special case of (12.3.5)). If a # A, then f ( a ) # g(a), and so there are disjoint neighborhoods U’ of f ( a ) and V’ of g(a). Since f -‘(U’) and f -‘(V’) are neighborhoods of a in E, the same is true of their intersection W , and it is clear that f ( x ) # g(x) for all x E W . Hence E - A is open. The principle of extension of identities (3.1 5.2) therefore remains true for continuous mappings of any topological space into a Hausdorfs space. Proposition (3.1 5.3) and its corollary (3.15.4) (the “principle of extension of inequalities ”) are also valid-the proofs are the same-for arbitrary topological spaces. (12.3.6) Let E be a compact metrizable space, F a Hausdorff space, and f a continuous mapping of E into F . Thenf ( E ) is closed in F. I f f is injective, it is a homeomorphism of E onto the subspace f ( E ) of F.

Let y be a point of the complement off (E). For each z € f ( E ) , there exist disjoint open neighborhoods V ( z ) of z and W ( z ) of y . The inverse images f - ’ ( V ( z ) ) form an open covering of E as z runs through f ( E ) (3.11.4). Hence by compactness there exists a finite number of points zi Ef ( E ) such that the V ( z i )form an open covering of f ( E ) in F. But then U = W ( z i )is an open i

neighborhood of y which does not intersect f ( E ) . This shows that f ( E ) is closed in F. It follows that if A is any closed subset of E, thenf(A) is closed in F (and therefore also in f ( E ) ) , because A is compact (3.17.3). Iff is injective, this establishes the continuity of the inverse mapping f (E) .+ E (3.11 -4). (12.3.7) I f a Hausdorff topology is coarser than the topology of a compact metrizable space, then the two topologies coincide. (12.3.8) Let E be a compact metrizable space, F a HausdorfSspace, f a continuous mapping of E into F . If b is any point o f f ( E ) and U is any open set in E containing f -‘(b), then there exists a neighborhood V of b in F such that f-yv) c

u.

8

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

For E - U is closed in E, hencef(E - U) is closed in F by (12.3.6). We have b #f(E - U) by definition, hence the complement V off(E - U) in F is an open neighborhood of b, and clearly U xf-'(V).

PROBLEMS

1. Find all possible topologies on a set of 2 or 3 elements. 2.

Let 5 be a set of subsets of a set E. Show that 5 is the set of closed sets for a topology on E if and only if 5 satisfies the following two conditions: (1) every intersection of a family of sets belonging to 8 belongs to 8 ; (2) the empty set belongs to 5, and the union of two sets belonging to 8 belongs to 8.

3. Let E be a set and for each x E E let B(x) be a set of subsets of E. Then there exists a topology F on E such that, for all x E E, B(x) is the set of neighborhoods of x with respect to Y, if and only if the sets B(x) satisfy the following conditions: (V,) Every subset of E which contains a set belonging to B(x) belongs to B(x). (V,,) The intersection of two sets belonging to %(x) belongs to B(x). (VIII) For all x E E and all V E B(x), we have x E V. (VIv) For all x B E and all V E B(x), there exists a set W E B(x) such that V E B(y) for all y E W. The topology Y is then unique. 4.

Let A be a commutative ring with an identity element 1 # 0. An ideal p of A is prime if A/@is an integral domain (and therefore #O). The set of prime ideals of A is called the spectrum of A and is denoted by Spec(A) (it can be shown to be nonempty). If a is an ideal in A, the set of prime ideals p 2 a is denoted by V(a). Show that V(a n b) = V(a) u V(b) for any two ideals a, b in A. Deduce that the subsets V(a) of Spec(A) are the closed sets in a topology, called the spectral topology, on Spec(A). If x , y are two distinct points of Spec(A), show that either there is a neighborhood of x which does not contain y, or else there is a neighborhood of y which does not contain x. Under what conditions is a set ( x } consisting of a single point closed in Spec(A)? Consider the case where A = Z, and the case where A is a discrete valuation ring. Let A' be another ring and let h: A+ A' be a ring homomorphism such that h(1) = 1. Show that the mapping p'wh-'(@') of Spec(A') into Spec(A) is continuous with respect to the spectral topologies.

5. (a) The following conditions on a nonempty topological space E are equivalent: (1) the intersection of any two nonempty open sets in E is nonempty; (2) every nonempty open set is dense in E; (3) every open set in E is connected. The space E is then said to be irreducible. A nonempty subset F of a topological space E is said to be an irreducible set if the subspace F is irreducible. (b) Show that in a Hausdorff space every irreducible set consists of a single point.

3 HAUSDORFF SPACES

9

(c) In a topological space E, a subset F is irreducible if and only if its closure F is irreducible. In particular, for each x E E, the set is irreducible. If an irreducible then x is said to be a generic point of F. set F in E is of the form (d) Let A be an integral domain. Show that the topological space Spec(A) (Problem 4) is irreducible and that {0}is its unique generic point. In Spec(Z), show that the complement of the generic point is an irreducible set which has no generic point. (e) If E is an irreducible space, every nonempty open set in E is irreducible. (f) Let (Urn),A be an open covering of a topological space E, such that U. n UB# 0 for all pairs of indices (a,6). Show that if the sets U, are irreducible, then E is irreducible. (g) Let E, F be two topological spaces and letfbe a continuous mapping of E into F. If A is an irreducible subset of E, show thatf(A) is an irreducible subset of F. (h) Let E, F be two irreducible spaces, each of which has at least one generic point. Suppose also that F has a unique generic point b. Let f be a continuous mapping of E into F. Show thatf(E) is dense in F if and only iff(x) = b for each generic point x of E.

E,

a

6. A topological space E is said to be quasi-compact if it satisfies the Borel-Lebesgue axiom (3.16) and compact if it is quasi-compact and Hausdorff. A subset F of a topo-

logical space E is said to be a quasi-compact set (resp. a compact set) if the subspace F of E is quasi-compact (resp. compact). Every finite set is quasi-compact. (a) In a quasi-compact space, every closed set is quasicompact. (b) Let E be a Hausdorff space and let A, B be two disjoint compact sets in E. Show that there exist two disjoint open sets U, V, such that A c U and B c V. (Consider first the case where A consists of a single point.) Deduce that a compact set in a HauSdorff space is closed. (c) In a topological space E, any finite union of quasi-compact sets is quasi-compact. (d) If E is a quasi-compact space andf: E + F is a continuous mapping, then the set f(E) is quasi-compact. (e) A filter base on a set E is a set 8 c b(E) of nonempty subsets of E such that, whenever X and Y belong to 8, there exists 2 E 8 such that Z c X n Y. Show that if E is a quasi-compact space and all the sets of 8 are closed, then the intersection of the sets of B is nonempty (consider the complements of the sets of 8, and argue by contradiction). (f) If A is a commutative ring with an identity element 1 # 0, show that the topological space Spec(A) (Problem 4) is quasi-compact. (g) Let E be the union of the interval 10, I ] and two elements a, 6. Let 8 be the set of finite intersections of sets of the forms ]a, I], {a}u 10, a [ , @ } u 10, a [ , where 0 < a < 1 . Show that 8 is a basis of a non-Hausdorff topology on E, for which E is quasi-compact and every set consisting of a single point is closed in E. Every point of E has a compact metrizable neighborhood, but there are such neighborhoods of CL (or 8) which are not closed in E. Show that the intersection of a compact metrizable neighborhood of a and a compact metrizable neighborhood of is not quasi-compact. The topology of E has a denumerable basis consisting of separable metrizable subspaces. (h) Let E be the sum (1.8) of N and an infinite set A. For each x E N, write Q(x) = ( x } . For each x E A, let Sx+ denote the union of { X I and the set of integers z n , and let G(x) denote the set of sets Sx, as n runs through N. Show that there exists a non-Hausdorff topology on E such that G(x) is a fundamental system of neighborhoods of x , for each x E E. In this topology, every set consisting of a single point is closed, and E has a dense compact subspace, although E itself is not quasi-compact.

10

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

4. UNIFORMIZABLE SPACES

A pseudo-distance on a set E is a mapping d of E x E into R which is such that d(x, x ) = 0 for all x E E, and satisfies axioms (I), (HI), and (IV) of (3.1), but not necessarily axiom (11). It is clear that a pseudo-distance satisfies the inequalities 4 x 1 , xn) 5 4 x 1 ,

~

2

+) 4 x 2

3

~

14x9 2 ) - d(y, 41

3

+ )

* * *

+

s 4x9 Y )

4 X n - 1 , Xn),

for all x i and x , y , z in E. Examples (12.4.1)

Iffis any mapping of a set E into R,the mapping ( x , r)b IfW - f(Y>I

is a pseudo-distance. Let cp be a mapping of the interval [0, co[ into itself which satisfies the following three conditions : (1) cp(0) = 0 ; (2) cp is increasing; (3) cp(u v ) 5 cp(u) cp(u) for all u >= 0 and v 2 0. Then if d is a pseudo-distance on a set E, the same is true of the composite mapping cp d : ( x , y ) ~ c p ( d ( xy ,) ) . We have only to check the triangle inequality, and by conditions (2) and (3) we have

+

+

+

0

cp(d(x, 4) 5

cp(d(X,

Y ) + d(Y, 4) 5 cp(&

Y>)+ cp(d(x, 2)).

If (d,) is a sequence of pseudo-distances on a set E, such that the series E E x E, then its sum d(x, y ) is a pseudo-

1dn(x,y ) converges for all ( x , y ) n

distance on E (3.15.4). Now consider a family (da)aE , of pseudo-distances on a set E. For each a E E, each finite family ( a j ) l j j m of elements of I and each finite family (rj)l s j s m of real numbers >O, put

B(a; (aj), (r,)) = { x E E I daj(a,x ) < r j for 1 5 j 5 m}, B’(a; ( a j ) ,( r j ) ) = { x E E I da,(a, x) S r j for 1 S j 5 m}. Let r) denote the set of subsets U of E such that, for each x E U, there exists a finite family ( c t j ) , S j s m of elements of I and a finite family ( r i ) l s j S m of strictly positive real numbers such that B(x; (mj),( r j ) ) c U. It is immediately verified that D is a fopology on E. This topology is said to be defined by the

4 UNIFORMIZABLE SPACES

11

family of pseudo-distances (d,), I . A topology which can be defined by a family of pseudo-distances is said to be uniformizable, and a space equipped with such a topology is said to be uniformizable. It is obvious that a metrizable space is uniformizable, but the converse is false (for example, the family consisting only of the pseudo-distance d = 0 defines the chaotic topology). There exist Hausdorff topologies which are not uniformizable (Problems 3 and 4); but throughout this book (and in the majority of situations in analysis) uniformizable spaces are the only ones which we shall have to consider. (12.4.2) Let E be a uniJormizable space whose topology is defined by a family of pseudo-distances (d,), E,. Then the sets

B(a; ( a j ) , (rj))

(resp. B‘(a;( a j ) , (rj)))

are open (resp. closed) in E.

Let x E B(a; (aj), (rj)),and put sj = d,,(a, x ) . Then sj c r j for 1 s j 5 m, hence B(a; ( a j ) , ( r j ) ) contains the set B(x; ( a j ) , ( r j - s j ) ) by virtue of the triangle inequality for each of the d., . This shows that B(a; (aj), ( r j ) )is an open set. If x 4 B‘(a;(aj), ( T i ) ) , then there exists an index k such that 1 k g m and d,,(a, x ) = s, > r, , and hence the triangle inequality for d,, shows that the set B(x; ak , s, - r,) does not meet B’(a; (aj), (rj)). This shows that B’(a;(aj),(rj)) is closed.

s

(1 2.4.3) In a uniformizable space, the closed neighborhoods of a point form a fundamental system of neighborhoods of the point.

This follows from (12.4.2) and the fact that

B’(a; ( a j ) , (rj/2))c B(a; ( a j ) , (rj))* (12.4.4) Let E be a uniformizable space and let (d,), I be a family of pseudodistances defining the topology of E. Then E is Hausdorfifand only $for each pair of distinct points x, y in E , there exists p E I such that d,(x, y ) # 0.

If E is Hausdorff, then by definition there exists a set B ( x ; (aj), ( r j ) ) not containing y , and therefore for at least one indexj we have d,,(x, y ) 2 rj > 0. Conversely, if d,(x, y ) = t > 0, then the open neighborhoods B(x; p, t / 2 ) and B(y; p, t/2) of x and y , respectively, are disjoint, by virtue of the triangle inequality for d, .

I f (d,) is a family of pseudo-distances defining a Hausdorff topology on a set E, then to say that a sequence (x,,)of points of E has a point a as limit with respect to this topology means that lim d,(a, x,,) = 0 for all LY. n-rm

12

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

Two families of pseudo-distances (dJ, (di)A on the same set E are said to be topologically equivalent if they define the same topology on E. (12.4.5) If ( d J U E I is any family of pseudo-distances on a set E , there exists a topologically equivalent family (d;), E , such that 0 5 dl 5 1 for all 01 E I .

+

Let cp(u) = inf(u, 1) for 0 5 u < co.Then cp(0) = 0, and cp is an increasing function on [0, + co[ and satisfies the inequality

+ cp(4.

cp(u + 0) 5

=+

+

(This last assertion is clear if u 1 or u > 1 or if u v 5 1 ; and if u 5 1 and v 5 1 and u v > 1 we have cp(u v ) = 1 < cp(u) + cp(v).) It follows rlow from (12.4.1) that di = cp d, is a pseudo-distance on E, for each 01 E 1. That the topologies defined by the families (d,) and (d;) are the same follows from (12.2.1) and the fact that, when 0 < r < 1, the relations d,(x, y ) < r and di(x, y ) < r are equivalent.

+

0

(1 2.4.6) A Hausdorfluniformizable space E , whose topology can be deJined by a sequence (d,,) of pseudo-distances, is metrizable.

Without loss of generality we may assume that the sequence (d,) is infinite, and by (12.4.5) that 0 5 d,, 5 1 for all n. Then the series (12.4.6.1)

1 d(x, y ) = 2 di(x, y )

1

1

+ 27 d,(x, y ) + + 2. d,,(x, y ) + * *

* * *

converges for all pairs (x, y ) of elements of E, and d is a pseudo-distance on E (12.4.1). Moreover, d is a distance, because d(x, y ) = 0 implies that d,,(x, y ) = 0 for all n, and therefore that x = y because E is Hausdorff (12.4.4). If Bo(x;r ) is the open ball with center x and radius r with respect to the distance d, then it follows immediately from (12.4.6.1) that B,(x; r ) c B(x;n,2"r) for all n. Conversely, if n is so large that 2"-' 2 l/r, then we have W

C 2-"-k k= 1

d,,+k(x,y ) 5 2-" 5 +r

for all x , y in E , and therefore

B(x; (192, . .

9

4,( t r , . . ,h))= B o b , r). *

By (12.2.1), the proof is complete. (12.4.7) Let E be a topological space and (U,) aJinite or denumerable open covering of E such that the subspaces 0, are separable and metrizable. Then E is separable and metrizable.

4 UNIFORMIZABLE SPACES

13

We begin by showing that the topology of E is Hausdorf. Let x , y be two distinct points of E. If there exists an index n such that x E U, and y 4 On, then V = U, and W = E - 0, are open neighborhoods of x and y respectively which do not intersect. If on the other hand x E U, and y E 0, for some n, then in the metrizable subspace 0, there exists an open neighborhood V, of x and an open neighborhood W, of y which do not intersect. The set V = V, n U, is an open neighborhood of x in E, and there exists an open set W in E such that W n 0, = W, . Hence W is an open neighborhood of y in E, and VnW=@. For each n, let d, be a distance defining the topology of O n ,and let (Vrn,JmLbe a basis for the topology of U, , where V,, is an open ball with center a,, and radius r,, (3.9.4). Let A,,,, be the function which is equal to rmn- d,(a,,, x ) in V,, and is 0 in the complement of V,,,, in E. Then.f,, is continuous on E, since it vanishes at the frontier points of V,, . Now let &n(X,

Y ) = Ifmn(x) -fmn(Y)l

for all x, y in E. By virtue of (12.4.6), the proof will be complete if we show that the pseudo-distances d,,, define the topology of E. For each x , E E , every set defined by an inequality of the form d,,(xo, x ) < CI is open in E (3.11.4). Conversely, there exists an integer n such that x , E U, , and for each neighborhood W of x , in E there exists an integer m such that V,, is an open neighborhood of xo contained in W (3.9.3). Hence fmn(xo)= B > 0, and the set of all x E E such that d,,(xo, x ) < +B is therefore contained in W. By virtue of (12.2.1), this completes the proof. We remark that the conclusion of (12.4.7) is not valid without the hypothesis of denumerability on the open covering (U,) (Section 12.16, Problem 22); again, the hypotheses cannot be weakened by assuming merely that the open sets U, are separable and metrizable (section 12.3, Problem 6(f)).

PROBLEMS

1. Let E be a topological space such that, for each x E E, the neighborhoods of x which are both open and closed in E form a fundamental system of neighborhoods of x. Show that E is uniformizable.(Observe that the characteristicfunction of a set which is both open and closed in E is continuous on E.)

2. Let E be a denumerable Hausdorff uniformizable space. Show that, for each x E E, the open-and-closed neighborhoods of x form. a fundamental system of neighborhoods of x.

14

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

3. Let 6 be a positive irrational number. For each point (x, y ) E Q x Q+ (where Q+ = Q n R,) and each integer n > 0, let B.(x, y ) denote the set consisting of (x, y ) and the points (z, 0) E Q x Q + such that Iz - (x 6y)l < I/n or Iz - (x - 6y)l < I/n. As (x, y ) runs through Q x Q ,and n runs through the set of strictly positive integers, show that the sets B.(x, y ) form a (denumerable) basis of a topology .T on Q x Q + Show that .T is Hausdorff but that, if a and b are any two points of Q x Q + , every closed neighborhood of a meets every closed neighborhood of 6. Deduce that Q x Q ,, endowed with the topology .T, is connected and that every continuous mapping of Q x Q, into R is constant. This shows that the topology .T is not uniformizable.

+

.

4.

Let Q be the set of rational numbers endowed with the topology I .induced by that of R. Let !Dl be the set of subsets A of Q such that the closure A of A (in the topology .To)has only finitely many nonisolated points (3.10.10). Let B be the set consisting of all open intervals in Q, all complements of sets belonging to Fm and all intersections of these complements with open intervals in Q. Show that B is a basis for a topology F on Q, and that .T is finer than .To and therefore Hausdorff. Show that, with respect to this topology, a convergent sequence in Q has only finitely many distinct terms, although the topology I is not discrete. In the topology I , no point of Q has a denumerable fundamental system of neighborhoods, although Q is denumerable and every point of Q is the intersection of a denumerable family of neighborhoods of the point. By using (1 2.4.3), show that I is not uniformizable. although it is finer than a metrizable topology.

5. Let E be a topological space satisfying the following condition: for each x ‘E E and each neighborhood V of x in E, there exists a continuous mapping f: E -+ [O,11 such that f(x) = I and f(y) = 0 for all y E E - V. Show that E is uniformizable. Deduce that if E is a topological space such that every point of E has a closed neighborhood in E which is a uniformizable subspace, then E is uniformizable. Can the word “closed” be omitted from this proposition? (Cf. Section 12.3, Problem 6(f).) 6.

For each x E R and each integer n > 0, let U.(x) denote the union of the intervals [x, x l/n[ and ] - x - I/n, -XI.Show that there exists a topology 9on R such that the U.(x) form a fundamental system of neighborhoods of x, and that .T is not comparable with the usual topology of R. Show that I is uniformizable (use Problem 5). In this topology, every point has a denumerable fundamental system of neighborhoods, and there exists a denumerable dense subset. But there exists no denumerable basis of open sets for I and , consequently I is not metrizable. Show also that the topology induced by .T on the interval E = [ - I , 11 of R is not metrizable, although E is quasicompact and Hausdorff in this topology.

+

5. PRODUCTS O F U N I F O R M I Z A B L E SPACES

Let El, E, be two topological spaces. In the product set E = El x E, let 0 be the set of (arbitrary) unions of sets of the form Al x A , , where A, is open in El and A, is open in E, . The set 0 is a topology on E, because it clearly satisfies axiom (Of), and it satisfies (0,Jby reason of the relation

5

PRODUCTS OF UNIFORMIZABLE SPACES

15

where Ai and B, are open sets in E i ( i = 1,2). This topology 0 is called the product of the topologies on El and E,, and the set E endowed with this topology is called the product of the topological spaces El, E, . If x = ( x , , x,) is any point of E, the sets V, x V, (where V i runs through a fundamental system of neighborhoods of xi, f0r.i = 1, 2) form a fundamental system of neighborhoods of x in E (3.9.3). From this it follows that the continuity criterion (3.20.4)is valid for the product of two arbitrary topological spaces. The relation A, x A, = A, x A, (where A, c El and A, c E,) also remains true; for if (a,, a,) E E and if Vi is a neighborhood of a, in Ei (i = 1,2), the set (V, x V,) n (A, x A,) = (V, n A,) x (V, n A,) is empty if and only if one of the sets V in Ai is empty. For each a, E El, the set ((a,} x E2) n (A, x A2) is either empty or equal to {a,} x A , , and therefore the mapping x 2 ++(a,, x 2 ) is a homeomorphism of E, onto the subspace (a,} x E, of E. Likewise, if a2 E E,, the mapping x,H(x,, a2) is a homeomorphism of El onto the subspace El x { a 2 }of E. From this it follows immediately that the propositions (3.20.12)and (3.20.13), and the continuity criteria (3.20.14)and (3.20.15),are valid in general. If El and E, are Hausdorf then so is El x E, . For if x = (x,, x 2 ) and y = (yl, y 2 )are distinct, then either x, # y , or x, # y , . If the latter, there is a neighborhood U of x2 and a neighborhood V of y , in E, which do not intersect, and then El x U and El x V are disjoint neighborhoods of x and y , respectively, in E. IfE, = El, the canonicalsymmetry (x,, x , ) ~ (x, , xl) is a homeomorphism of El x El onto itself, and is equal to its inverse. The product of any finite number of topological spaces is defined in the same way. It follows immediately from the definition that the canonical “associativity” mappings such as (El x E,) x E, 4 El x (E, x E,) are homeomorphisms. We shall be concerned especially with the case in which El and E, are uniformizable. In this case the product space E is also uniformizable. To be precise, let (d$l))LL, ( d r ) ) r be families of pseudo-distances which define the topologies of’ El, E, respectively, and let

Then the el1) and )e: are pseudo-distances on E, and the definition of neighborhoods given above shows that the family of pseudo-distances ey) and ff), where I E L and p E M, defines the product topology on E. This leads to a generalization of the notion of a product to arbitrary (not necessarily finite) families of topological spaces. We shall restrict ourselves to uniformizable spaces.

16

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

Let (E,),

I

be any family of uniformizable spaces, and let E =

For each a E I, let (d,, topology of E, . If we put (12.5.1)

fl E, .

,El

be a family of pseudo-distances defining the

eu, Ax, Y ) = d,, n(P'uX7 PlhY)

for each pair (x, y ) of elements of E =

nE,, then it is immediately checked

. E l

that the e,, are pseudo-distances on E. The product of the topologies of the E, is then the topology on E defined by the e,, as a runs through I and, for each a E I, Iz runs through L, . The set E endowed with this topology is called the product of the spaces E, . When I = { 1,2} this definition agrees with that given above for two spaces El, E, . of For every finite family ( a i , Ai), and every finite family (ri)l strictly positive real numbers, the set B(x; ( ( a i ,Ai)), (ri)) can be written in B, where, if a = ai, we have B, = B(pr,, x ; (Aj), (rj)),where j the form

n

UEI

runs through the indices for which aj = t l i , and B, = E, if a is not equal to any ai. From this and (3.6.4) we deduce immediately: (12.5.2) For each a E I, theprojection pr, is a continuous mappingof E onto E, , and the image under pr, of any open set in E is an open set in E, .

(,VH )

(12.5.3) For each finite subset H of I, and each family (UJUEH,where U, is U, x ( , B H E . ) is open in E. open in E, for each a E H, the set

For it is the intersection of the sets pr;'(U,), are open (12.5.2) and finite in number.

where a E H, and these sets

The sets described in (12.5.3) are called elementary sets in E. The description given above of the neighborhoods shows that (having regard to (3.9.3)) these sets form a basis for the topology of E. Incidentally, this shows also that the topology of E depends only on the topologies of the E, and not on the families of pseudo-distances (d,, Moreover, if b, is a basis for the topology of E,, the elementary sets such that U, E b, for all a form a basis for the U, with topology of E. It should be noted that, if I is infinite, a product

n

u61

U, open in E,, nonempty and # E, for a f fc1 E I, is not open in E. (12.5.4) Let A, be a subset of E,, for each a, and let A

A=

=

n A,.

Then

n A,. In particular, $each A, is closed in E, , then A is closed in E.

,El

QCI

5

PRODUCTS OF UNIFORMIZABLE SPACES

17

We have pr,(A) c pr,(A) = A, for all u E I ((12.5.2) and (3.11.4)), hence A,. Conversely, if a = (a,) E A,, and if we consider an ele-

Ac

n

n

as1

mentary set U =

n U, n E, containing a, its intersection with A is n n A,. as1

aeH

x

asl-H

aeH

(Aan U a ) x

asl-H

Since none of the A, is empty, no A, is empty and therefore U n A # which shows that a E A.

a,

(12.5.5) Let z~ f ( z ) = (f,(z)), I be a mapping of a topological space F into E. For f to be continuous at a point z, E F , it is necessary and sufficient that each f, should be continuous at zo .

For any elementary set U =

n f,-1(uU).

n U, n E,,

asH

x

asl-H

we have f-'(U) =

aeH

(12.5.6) Let (x("))be a sequence of points of E. For a = (a,) to be a limit of the sequence (x'")), it is necessary and suficient that a, should be a limit of the sequence (prax(n)),,tl, - for each CI E I .

This is an immediate consequence of (12.5.5) and the definition of a limit. In particular, if all the E, are equal to the same space F, then to say that a mapping u E F' of I into F is a limit of a sequence of mappings ( u , , ) , , of ~ ~I into F, with respect to the product topology, means that for each c1 E I the sequence (u,,(M))has u(a) as a limit in F. In this situation we say that the mapping u is a simple limit of the sequence (u,,), or that the latter converges simply to u. (12.5.7) v e a c h space E, is Hausdorfl, then so is the product space

E.

For if x # y , there exists an index u E I such that prax # p r a y ; hence, using (12.4.4), there exists A E La such that d,, I(pr,x, p r a y ) # 0, and therefore e,,,(x, y ) # 0. The result now follows from (12.4.4). (12.5.8) The product E of a denumerable family of metrizable (resp. separable metrizable) spaces is metrizable (resp. separable and metrizable).

First of all, E is Hausdorff by (12.5.7). To show that E is metrizable, apply (12.4.6) and the definition of pseudo-distances on a product. Now suppose that E =

fr En,where the Enare separable and metrizable. Then each

n= 1

18

XI1 TOPOLOGY A N D TOPOLOGICAL ALGEBRA

Enhas a denumerable basis (Um,JmLO of open sets. For each n

n uf(j), n n

the set of elementary sets in E of the form

x

j= 1

a

k=n+ 1

1, let 23, be Ek, where f is

any mapping of { 1, 2, . . . ,n} into N. Since N" is denumerable (1.9.3), the set 23, is at most denumerable (1.9.2). We have seen that the union 23 of the sets 23, is a basis for the topology of E (12.5.3); since $'3 is denumerable (1.9.4), it follows that E is separable (3.9.4). The product of a denumerablefamily of compact metrizable spaces is compact and metrizable. (1 2.5.9)

Let (En) be a sequence of compact metric spaces. Since we know already (12.5.8) that E =

m

fl E,

,= 1

is metrizable, it is enough to show that every

sequence (x,) of points of E has a cluster value (3.16.1). We define by induction on m 2 0 a family of sequences ( x ? ) ) , , ~ of ~ points of E, as follows: x:)' = x,; for m 2 1, the sequence ( x : ~ ) ) , ,is-a ~ ~ subsequence of the sequence (xp- I)),, (in other words, there exists a strictly increasing mapping q m : N + N such that xkm)= x$",;nf)) such that the sequence ( p r m x ~ ) ) n z l converges in Em to a point a,,,. This construction is possible because Em is compact. Now consider the sequence (y,) in E, where y, = xf) (Cantor's q l , we have y , = x,Jn); "diagonal trick"). If we put $,, = (P, q n - l 0 since $,(n) > $n-l(n - 1) by definition, it follows that (y,) is a subsequence of (x,). Furthermore, for each m, the sequence (y,Jn2, is a subsequence of the sequence ( x : ~ ) ) , , ~and ~ , therefore the sequence (prmy,JnLm converges to a,". Hence so does the sequence (prmy,),;L,, since it differs from the former by only a finite number of terms. Hence the sequence (y,) converges to the point a = (a,,,) (12.5.6). 0

0

PROBLEMS

1. Show that the product of two quasi-compact topological spaces (Section 12.3, Problem 6) is quasi-compact. 2. Show that a topological space E is Hausdorff if and only if the diagonal (1.4.2) of E X E is closed in E x E.

3. Let (Ea)as, be a family of arbitrary (not necessarily uniformizable) topological spaces, and let E = E.. Show that the elementary sets, defined as in (12.5.3), form a basis

n

OlEl

of a topology on E. This topology is called the producr of the topologies of the E, Generalize (12.5.2) to (12.5.7) to this situation.

.

5 PRODUCTS OF UNIFORMIZABLE SPACES 4.

Let E =

n E.

19

be a product of Hausdorff topological spaces, such that each E.

= E l

contains at least two distinct points a , , 6 , . For each (1: E I, let c. be the point of E such that pr.c. = b. and prac. = aa for all /3 # a. Show that every point of the set { c ~ } is~ an ~ , isolated point. Deduce that the topology of E has a denumerable basis if and only I is denumerable and each of the E. has a denumerable basis. Show that if I is not denumerable, the point a = (a,) has no denumerable fundamental system of neighborhoods. If E. = {arn, ba}for each a, and if F c E is the set of all x t E such that pr,x = b. for all but denumerably many indices a, show that F is dense in E and that a = (an)is not a limit of any sequence of elements of F.* 5. Let K be the discrete space {0, l}, let A be an infinite set, and let E be the product space KA. Let V be a nonempty elementary set in E. If h is the (finite) number of indices o! such that pr,V # K, let p(V) = 2 - h (cf. (13.21)). (a) Show that, if U,, . . . , U. are nonempty mutually disjoint elementary sets, then

"

&=1

p(Uk)5 1. (Write Urin the form Wx x KB,where B is the same for all k and is the

complement of a finite subset of A.) (b) Deduce from (a) that if (U,A6 is a family of nonempty pairwise disjoint open sets in E, then L is at most denumerable, although there exist sets C of arbitrary cardinal in E (for a suitable choice of A) all of whose points are isolated. Show also that if the cardinal of A is strictly greater than that of B(N), then E contains no denumerable dense subset.

6. With the notation of Problem 5, let 9 be the set of all subsets of KA of the form M a , where M. = K except for ar most denumerably many indices or. Show that 8

n

.EA

is a basis for a Hausdorff topology on KA which is finer than the product topology, and which is not discrete provided that A is infinite and non-denumerable.In this topology, every denumerable intersection of open sets is open; no point has a denumerable fundamental system of neighborhoods; and every quasi-compact subset is finite. The projections pr. are continuous with respect to this topology. Deduce that KA is uniformizable with respect to this topology (Section 12.4, Problem 5).

7. Let 1 be the interval [0, 11 in R,with the induced topology. Show that every separable metrizable space E is homeomorphic to a subspace of the product IN. (Reduce to the case where the distance d on E is 5 1, and consider a sequence (0.) which is dense in E, and the functions x H d ( a . , x).) 8. With the notation of Problem 7, show that in the uniformizable product space I', the subspace of continuous mappings of I into I is dense. Deduce that I' has a denumerable dense subset, although there is no denumerable basis of open sets (Problem 4). 9.

Show that if I is a nonempty open interval in R,there exists no nonconstant mapping f o f I into the product space NN with the following property: for each x E I and each

*This example shows that in general topological spaces (and even in Hausdorff uniformizable spaces) the convergent sequences do not determine the topology, as they do in metrizable spaces (cf. (3.1 3.3) and (3.1 3.4)). To get corresponding results in general, it is necessary to replace the notion of sequence by the more general notion of a filter (cf. [5]).

20

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

integer n > 0, there exists a neighborhood V of x in I such that, for all y E V, the first n terms of the sequence f(y) are the same as the first n terms of the sequence f ( x ) . (Show that this condition implies that f i s continuous, and use (3.19.7).) 10. (a) Let E be a topological space and A a nonempty closed subset of E. Let E’ be the sum of E - A and a set { w } consisting of a single element, and let 0’be the set of subsets of E’ which are either of the form U where U is an open set in E - A, or of the form (V - A) u { w } , where V is an open set in E which contains A. Show that 0’is a topology on E’, and denote by E/A the set E’ endowed with this topology. Let p(x) = x if x E E - A, and ~ ( x = ) w if x E A. Show that y is a continuous mapping of E onto E/A. Every continuous mapping f of E into a topological space F, which is constant on A, can be written uniquely in the form f = g 0 p, where g: E/A -+ F is continuous. (b) Suppose that E is metrizable and compact; let d be a distance defining the topology of E, and let (Wn)nblbe a basis for the topology of E - A. Let f o ( x )= d(x, A), f.(x) = d(x, E - W,) for n 2 1, and f ( x ) = ( f , ( ~ ) ) e. R~ N~ . Show that f(E) is a compact subspact: of RN, homeomorphic to E/A. This shows that E/A is compact and metrizable. (c) Show that if we take E = R and A = Z, then the space E/A defined in (a) is not metrizable (cf. Section 3.6, Problem).

6. LOCALLY FINITE COVERINGS A N D PARTITIONS O F U N I T Y

In a topological space E, a family (A,),., of subsets of E is said to be locallyfinite if for each point x E E there exists a neighborhood U of x in E such that U n A, = $3 for all but a finite number of indices c1 E I . If E is metrizable and K is a compact subset of E, it follows that there is a covering of K by afinite number of neighborhoods of points of K in E, each of which meets only afinite number of the sets A,. In particular, K meets only afinite number of the A,. If ( A J L E L ,(B,),EM are two coverings of a topological space E, the covering (B,) is said to be finer than (A,) if, for each p E M, there exists A E L such that B, c A,. (12.6.1) Let E be a separable, locally compact, metrizable space and let 23 be.a basis of open sets in E. If (A,),. is any open covering of E, there exists a denumerable locallyfinite open covering (B,) of E which isfiner than (A,), and such that the sets B, are relatively compact and belonq to 23. Consequently each B, meets only finitely many of the sets B, .

From (3.18.3) we know that there exists an increasing sequence (U,JnLO of relatively compact open sets such that 0, c U,,, for all n, and E = U,.

u n

6

LOCALLY FINITE COVERINGS AND PARTITIONS OF UNITY

21

Put K, = 0, - Un-l, which is a compact set for all n 2 0 (we put U,, = for all n < 0). For each n 2 0, the open set U,+, IS a neighborhood of K, . Hence, for each x E K,, there exists an open neighborhood W v ) E 23 of x, contained in Un+l- On-2and contained in one of the sets A,. There exists a finite number of points x i E K,, (I 5 i 5 p,) such that the sets WE) cover K, . Arrange the sets WLy) (m 2 0, 1 5 i 5 p m for each m) in a sequence in any way, and let (B,) denote this sequence. It is clear that (B,) is an open covering of E which is finer than (A,), L , and that the sets B, are relatively compact. Thus all that has to be checked is that the covering (B,) is locally finite. Let z be any point of E, and let n be the least integer ~ U hX that z E U, . Since z 4 U,-l, there exists a neighborhood T of z contained in U, and not meeting Consequently T can intersect only the sets W:) for which n - 2 m 5 n + 1, and the number of these sets is finite. (12.6.2) Let (A,,) be a denumerable locally$nite open covering of a metrizable space E. Then there exists an open covering (B,) of E such that B, c A, for all n.

B,

We shall define the family (B,) by induction on n, in such a way that

c A,, for each n, and such that for each n the family consisting of the B, with k 5 n and the A j with j > n is an open covering of E. Suppose that the

sets B, have been defined for n < m. Then the B, with n < m and the A j with

j 2 m cover E. Let C be the open set which is the union of the B, with n < m

and the Aj withj > m + I , so that we have E - A, c C. We shall show that there exists an open set V such that E - A,, c V c 0 c C. If E = A,, we may take V = 0. If C = E, we may take V = E. If neither E - A,, nor E - C is empty, there exists a continuous mapping f of E into the interval 10, 11of R,which is equal to 0 on E - A,, and to 1 on E - C (4.5.2).We then take V to be the open set of points y such that f ( y ) < f , and then 0 is contained in the closed set of points y such that f(y) 5 +, and hence ? c C. Put B, = E - 0. Then we have B,, c E - V c A,,, and B,, u C = E, so that the sets B,with n 5 m satisfy the required conditions, and the induction can proceed. For each x E E, there exists an integer n such that x 4 A,, for all m > n, and therefore x E B, for some k 5 n. Hence the sets B, cover E.

It is clear that the same argument will apply if the covering (A,) isfinite.

I f E is a topological space andf'is a mapping of E into a real vector space F or into R, the support off (denoted by Supp(f)) is defined to be the smallest closed set S in E such thatfvanishes on E - S. In other words, Supp(f) is the closure in E of the set of points x E E such that f(x) # 0 ; or again it is the set of points x E E with the property that every neighborhood of x in E contains a point y such that f ( y ) # 0.

XI1 TOPOLOGY A N D TOPOLOGICAL ALGEBRA

22

Let (f,),. I be a family of mappings of E into F (resp. R) whose supports form a locally finite family. Then the sum C f a ( x ) is defined for all x E E, aa I

because it contains at most finitely many non-zero terms. We denote by f , the function x + + CfR(x).If F is B or a normed space (or more generally

c

as1

aE1

a topological vector space (12.13)), and if each f , is continuous on E, then so is f = f a :for given x E E there exists a neighborhood V of x which meets only

c

as1

finitely many of the Supp(f,), and hence there is a finite subset H of I such that f ( y ) = f a ( y ) for all y E V.

c

asH

A continuous partition of unity on E is by definition a family ( f a ) , ,I of continuous mappings of E into [0, I], such that the supports of the f a form a locally finite family, and such that f a ( x )= 1 for all x E E. If (Aa)aE I is an

1

aal

open covering of E indexed by the same set I, then the partition of unity (fa), I is said to be subordinate to the covering (AJa I if we have Supp(f,) c A, for all u E I. (12.6.3) Let (A,) be an at most denumerable 1ocally.finite open covering of a metrizable space E. Then there exists a continuous partition of unity (f,)on E subordinate to (A,,).

Let (B,) be an open covering of E, such that B, c A, for all ii (12.6.2). It is clear that the covering (B,) is locally finite. By (4.5.2), for each iz there exists a continuous mapping h,: E -+ [0, I] such that h, is equal to 1 on B, and equal to 0 on E - A,. If we put g , = (h, - +)+, then Supp(g,) is contained in the set of points x such that h,(x) 2 t,and hence is contained in A,. Let g = g, .

1 n

Since the sets B, cover E, we have g ( x ) > 0 for all x E E, and therefore the functionsf, = g,/g are defined and continuous on E, and form a partition of unity with the required properties. (12.6.4) Let E be a metrizable space, K a compact subset of E and afinite covering of K by open sets of E. Then there exist m continuous mappings f,: E --* [O, I] such that supp(fk) c A, for 1 5 k 5 m, and such that m

1

fk(x)

k= 1

5 1 for all x E E and

m

fk(x) = 1 for all x E K.

k= 1

Take a continuous partition of unity (fk)Odksn, subordinate to the open covering of E consisting of A,, = E - K and the A, (1 5 k 5 m).

7 SEMICONTINUOUS FUNCTIONS

23

Remark (12.6.5) Let F be a subset of the set of continuous mappings of E into R,with the following three properties :

(1) for each pair of disjoint nonempty subsets M, N of E, with M compact and N closed, there exists a function f 2 0 in F which is equal to 0 on N and 21 on M ; ( 2 ) if (f U ) E, , is a family of functions in 9 whose supports form a locally finite family, then f, belongs to 9; U E l

(3) i f f € 9 is such thatf(x) > 0 for all x E E, then g/fe 9for all g E 9.

Then the conclusions of (12.6.3) and (12.6.4) are still valid, when the A, are relatively compact, if we impose the extra condition that the functionsh should belong to 9. The proofs are unaltered.

7. SEMICONTI NUOUS FUNCTIONS

In this section, and above all in Chapter XIII, we shall need to consider mappings of a set A into the extended real line 8. Such mappings we shall call (by abuse of language) real-valuedfunctions on A. Such a function f is said to be finite if its value at each point a E A is finite, i.e., iff(A) c R.The functionf is said to be majorized or bounded above (resp. minorized or bounded below) if there exists a finite majorant (resp. a finite minorant) of f ( A ) . Iff is both majorized and minorized, it is said to be bounded (which clearly implies that f i s finite). We recall ((4.1.8), (4.1.9), and (3.15.5)) that in R x R the function (x, y ) x +~y (resp. (x,y ) H x y ) is defined and continuous except at the points (+ co, - co) and (- co, 00) (resp. (+ co, 0), (- co,O), (0, co), (0, - co)). The function X H I / X on R is defined and continuous except a t the point x = 0. In the interval [0, + co] of R, the function X H l/x (defined on 10, +a])can be extended by continuity by giving it the value f c o at the point 0. Let E be a topological space and letfbe a mapping of E into the extended real line R. The function f is said to be lower (resp. upper) semicontinuous at a point x o E E if, for each a E R such that a c f ( x o ) (resp. a > f ( x , ) ) , there exists a neighborhood V of xo in E such that, for all x E V, we have a < f ( x ) (resp. c( > f ( x ) ) . The functionfis said to be lower (resp. upper) semicontinuous on E if it is so at every point xo E E.

+

+

24

XI1 TOPOLOGY A N D TOPOLOGICAL ALGEBRA

Clearly, iff is lower semicontinuous at xo , then -f is upper semicontinuous at this point. Hence we need consider only lower semicontinuous functions. If (P is a mapping of a topological space F into E which is continuous at a point yo E F, and iff is lower semicontinuous at the point xo = (~(y,,),then f (P is lower semicontinuous at y o . In particular, if F is a subspace of E, and iff is lower semicontinuous at a point yo E F, then so is the function f I F. 0

Examples

(12.7.1) A mapping f : E .+ a is said to have a relative minimum at a point xo E E if there exists a neighborhood V of xo such that f ( x ) 2 f ( x o ) for all x E V. If so, thenfis lower semicontinuous at the point xo . This is clearly the case whenever f ( x o ) = - 03. For each x E R, put f ( x ) = 0 if x is irrational, and f ( x ) = l/q if x is rational and equal to p / q (where p , q are coprime integers and q > 0). For each integer n 2 0, the subspace of rational numbers p / q with q S n is closed in R and discrete. Hence for each irrational x there exists a neighborhood V of x such that f ( y ) l / n for all y E V, and therefore f is continuous at the point x . On the other hand, f has a relative maximum at each rational point, and is therefore upper semicontinuous on R. (12.7.2) A mapping f : E + is lower semicontinuous on E if and only $ for each a E R, the set f - ' ( ] a , 031) of points x at which f ( x ) > a is open in E (or, equivalently, the set f -I([ - 03, a ] ) of points x such that f ( x ) 5 a is closed in E).

+

For to say that f is lower semicontinuous on E signifies that, for each a E W, the set f - ' ( ] a , +a])is a neighborhood of each of its points (3.6.4).

If A is any subset of a set E, the characteristic function of A (usually denoted by qA)is the mapping of E into R such that ( P ~ ( x= ) 1 for all x E A and pA(x) = 0 for all x E E - A. So we have q E = 1, ( P = ~ 0, and the formulas (PE-A=

I1 2.7.3)

(PA

-(PA,

+ (PB = (PA v B + (PA

nB

(PA(PB=(PAnB,

,

(PA n CB

= (PA

- (PA (PB,

where A, B are any two subsets of E, and (A,) is any family of subsets of E.

7 SEMICONTINUOUS FUNCTIONS

25

(12.7.4) A subset A of a topological space E is open (resp. closed) in E ifand only if qAis lower (resp. upper) semicontinuous on E.

This follows immediately from (12.7.2). (12.7.5) Let f, g be two mappings of a topological space E into R, each of which is lower semicontinuous at a point xo E E . Then the functions sup(f, g ) and inf(f, g ) are lower semicontinuous at xo . The same is true ,forf + g if the sum f ( x ) + g(x) is defined for all x E E (4.1.8), and for f g i f f and g are both 20 and the product f (x)g(x)is defined for all x E E (4.1.9).

We shall give the proof for f + g ; the other cases are analogous. The result is obvious if f ( x o ) or g(xo) is equal to -03. If not, then we have f ( x o ) + g ( x o ) >- co. Every number u E R such that u X

for all sequences (y.) such that

2 x for all n. Show that the set of all x E E such that lirn inf f ( y ) > lirn inff(y)

Y-x,

Y-IX

Y>X

is at most denumerable. (For each pair ( p , q) of rational numbers with p > q. show that the set of points x E E such that

lirn inf f(y) > p > q > lirn inff(y)

Y-x.

Y>X

Y+X

is at most denumerable, by following the method of Section 3.9, Problem 3.) 9. A Dirichlet series is a series whose general term is of the form a.e-'"S, where (h.) is an increasing sequence of real numbers, tending to co, and (a")is any sequence of complex numbers, and s is complex. (a) Show that if the series is convergent for s = s o , then it is uniformly convergent in the angular sector consisting of the points s = so pele with p 2 0 and -a 2 6 a, where a is such that 0 < a < n/2. (Reduce to the case so = 0; show first that if 9 s = u, then

+

+

le-os-e-b*l

5

I~Io-'(e-O"--e-b")

whenever a and b are real and a < b, by considering the integral

+

s."

cXs dx. For each

integer m and each n l m , put S,,,= ~ , , + a , + ~* . . + a , and use the identity (Abel's partial summation formula)

7 SEMICONTINUOUS FUNCTIONS

31

(b) Deduce from (a) that there are the following three alternatives: either the series ( ~ " e - ~ does " ' ) not converge for any value of s, or it converges for all values of s, or else there exists a real number uo such that the series converges for 9 s > uo and does not converge when 9 s < uo. In the first (resp. second) case we put uo = m (resp. - 03). In all cases, the (extended real) number uo is called the abscissa of convergence of the series. The sum of the series is an analytic function of s in the region 92s > uo. (c) Show that if uo 2 0, then uo = lirn sup (log ISo. .I)/&. (Show first that if the

+

n-m

series with general term a.e-"' = b, is convergent (with s real and >O), then we have ISo, 5 KeAnswhere K is a constant, by writing a, = b.eAnS.Then show that if y = lim sup (log ISo, "\)/An,the series (ane-Afls) converges when s = y 6, where 6 is

+

n-m

real and >0, by arguing as in (a).) (d) Let u1 be the abscissa of convergence of the Dirichlet series with general term la.le-Ans.Then u1 2 uo. If uo < m , show that

+

log n u1 - uo 5 lim sup n-m

A.

I

(Remark that, given E > 0, we have lan\5 e'"o+e'Anfor all sufficientlylargen.)Consider the case where A,, = n (cf. Section 9.1, Problem 1). Show that, for the series with general term

wehaveuo=-mandal=+co. 10. Letfbe a real-valued function defined on the interval [0,

+

m[, satisfying the following conditions: (1) f ( s + t ) Z f ( s ) + f ( t ) ; (2) there exists a number M > 0 such that If(t)l 5 Mt for all t. Show that the limits

exist and are finite, and that at sf(r)5 /3r for all t 2 0. (To establish the existence of a it is enough to show that, if we take a = limsupf(r)/t (Problem 8), then 1-0

sf(r)/rfor

all t > 0. Take a decreasing sequence (1.) tending to 0, such that limf(rn)/tn = a, and let k, be the integral part of r/t.. Show that a

The proof of the existence of lim t-+m

f(r)/t

is similar; put

/3 = lim inff(r)/t.) r-+m

11. Let f b e a continuous mapping of an open set U in a Banach space E into a Banach space F. For each x E U, let Ilf(Y) -f(x)ll

D+f(x) = lim sup y-x.y+x

D - f ( x ) = lim inf Y-x.Y#x

We have 0 2 D-f(x)

5 D+f(x)

+ m.

IlY-XI1

/I f(Y)- f ( x )II IlY-XI1

'

.

32

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

(a) If the function f is differentiable at a point x E U, then D+f(x) = Ilf’(x)ll. If f:(x) is not a linear homeomorphism of E onto a subspace of F,we have D-f(x) = 0 . In the contrary case, we have D-f(x)= ~ ~ ( J ’ ( x ) ) - ~where ~ ~ - ~(,f ’ ( ~ ) ) - ~denotes the inverse homeomorphism. (b) Suppose that the segment with endpoints a, b is contained in U, and that D+f(x) M for all points x in this segment. Prove that llf(b) -f(a)ll 5 Mllb - all (see the proof of 8.5.1).) (c) Take U = E =R3 and F =R3,and letfbe defined as follows:

if (fl, &) # (0,O); and f(0,O) = 0. Show that the greatest lower bound of D-fon U is strictly positive, but that there is no neighborhood of 0 in U on whichfis injective. (d) For the remainder of this problem, suppose that there exist two numbers m, M with 0 < m < M < co, such that m 5 D-f(x) 5 M for all x E U. Suppose also that for each x E U there exists an open neighborhood V of x*in U such that f l V is a homeomorphism of V onto an open set in F ; thenf(U) is open in F. Let a be a point of U, and for each line D c F containing f ( a ) let ID be the connected component of the point f(a) in the open set D n f ( U ) in D. The union S, of the sets I D is the largest star-shaped open set with respect to f ( a ) contained in f(U). For each line D in F passing through f ( u ) , tnere exists a unique continuous mapping go : ID--f U such that g o ( f ( a ) )= a and f ( g D ( y ) )= y for all y E ID (the proof is the same as in Section 10.2, Problem 6(c).) If y, y’ are points of ID, we have I ~ ~ D ( Y ‘ ) - S D ( Y ) ~S~ m--’Ily’ - Y I I . Deduce that as y tends to an endpoint of ID (when there is one), g&) has a limit belonging to Fr(U). (e) Let y : J + U be a path in U with origin a and extremity b. Show that, if f(y(J)) C S., then we have b =gD(j(b)), where D is the line throughf(a) and f(b), and II f ( b )-f(a)ll 2 mll b - all. (f) Deduce from (d) and (e) that if U = E then S, = F, and f is a homeomorphism of E onto F. (9) Let k = M/m. Let a, b be two points of U such that the set Ek7a,b of points z E E such that I/z- all llz - bl/ 5 k/l b - all is contained in U. If L is the closed segment with endpoints a, b, show that f(L) c S,, and hence that

+

+

IlfW -f(a)ll 2 mll b - all.

+

(Proof by contradiction: consider the least I E [0, I ] such that y = f ( u t(b - a)) $ S., If D is the line throughf(u) and y, there is an endpoint u of ID belonging to the open segment with endpoints !(a) and y . When f ‘ < t tends to t, there is a point u‘ of the open segment with endpoints f(a) and y‘ = f ( a r’(b - a)) which tends to u. Let D’ be the line through f ( a ) and y’, and let z’ = go&’). Using (e), show that z’ E Ek,(I,b , and obtain a contradiction by making r’ tend to I and using (d).) (h) Suppose that E and F are Hilbert spaces and that U is the ball llxll < 1. Deduce (kZ- 1)’l2), the restriction off to B is a from (g) that if B is the ball llxll < 1/(1 homeomorphism onto f(B),and that Ilf(x’) -f(x)il 2 m ljx’ - XI( for any two points x, x’ in B. ( 9 With the hypotheses of (h), suppose also t h at4 < ((1 +2/5)/2)1/2. Show that f is then injective on U and, more precisely, that for any two points x, x’ in U we have Ilf(x’) -f(x)II 2 p Ilx’- x/I, where - M(M2 - m2)1/2 p = m + (Mz - m2)1/2 ‘

+

+

8 TOPOLOGICAL GROUPS

33

8. TOPOLOGICAL GROUPS

If G is a group, in which the law of composition is written (for example) multiplicatively, a topology on G is said to be compatible with the group structure if the two mappings ( x , y ) H x y of G x G (endowed with the product topology) into G, and X H X - ' of G into G, are continuous. A group endowed with a topology compatible with its group structure is called a topological group. We leave it to the reader to transcribe this definition (and all the results which follow) into additive notation. An isomorphism of a topological group G onto a topological group G' is by definition an isomorphism of the group G onto the group G' which is bicontinuous. If G' = G, we say automorphism instead of isomorphism. If G is any group, the law of composition ( x , y ) ~ y defines x a group structure on the set G (and this structure is different from the given one unless G is commutative). The group so defined is denoted by Go and is called the opposite of G. Any topology which is compatible with the group structure of G is also compatible with that of Go, and hence makes G o into a topological group. The mapping XHX-' is then an isomorphism of G onto Go.

Examples (12.8.1) The discrete topology and the chaotic topology (12.1.1) are compatible with the structure of any group. The topology of a normed space (in particular R or C) is compatible with its additive group structure. On the additive group Q of rational numbers, the topology defined by the p-adic distance d (3.2.6) is compatible with the group structure, because by virtue of the definition of this distance and (3.2.6.4) we have

4x0

+ yo x + Y ) s MaxMxo 4, dcvo ,v)) 5

Y

and

d(-xo, -x)

= d ( x o ,x ) .

If E is a (real or complex) Banach space and GL(E) the set of linear homeomorphisms of E onto E, then the topology induced on GL(E) by that of Y(E; E) is compatible with the group structure ((5.7.5) and (8.3.2)).

34

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

In particular, the topology induced on the multiplicative group R* (resp. C*)of real (resp. complex) numbers # O by the topology of R (resp. C) makes R* (resp. C*)into a topological group. If G is a topological group (in multiplicative notation), the mappings

(x, y ) H xy-' and ( x , y ) H x - ' y are continuous mappings of G x G into G (3.11.5). For each a E G , the left and right translations x H ax and x H xa

are homeomorphisms of G onto G, because they are bijective and continuous

((3.20.14) and (12.5))and so are the inverse mappings x H a - ' x a n d x H x a - ' .

For any a, b in G, the mapping x H a x b (and in particular the inner automorphism x ~ a x a - ' is ) therefore a homeomorphism of G onto G (3.11.5). Since the mapping X H X - ' is bijective and equal to its inverse, it also is a homeomorphism of G onto G. (12.8.2) Let G be a topological group.

(i) For each open (resp. cloJed) subset A of' G, and each x E G, the sets x A , Ax and A - ' (the set of ally-', where y E A) are open (resp. closed) in G. (ii) For each open set A in G and each subset B of G , the sets AB (the set of all products yz, where y E A and z E B) and BA are open in G. The assertions in (i) follow immediately from the preceding remarks, and (ii) follows from (i), since AB = A z is a union of open sets.

u

ZEB

(12.8.2.1) On the other hand, if A and B are closed in G, it does not necessarily follow that AB is closed (cf. (12.10.5)). For example, if 9 is an irrational number, consider the subgroups Z and OZ of R . Then the subgroup Z 8Z is not closed in R. To see why, observe that this subgroup is denumerable, and therefore is not the whole of R (2.2.17). Hence it is enough to prove the following result:

+

The only closed subgroups of R are R itseyand the subgroups of the form crZ, where cr E R. (12.8.2.2)

Granted this, we cannot have both 1 = nz and 9 = ma with m and n integers, because 8 is irrational, and the assertion of (12.8.2.1) then follows. To prove (12.8.2.2), we shall first show that a subgroup H of R is either discrete or dense in R. If H is not discrete, then for each E > 0 there exists x # 0 in the set H n [ - E , S E ] .Since the integer multiples n x ( n E Z) belong to H, every interval of length greater than E in R contains one of these points and therefore H is dense in R.

8 TOPOLOGICAL GROUPS

35

To complete the proof of (12.8.2.2), it remains to be shown that if H is discrete it is of the form a Z . We may assume that H # ( 0 ) . Since H = -H, the intersection H n 10, +a[is not empty. If b > 0 belongs to H, the intersection H n [O, b ] is compact and discrete and therefore$nire (3.16.3). Let a be the smallest element > O in this set, and for any X E H let m = [ x / a ] be the integral part of x / a . Then x - ma E H, and 0 5 x - ma < a. This implies that x - ma = 0 and so we have H = a Z . (12.8.3) Let G be a topological group. (i) Let a E G . r f V runs through a fundamental system of neighborhoods of the neutral element e of G , the sets a V (resp. V a )form a fundamental system of neighborhoods of a. (ii) For each neighborhood U of e, there exists a neighborhood V of e such that VV-' c U. (iii) For each neighborhood U of e and each a E G , there exists a neighborhood W of e such that aWa-' c U. (iv) G is Hausdorflifand only if the set { e } is closed in G . The assertion (i) follows from the fact that translations are homeomorphisms; (ii) expresses that the mapping (x, y ) ~ x y - 'is continuous at ( e , e ) , having regard to the definition of open sets in G x G (12.5); (iii) expresses the continuity of the mapping X H axa- at the point e . As to (iv), it is clear that if G is Hausdorff the set { e } is closed (12.3.4). Conversely, if { e } is closed and if x , y are distinct points of G, then there exists a neighborhood V of e such that e # x-'yV, i.e., such that x # y V . If W is a neighborhood of e such that WW-' c V (which is possible by (ii)), then we have XW n yW = 0, because the relation xw' = yw" with w' and W" in W would imply x = yW"W'-1

Eyww-'

cyv

Hence G is Hausdorff. A neighborhood V of e is said to be symmetric if V-' = V. The symmetric neighborhoods of e form a fundamental system of neighborhoods of e in G, because if U is any neighborhood of e, then so is U-' (12.8.2) and therefore U n U-' is a symmetric neighborhood of e contained in U. For each integer n > 0 and each subset V of G we define V" inductively by the rule V" = V"-' V = V V"-'. (The set V" is not the set of X" with x E V.) If V is a neighborhood of e , then so is V" 3 V for all n 2 I , and it follows from (12.8.3(ii)) that if U is any neighborhood of e and n is an integer > 1, there exists a symmetric neighborhood V of e such that V" c U.

36

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

(12.8.4) For a homomorphism f of a topological group G into a topological group G' to be continuous, it is necessary and suficient that f should be continuous at one point.

For iff is continuous at a E G and if V' is a neighborhood of f ( a ) , then f -'(V') = V is a neighborhood of a. For each x E G, we have

f(xa-'v) = f ( x > ( f ( a ) > - ' f ( Vc)f ( x > ( f ( a ) > - ' v ' , which establishes the continuity off at the point x , by virtue of (12.8.3(i)). If H is a subgroup of a topological group G, the induced topology on H is clearly compatible with the group structure of H. Whenever we consider H as a topological group, it is to be understood in this sense unless the contrary is explicitly stated. (12.8.5) The closure R of a subgroup (resp. normal subgroup) H qfa topological group G is a subgroup (resp. normal subgroup) of G. If G is Hausdorff and H is commutative, then R is commututive.

'

The image of R x R = H x H under the continuous mapping (x,y ) Hxy of G x G into G is contained in R, because the image of H x H under this mapping is contained in H (3.11.4). Hence R is a subgroup. Likewise, if H is normal and a E G, the image of H under the mapping x H a x a - ' is contained in H, hence the image of R is contained in R, and therefore R is normal. Finally, if G is Hausdorff and H is commutative, the continuous functions x y and y x are equal on H x H and therefore also on R x R by virtue of the principle of extension of identities ((3.15.2), 12.3, and 12.5). (12.8.6) ( i ) In a topological group G, the normalizer N ( H ) of a closed subgroup H (i.e., the set of all x E G such that xHx-' c H) is a closed subgroup. (ii) In a Hausdorff topological group G , the centralizer d ( M ) of any subset M ofG (i.e., the set of all x E G which commute with every element of M ) is a closed subgroup of G. In particular, the center of G i s closed.

For each z E H, the set of elements x E G such that xzx-' E H is the inverse image of H under the continuous mapping X H X Z X - I , hence is a closed set (3.11.4). Since N ( H ) is the intersection of these sets as z runs through H , it follows that N(H)is closed (3.8.2). Again, if G is Hausdorff then for each z E M the set of x E G such that zx = xz is closed (12.3.5) and hence so is S ( M ) since it is the intersection of these sets. (12.8.7) (i) In a topological group G, every Iocally closed subgroup is closed. Every subgroup which has an interior point is both open and closed. (ii) In a HausdorfSgroup, every discrete subgroup is closed.

8

TOPOLOGICAL GROUPS

37

(i) Let H be a locally closed subgroup of G. Then R is a subgroup of G, and H is an open subgroup of R (12.2.3). Hence it is enough to prove the second assertion. Now if H has an interior point, then by translation it follows that every point of H is interior, and therefore H is open. Hence the left cosets x H are also open sets, and therefore CH is open in G (because it is a union of cosets xH). Consequently H is closed in G. (ii) If G is Hausdorff and H is a discrete subgroup of G, then there exists a symmetric open neighborhood V of e such that V n H = { e } . If X E R ,we have xV n H # fa. Now if y ~ x V n H, then x ~ y V and the set { y } is closed in the open set yV, because G is Hausdorff. Since yV n H = { y } (because y E H), we must have x = y , and therefore R = H. (12.8.8) If G is a connected group and V is a symmetric neighborhood of e, then G is equal to the union V" of the sets V"for n > 0.

The set V" is clearly symmetric, and since V"V" = Vm+"it follows that V"V" c V". Hence V" is a subgroup of G. Since e is an interior point of V", this subgroup is both open and closed (12.8.7) and therefore is the whole of G . (12.8.9) In a topological group G , the connected component K of the neutral element (3.19) is a closed normal subgroup (called the neutral component or identity component of G). For each x E G , the connected component of x in G is XK = Kx.

If a E K, the set a - ' K is connected and contains e, hence K-'K c K. This shows that K is a subgroup of G. It is invariant under all automorphisms of the topological group G, in particular under all inner automorphisms, hence K is normal in G. Also K is closed in G (3.19). Finally, the last assertion follows from the fact that left and right translations are homeomorphisms of G onto G. (12.8.10) If G,, G, are two topological groups, then the product topology on the product group G = G, x G, is compatible with the group structure of G. For, identifying canonically the product spaces G x G and (GI x G,) x (G, x G,) (12.5), the mapping

((x19 x2)9 ( Y l , Y 2 ) ) +-+(XlY13 x2 Y2) is continuous ((3.20.15) and (12.5)), and the mapping (xl, x,)H(x;~, x;') is continuous for the same reason. The group G, x G , , endowed with the product topology, is called the product of the topological groups G , and G , . If G is a commutative topological group, the mapping (x, y ) ~ x is y a continuous homomorphism of G x G into G.

38

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

PROBLEMS

1. Let G be a group and let 'x1 be a set of subsets of G , satisfying conditions (V,) and (V,,) of Section 12.3, Problem 3, together with the following:

(GV,) For all U E % there exists V E 3 such that V . V c U. (GV,,) For all U E ti we have U - ' E %, (GV,,,) For all U E % and a E G, we have aUa-I E ti. Show that there exists a unique topology on G , compatible with the group structure of G, for which % is the set of neighborhoods of the neutral element e . 2.

Every topology compatible with the group structure of a finite group G is obtained by taking as neighborhoods of the neutral element the sets containing a normal subgroup H of G. Give an example of a non-Hausdorff topological group in which the center is not closed and has as its closure a noncommutative subgroup.

3.

Let G be a connected topological group. Show that every totally disconnected normal subgroup D of G is contained in the center of G. (If a E D, consider the mapping XHXUX-' of G into D.)

4.

Show that the commutator subgroup of a connected topological group is connected (use (3.19.3)and (3.19.7)).

5.

Let G be a topological group and let H , K be subgroups of G such that H 3 K 2 H', where H' is the commutator subgroup of H. Show that R contains the commutator subgroup of R . Deduce that if G is Hausdorff, the closure in G of a solvable subgroup is solvable (induction on the length of the derived series).

6. Let G be a topological group and let H be a closed normal subgroup of G , containing the commutator subgroup of G. If the identity component K of H is solvable, show that the identity component L of G is solvable. (Show by using Problem 4 that K contains the commutator subgroup of L.)

T be the canonical homomorphism and let 6 be an irrational number. O n the topological space G = RZ x T2,a group law is defined by

I. Let cp : R

--f

In this way G becomes a locally compact group (even a Lie group). Show that the commutator subgroup of G is not closed in G. 8.

Let (G&cl be any family of topological groups. Show that the product topology on G = G , is compatible with the product group structure (Section 12.5, Problem 4).

n

.El

The topological group so defined is called the product of the topological groups G, . Let H be the normal subgroup of G cocsisting of all x = (x,) such that for all but a finite number of indices x. is the identity element of C , . Show that H is dense in G .

9

METRIZABLE GROUPS

39

9. METRIZABLE G R O U P S

If G is a group, a function f on G x G is said to be left (resp. right) invariant if f ( x y , x z ) =f ( y , z ) (resp. f ( y x , zx) =f ( y , z ) ) for all x, y , z in G. When G is commutative, these two conditions coincide, andfis then said to be translation-invariant. A distance d o n G is left (resp. right) invariant if and only if the left (resp. right) translations are isometries with respect to d. Iff is a left-invariant function on G x G, then the function (x,y ) ~ f ( x - ly,- ' ) is right-invariant, and vice versa. For example, if E is a normed space, the distance / / x- y/1 on E is translation-invariant. (12.9.1 ) In order that the topology of a topological group G should be metrizable (in which case G is said to be a metrizable group) it is necessary and suficient that there should exist a denumerable fundamental system of neighborhoods of the neutral element e, whose intersection consists of e alone. When this condition is satisfied, the topology of G can be defined by a leftinvariant distance, or by a right-invariant distance.

It is enough to show that if there exists a denumerable fundamental system of neighborhoods (U,) of e in G such that U, = { e } ,then the topology of G n

can be defined by a left-invariant distance. We define inductively a sequence (V,) of symmetric neighborhoods of e in G such that V, c U, and

VL, v,

n

u,

for all n 2 1 (12.8.3), so that (V,) is also a fundamental system of neighborhoods of e. Now define a real-valued function g on G x G as follows: g(x, x ) = 0 ; if x # y , then either x - ' y 4 V,, in which case we takeg(x, y ) = 1; or else there exists a greatest integer k such that x - ' y E Vk (because x - ' y # e cannot belong to all the V,), in which case we define g(x, y> = 2-k. It is clear from this definition that g ( x , y ) = g(y, x), that g(x, y ) 2 0, and that g(zx, z y ) = g(x, y ) for all x , y , z in G. Now let (12.9.1 . I )

where the infimum is taken over the set of all finite sequences (zo , z l , . . . , z,,) (with p arbitrary) such that z,, = x and z p = y . We shall show that d is a leftinvariant distance on G and satisfies the inequalities (1 2.9.1.2)

M X ?Y)

5 4x9 Y ) I g(x, Y).

40

XI1 TOPOLOGY A N D TOPOLOGICAL ALGEBRA

From the definition of d, it follows immediately that d is left-invariant (because g is), satisfies the triangle inequaiity, is symmetric and positive. Moreover, the right-hand inequality in (12.9.1.2) is obvious, and shows that d(x, x ) = 0 for all x E G. Hence d is a pseudo-distance on G. T o prove the lefthand inequality of (12.9.1.2) we shall show by induction on p that, for each finite sequence ( z i ) o s i s po f p 1 points in G such that zo = x and z p = y , we have

+

(12.9.1.3)

P- 1

1 g(zi

i=O

7

This inequality is obvious if p

zi+1) 2 +g(x, Y ) . P- 1

1. Let us write a =

=

1g(zi, ziti). The

i=O

inequality (12.9.1.3) is true if a 2 +, because g(x, y ) 1, If a = 0, then zi = zi+l for 0 5 i S p - 1, hence x = y and so (12.9.1.3) is trivially satisfied. So suppose that 0 < a < and let h be the greatest index such that g ( z i , zi+l>5 +a. Then we have g ( z i , z ~ +>~+a, ) so that

1

+,

i O such that 2 - k 5 a. Then k 2 2, and the elements X-'zh, Z ; ' Z ~ + ~ ,Z;>~Y are all in Vk, by virtue of the definition ofg. Hence x - ' y E V l t Vk-l, which implies that g(x, y ) 5 2l - k 5 2a, and proves (12.9.1.3). Hence (12.9.1.2) is established, and shows first of all that d is a distance on G. Also, if r > 0, the ball B'(e; r ) (with respect to d ) contains vk for all indices such that 2 - k < r, and conversely each v k contains the ball B'(e; 2-k-1). Since d is left-invariant, it now follows from (12.8.3(i)) and (12.2.1) that d defines the topology of G. In general it is not possible to define the topology of G by means of a distance which is both left- and right-invariant (Problem 1 and Section 14.3, Problem 11). (12.9.2) Let G be a metrizable group. A necessary and su8cient condition for a sequence (x,) in G to be a Cauchy sequence with respect to a left-invariant distance defining the topology of G is that, for each neighborhood V of e, there exists an integer no such that x i ' x , E V for all m 2 no and n 2 n o .

For if d is a left-invariant distance, we have d(x,, x,) = d(e, X ,

'Xrn),

and the proposition follows from the fact that d defines the topology of G.

9 METRIZABLE GROUPS

41

Hence the property of being a Cauchy sequence with respect to some leftinvariant distance is independent of the choice of distance, and depends only on the topology of G. We shall say simply that such a sequence is a left Cauchy sequence in G. A right Cauchy sequence in G is defined analogously by replacing x; ' x , in (12.9.2) by x, x i It can happen that a left Cauchy sequence is not a right Cauchy sequence (Problem 8); but if (x,) is a left Cauchy sequence, then (x,-') is a right Cauchy sequence. Since the mapping X I - + X - ~ is continuous it follows that, if every left Cauchy sequence in G converges, then so does every right Cauchy sequence. In this case G is said to be a complete metrizable group: it is complete with respect to every leftinvariant distance and every right-invariant distance defining the topology of G. (12.9.3) Let G, G' be two metrizable groups. Then every continuous homom0rphism.f of G into G' is unformly continuous with respect to left- (resp. right-) invariant distances on G and G'. Let d, d' be two such distances on G, G , respectively. By hypothesis, for each E > 0 there exists 6 > 0 such that the relation d(e, z ) 5 6 implies d'(e', f ( z ) )S E . Hence if d(x, y ) = d(e, x - ' y ) 5 6 , then d'(f(x), f ( y ) ) S E , because d ' ( f ( X ) , f ( Y ) ) = d'(e'9 ( f ( x ) ) - ' f ( y ) )= d'(e',f(x-'y))

since f is a homomorphism. (12.9.4) Let G,, G, be two metrizable groups such that G, is complete, and let H, (resp. H,) be a dense subgroup of G , (resp. G,). Then every continuous homomorphism u : H1 -+ H, can be extended uniquely to a continuous homomorphism ii : G , -+ G, . If G, is also complete and if u is an isomorphism (of topological groups) of H, onto H, , then ii is an isomorphism (of topological groups) q f G , onto G, . There exist left-invariant distances on G, and G, , and the existence of the continuous extension ii then follows from (12.9.3) and (3.15.6).The fact that ii is a group homomorphism follows from the principle of extension of identities applied to the two functions ( x , y ) ii(xy) ~ and (x,y)wE(x)U(y)on G, x G, (having regard to (3.20.3)). Finally, if G, is complete and if v : H, -+ H, is the inverse of the isomorphism u, then v can be extended to a continuous homomorphism fi of G, into G,. Since V 0 U and E fi agree with the identity mappings on HI and H, respectively, they are the identity mappings on G , and G, respectively, by virtue of the principle of extension of identitiesand (3.11.5). This completes the proof. 0

42

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

It should be carefully noted that, if we suppose only that u is injective (resp. surjective), then 6 is not necessarily injective (resp. surjective) (Problem 9). (12.9.5) Let G be a metrizable group in which there exists a neighborhood V of e which is complete (with respect to some left- or right-invariant distance). Then G is complete. In particular, a locally compact metrizable group is complete.

Let d be a left-invariant distance on G and let (x,,) be a left Cauchy sequence in G. Let E > 0 be such that theclosed ball B’(e; E ) is contained inV. By hypothesis there exists an integer no such that d(x,,, x,) 5 E for all m 2 no and n 2 n o , hence the sequence (x,,),~,,,, is contained in the closed ball B’(x,,,; E ) . But this closed ball is a complete subspace, because it is obtained by left translation from B’(e; E ) which is closed in V (3.14.5). Hence the seconverges in G. The last assertion follows from (3.16.1). quence (x,,),,~~ (12.9.6) In a Hausdorff topological group G , every locally compact metrizable subgroup H is closed.

Let x E R, let V be a neighborhood of e in G such that V n H is compact, and let W be a symmetric neighborhood of e in G such that W2 c V. Then XW n H is nonempty and relatively compact in H, because if yo E XW n H then for each y E xW n H we have y o ‘ y E Wz n H c V n H and therefore y E yo(V n H), which is a compact set. It follows (12.3.6) that the closure of XW n H in G is contained in H. and therefore x E H. (12.9.7)

Let G be a Hausdorff commutative topological group, written additively. All the material on series in Section 5.2 which involves only the topology of C remains valid without any change. The same is true of Cauchy’s criterion (5.2.1) if C is metrizable, by replacing the norm llxll by d(0, x), where d is an invariant distance on G. PROBLEMS

1. Let G = GL(2, R) be the multiplicative group of all real 2 x 2 nonsingular square

matrices. For each integer

n

> 0, let V. be the set of matrices X =

(z

t )

E

G such

that jx - I I 6 l/n, ly] 5 l / n , IzI 5 l/n, and jt - 11 5 I/n. Show that the family of sets V, is a fundamental system of neighborhoods of the neutral element I of G for a topology .Y compatible with the group structure of G (cf. Section 12.8, Problem I). The group G is locally compact in the topology .T, but 9-cannot be defined by a

9

METRIZABLE GROUPS

43

distance which is both left and right-invariant (cf. Section 14.3, Problem ll).) (Use the fact that if the topology of a metrizable group G can be defined by a distance which is both left and right-invariant, then for every neighborhood V of the neutral element e of G there exists a neighborhood W c V of e such that xWx-' c V for all X E G.) 2.

Let S be a compact subset of a metrizable group G, such that xy E S whenever x E S and y E S . Show that for each x E S we have xS = S. (Consider a cluster value y of the sequence (x")., in S, and show that yS is the intersection of the sets Y S; deduce that yxS = y S . ) Deduce that S is a subgroup of G.

3

Let G be a locally compact and totally disconnected metrizable group. (a) Show that every neighborhood of e in G contains an open compact subgroup of G. (Every neighborhood V of e contains a neighborhood U of e which is both open and closed (Section 3.19, Problem 9). If B = CU, show that there exists a symmetric open neighborhood W of e in G such that W c U and UW n BW = 125, and deduce that the subgroup generated by W is contained in U.) (b) Suppose that the topology of G can be defined by a left- and right-invariant distance. Show that every neighborhood of e in G contains a compact open normal subgroup of G (remark that every neighborhood V of e contains a symmetric open neighborhood W such that XWX-'c V for all x E G).

4.

Let p be a prime number. Consider the family of finite groups Z/p"Z (n 2 I), each with the discrete topology, and their product G (Section 12.8, Problem 8) which is compact and totally disconnected. For each n, let y.: Z/p"Z+Z/p"-'Z be the canonical homomorphism. = z. for all n is a closed (a) Show that the set of all z = (z.) E G such that q&.) (hence compact) subgroup Z, of G. If a+bnis the restriction to Z, of the projection pr, : G + Z/p"Z, then is a surjective homomorphism. (b) Foreachz=(z.)EZ,,put lz(,=Oifz=O,and ~ ~ ~ , = p ' - ~ i f m i s t h e s m a l l e s t integer such that z, # 0. Show that 1z - 2'1, is a translation-invariant distance which defines the topology of Z, . (c) For each n 2 1, let f n : Z + Z/pnZ be the canonical homomorphism. Show that the homomorphismf: Z H( ~ , ( Z) ) , , ~of Z into Z, is injective and that its imagef(Z) is a dense subgroup of Z,. If we put d(z, z') = I f ( z ) -f(z')l,, show that d is the p-adic distance on Z (3.2.6).The elements of Z, are called p-adic integers.

5.

Let p be a prime number, and let G be the compact group which is the product of an infinite sequence (G,),,, of groups G. all equal to T =R/Z. For each n, let vnbe the homomorphism XHPX of G . into G.-'. The compact subgroup of G consisting of all z = (2.) such that y,(z.) = zn-' for all n is called the p-adic solenoid and is denoted by Tp. (a) For each n , let fn:T, -+ G , = T be the restriction of pr. to T, . Show that f. is a surjective homomorphism with kernel isomorphic to the group Z, (Problem 4). (b) Let y : R -+T be the canonical homomorphism. For each x E R, show that the point 8(x) = (cp(x/pn)).,, belongs to T,. Prove that 6 is an injective continuous homomorphism of R into T,, and that 8(R) is a dense subgroup of T, . Deduce that T, is connected. (c) Let I be an open interval in R, with centre 0 and length < 1. Show that the subspace f 0, and hence get a contradiction.) 7.

Let G be a locally compact metrizable group with no small subgroups, and let V be a compact symmetric neighborhood of e containing no subgroup of G other than {el, and such that for all x , y E V the relation x 2 = y2 implies x = y (Problem 6(a)). (a) Show that if (an)is any sequence of points of V with e as limit, there exists a subsequence (b.) of (a,) and a sequence (k.) of integers > O such that the sequence (b:")converges to a point #e. (Consider the smallest of the integers k such that .:+I 6 V.) (b) Show that if r, s are two real numbers such that the sequences ( b p ' ) , (b:'"') converge to x, y, respectively, in G, then the sequence (b!,(""""') converges to xy. (Here [t] denotes the integral part of the real number t . ) (c) Using (a), (b), and the uniqueness of the square root, show that for every dyadic number r E [0, I ] the sequence (by'"') converges in G. (d) Let W c V be a neighborhood of e in G. Show that, for every real number r E [0, I], there exists a dyadic number s such that b[,'+s)knlE W for all n (use Problem 6(b)). Deduce that, for each r E [0, 11, the sequence (by'"') has a limit X(r). (e) If -1 2 r $0,put X ( r ) = ( X ( - r ) ) - I . Show that, if r, s and r s are all in the interval [- 1, 11, we have X ( r ) X ( s ) = X(r s) and that the mapping r t+X(r) of [-1, 11 into G is continuous. Deduce that this mapping can be extended to a nonconstant continuous homomorphism of R into G.

+

8.

+

Let I be the compact interval [0,1] in R,and let G be thegroup of all homeomorphisms of I onto I, which is contained in the Banach space '#,(I) (7.2.1). Show that the

10 SPACES WITH OPERATORS. ORBIT SPACES

45

(metrizable) topology induced on G by that of VR(I) is compatible with the group structure of G, and that right Cauchy sequences in G are the same as Cauchy sequences (7.1). Give an example of a right Cauchy (in G) with respect to the norm on ‘if& sequence in G which is not a left Cauchy sequence. 9.

(a) Let G’ be a complete metrizable group and let G o # G‘ be a dense subgroup of G’. Let G be the topplogical group obtained by giving Go the discrete topology. The identity mapping G + G o is a continuous bijective homomorphism, but its extension by continuity to a homomorphism G --t G’ (12.9.4) is not surjective. (b) Let G be the dense subgroup QZ of RZ,let 0 be an irrational number, and let u be the continuous homomorphism ( x , ~ ) H X 0y of G into R. Let G’ = u(G).The mapping u, considered as a homomorphism of G into G’, is bijective, but its continuous extension to a homomorphism of RZ into R is not injective. (c) Use (a) and (b) to construct an example of two complete groups G I , G2 and a continuous bijective homomorphism u of a dense subgroup HI of G I onto a dense subgroup Hz of Gz , such that the continuous extension of u to a homomorphism of G I into GZis neither injective nor surjective.

+

10. Let f be a continuous homomorphism of a subgroup H # {O} of R into a locally compact metrizable group G. Show that i f f is not an isomorphism of H onto the

subgroup f(H) of G, then f(H) is relatively compact in G. (Reduce to the case where H is closed in R andf(H) is dense in G . Begin by showing that, for each neighborhood W of the neutral element e in G, the set f-’(W) is unbounded, and deduce that f(H n R,) is dense in G. If V is a compact symmetric neighborhood of e in G, show that there exist a finite number of elements ti > 0 in H such that the neighborhoods f ( f l ) Vcover V. For each x E G, let A, be the set of all t E H such that f ( t ) E xV. Show that, if t E A,, there exists a ti such that t - ti E A,, and deduce that, if I is the largest compact interval with origin 0 in R containing all the t l , then I n A, is not empty. Deduce that G Cf(1) . V is compact.)

11. Let G be a commutative metrizable topological group, and let d be an invariant

distance defining the topology of G. Let h be the Hausdorff distance on %(G)corresponding to d (Section 3.16, Problem 3). Show that if M, N, P, Q are four bounded closed subsets of G, then h(MP, NQ) h ( M , N) h(P, Q).

+

10. SPACES WITH OPERATORS. ORBIT SPACES

Let G be a group and E a set. An action (or left action) of G on E is a mapping (s, x ) H S . x of G x E into E satisfying the following conditions:

-

(1) If e is the neutral element of G, then e x = x for all x E E. (2) For all s, t in G we have s ( t * x ) = ( s t ) x for all x E E.

-

- -

These conditions imply that s-’ (s x ) = x for all s E G and all x E E; hence for each s E G the mapping x ~ x sis a bijection of E onto E, and the inverse bijection is XHS-’ . x. For each x E E, the set G x of elements s . x where s E G is called the orbit of x (for the given action of G on E). The set S , of elements s E G such

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

46

that s * x = x is a subgroup of G called the stabilizer of x. The relation s * x = t * x is equivalent to t - ' s E S,. The mapping SHS x of G onto G x factorizes as follows: (12.10.1)

G A G/S,

3G

x

where G/S, is the set of left cosets sS, of S, in G ; p is the canonical mapping SHSS,; and cp is the bijection s S , w s x. The group G is said to act faithfitlly on E if the intersection of the stabilizers S,, as x runs through E, consists only of e. The group G is said to act freely on E if the stabilizer of every x e E consists only of e, or equivalently if for every x E E the relation s * x = t ' x implies s = t . The relation " y belongs to the orbit of x" is an equivalence relation on E, for which the equivalence classes are the orbits of the elements of E. The set of orbits is denoted by E/G (it is a subset of Cp(E)). If this set consists of a single element (in other words if, given any two elements x , y of E, there exists s E G such that y = s . x), then G is said to act transitively on E. The union G * A of the orbits of the elements of a subset A of E is called the saturation of A with respect to G ; the restriction to G x (G. A) of the mapping (s, X)HS * x is an action of G on G A. If : E -+ E/G is the canonical mapping (so that n(x) = G * x for all x E E), then G * A is equal to x-'(x(A)). The relation G A = A is equivalent to G ' A c A.

-

Now suppose that G is a topological group and E a topological space. Then G is said to act continuously on E if the mapping (s, X)HS * x of the product space G x E into E is continuous.

Examples (12.10.2) Let H be a subgroup of a topological group G . Then each of the actions (s, x)++sx and (s, X)HSXS-' of H on G is continuous. For the former action, the stabilizer of each x E G is the identity subgroup { e } , and the orbit of x is the right coset Hx; for the latter action, the stabilizer of x is the intersection H A b(x), where b(x) is the centralizer of x in G (12.8.6), and the orbit of x is the set of its conjugates hxh-' by elements h E H. If H, K are subgroups of G, the product group H x K acts continuously on G by ((s, t ) , x)Hsx~-'. The orbit of x is the double coset H x K of x with respect to H and K . If E is a real (resp. complex) topological vector space (12.13), the group R* (resp. C*) acts continuously on E by (A, x ) ~ A . xThe . orbits are {0} and the sets D - {0}, where D is a line (i.e. a one-dimensional subspace) in E.

10 SPACES WITH OPERATORS. ORBIT SPACES

47

If E is a Banach space, the group GL(E) of linear homeomorphisms of E onto E (12.8.1) acts continuously on E by (u, X ) H u(x) (5.7.4). Let G be a topological group and E a topological space. The mapping (s, X)HX of G x E into E is a continuous action of G on E, called the triuial action. Let G be a topological group acting continuously on a topological space E. If p is a continuous homomorphism of a topological group G' into G, then G' acts continuously on E by (s', x ) ~ p ( s '* )x. (12.10.3) r f G is a topological group acting continuously on a topological space E, then for each s E G the mapping x H s * x is a homeomorphism of E onto E.

For it is a continuous bijection, and the inverse bijection XHS-' also continuous.

x is

(12.10.4) If G is a topological group acting continuously on a Hausdorf topological space E, the stabilizer of each point of E is a closed subgroup of G . This follows immediately from (12.3.5). (12.10.5) Let G be a metrizable group acting continuously on a metrizable space E. Let A be a compact subset of G, and B a closed (resp. compact) subset of E. Then A * B is closed (resp. compact) in E.

The second assertion follows from the fact that A B is the image of the compact set A x B (3.20.16(v)) by the continuous mapping (8, X)HS * x (3.17.9). As to the first assertion, consider a sequence (s, * x,) of points of A * B (where s, E A and x, E B) with a limit z E E. By hypothesis, the sequence (s,,) has a subsequence (s,,,) converging to some point a E A . Since x,, = s";' (s,, * x,,,), the sequence (x,,,)converges to a-' * z. But B is closed in E and therefore a-' * z E B, hence z = a * (a-' * z ) E A * B. By (3.13.13), the proof is complete. Let G be a topological group acting continuously on a topological space E. Let E/G be the set of orbits and let TC : E + E/G be the canonical mapping, so that n(x) is the orbit G * x for each x E E. Let D be the set of subsets U of E/G such that n-'(U) is open in E. It follows immediately from the formulas (1.8.5) and (1.8.6) that r) is a topology on E/G. The set E/G, endowed with

48

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

this topology, is called the orbit space of the action of G on E. The mapping UH x-'(U) is a bijection of the set of open sets of E/G onto the set of saturated open sets of E. A subset F of E/G is closed in E/G if and only if n-'(F) is closed in E, because n-'((E/G) - F) = E - n-'(F).

(12.10.6) (i) The canonical mapping x : E -+ E/G is continuous. (ii) The image under n of every open set in E is open in E/G. (iii) A mapping f of E/G inzo a topological space E' is continuous if and only i f f n : E -,E' is continuous. 0

The first assertion follows from the definition of the topology of E/G and from (3.11.4(b)).To prove the second, it is enough to show that if V is open in E then its saturation G * V = n-'(n(V)) is open in E; and this is clear because G V = s * V, and each s V is open in E by virtue of (12.10.3). Finally, if

u

SEG

f:E/G + E' is continuous, then so is f

0 x by (i); conversely, i f f 0 x is continuous, then for every open set U' in E' the set n-'(f -'(U')) is open in E, hence f -'(U') is open in E/G. This proves (iii).

If x is any point of E and if V runs through a fundamental system of neighborhoods of x in E, the sets x(V) form a fundamental system of neighborhoods of the point x(x) in E/G.

(12.10.7) Let A be a subset of E and let A' = G A = n-'(n(A)) be its saturation with respect to G. Then the canonical bijecfion cp of the subspace n(A) of E/G onto the orbit space A'/G is a homeomorphism. The mapping cp is defined as follows: if x E A, the image under cp of the orbit G x is the same orbit considered as an element of A'/G. As U runs through the set of open subsets of E/G, the mapping cp takes U n n(A) to the canonical image of n-'(U) n A' in A'/G. Now U n n(A) runs through the set of open subsets of the subspace x(A) of E/G, and cp(n-'(U) n A') runs through the set of open subsets of A'/G. For if V' is a saturated open subset of .the subspace A' of E, then V' is of the form V n A', where V is open in E; but also V' = (G V) n A' = x-'(n(V)) n A', and x(V) is open in E/G by (12.10.6).

(12.10.8) Let G be a topological group acting continuously on a topological space E. In order that E/G should be Hausdorff it is necessary and suflcient that, in the product space E x E , the set R of pairs (x, y ) belonging to the same orbit be closed. When this is the case, every orbit is closed in E.

10 SPACES WITH OPERATORS. ORBIT SPACES

49

Let n(x) and n(y) be distinct points of E/G. If E/G is Hausdorff, there exist saturated open sets V, W in E such that V n W = fa and x E V and y E W. It is clear that the open set V x W in E x E contains ( x , y ) and does not meet R. Hence R is closed in E x E. Conversely, if R is closed in E x E, there exists an open neighborhood S of x and an open neighborhood T of y in E such that (S x T) n R = @. By (12.10.6), n(S) and n(T) are neighborhoods of n(x) and n(y), respectively. I f they intersected, there would exist s E S and t E T belonging to the same orbit, which means that (s, t ) E R ; and this is absurd. Hence E/G is Hausdorff. The last assertion follows from (12.3.4) and the continuity of n. (12.10.9) Let E be a metrizable space, G a topological group acting continuously on E, and n : E + E/G the canonical mapping. Suppose that E/G is metrizable. Then :

(i) ifE is separable, E/G is separable. (ii) ifE is locally compact (resp.' compact), E/G is locally compact (resp. compact); (iii) ifE is locally compact and K is any compact subset of E/G, there exists a compact subset L of E such that K = n(L).

If D is a denumerable dense subset of E, then n(D) is dense in E/G (3.11.4(d)). This proves (i). If V is a compact neighborhood of x E E, then n(V) is a compact neighborhood of n(x) in E/G ((3.17.9) and (12.10.6)); hence

(ii). Finally, for each z E K , let V(z) be a compact neighborhood of a point of n-'(z) in E, so that n(V(z))is a compact neighborhood of z. There exist a finite number of points zi E K such that the n(V(z,)) cover K. Let L, be the compact set V(zi) in E. We have K c n(L,), hence the set L = L, n n-'(K)

u i

is compact (because it is closed in L, : (3.11.4) and (3.17.3)) and we have x(L) = K. (12.10.10) Let E be a locally compact, separable metrizable space and G a topological group acting continuously on E. Let n : E --+ E/G be the canonical mapping. Suppose that (1) E/G is Hausdorff, (2)for each x E E there exists a compact subset K ( x ) of E , containing x and such that the restriction of n to K ( x ) is injective and n ( K ( x ) )is a neighborhood o f n ( x ) . Then E/G is metrizable (and therefore, by (12.10.9), locally compact and separable).

We shall first show that there exists a sequence (K,) of compact subsets of E such that the restriction of n to K, is injective and the interiors of the sets n(K,,) cover E/G. By (3.18.3)there exists an increasing sequence (A,,) of compact subsets of E, whose union is E. For each z E n(A,), let x be a point of

50

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

n-’(z), and let K(x) be a set with the properties enunciated above. Then V(z) = (n(K(x)))O is an open neighborhood of z, and therefore n-’(V(z)) is an open neighborhood of n-‘(z) n A, . Since A,, is compact, there are a finite number of points zi E n(A,,) such that the sets n-’(V(z,)) (1 g i 6 p,,, say) cover A,. Let Ki,, be the set K(x) corresponding to V(zi), so that the interiors of the sets n(K,,,) form an open covering of n(A,,)in E/G. Then it is clear that the K i n( n 2 1 , 1 5 i 5 p , for each n ) satisfy the required conditions. This being so, the restriction of n to K, is a homeomorphism of K, onto the subspace n(K,,) of E/G, because E/G is Hausdorff (12.3.6), and this subspace is therefore compact and metrizable. The result therefore follows from (12.4.7). (12.10.11) Let G (resp. G‘) be a topological group acting continuously on a topological space E (resp. E’). Then G x G’ acts continuously on E x E‘ by (s,s‘) (x, x’) = (s x, s’. K‘), and the mapping w defined by

-

-

(G x G’) (x, x’) H ((G . x), (G’ . x’)) is a homeomorphism of(E x E’)/(G x G’) onto (E/G) x (E’/G’).

It is immediately checked that o is bijective, and it is continuous by virtue of (12.10.5). Moreover, every open set in (E x E’)/(G x G’) is the image, under the canonical mapping p : E x E’ + (E x E’)/(G x G’),of an open set U in E x E‘, and we have o ( p ( U ) ) = n(prl U) x n’(pr, U ) , where n : E -+E/G and 71‘ : E’ -+ E’/G’ are the canonical mappings. The set w ( p ( U ) ) is therefore open in (E/G) x (E’/G’), and the proof is complete. (12.10.12) Let G be a connected topologicalgroup acting continuously on a topological space E. If the orbit space E/G is connected, then E is connected.

Since the mapping s w s x of G onto G x is continuous, it follows

(3.19.7) that every orbit is connected. Suppose that there exist two non-empty open sets U, V in E such that U n V = @ and U u V = E. For each x E E, the sets U n (G * x) and V n (G * x) are open in G * x; their union is G * x

and their intersection is empty. Hence one of them is empty; in other words, U and V are saturated. But this implies that n(U) and n(V) are nonempty open sets in E/G whose intersection is empty and whose union is E/G, and this contradicts the hypothesis that E/G is connected. Remark (12.10.13) A right action of a group G on a set E is a mapping (s, x) Hx * s of G x Einto E such that x . e = xfor all x E E, and(x * t ) s = x . (ts)for all

10 SPACES WITH OPERATORS. ORBIT SPACES

51

x E E and all, s, t in G. Everything we have said can be immediately transposed to this situation; the set of orbits is sometimes denoted by G\E. For example, a subgroup H of a topological group G acts continuously on the right on G by the action (s,x) Hxs.

PROBLEMS

1. Let G be a locally compact metrizable group and E a locally compact metrizable space on which G acts continuously. For each pair of subsets K, L in E let P(K, L) denote the set of all s E G such that (s . K) n L # 0. (a) Show that if K is compact and L is closed in G, then the set P(K, L) is closed in G. The group G is said to act properly on E if P(K, L) IS a compact subset of G whenever K and L are compact subsets of E. (This will always be the case if G is compact.) (b) Show that, if G acts properly on E, then F . K is closed in E whenever F is closed in G and K is a compact subset of E. In particular, for each x E E, the orbit G * x is closed in E. (c) Under the same hypotheses, for each x E E the stabilizer S, of x is a compact subgroup of G, and the canonical map G/S, + G . x is a homeomorphism. (d) Under the same hypotheses, the orbit space E/G is Hausdorff. 2.

For each pair (a,t) of real numbers such that a 2 1, the pointf.(r) follows : if

t


0, where Db is the set of points (t, -1.0) with t E R , and D:(is the set of points ( t , 1, z) with t E R . The additive group R acts continuously on E as follows: (1) s . ( t , - 1,O) = (s f , - 1,O); (2) s . (f.(t), z)= (f.(s -tt ) , z ) ; (3) s . ( t , 1, z ) = ( t - s, 1, z). Show that the orbits of this action have the same properties as in Problems 2 and 3, and that the orbit space E/R is Hausdorff but not metrizable.

+

5.

Let E be a locally compact metrizable space and let G be a topological group acting continuously on E. Let w : E + E / G be the canonical map. Suppose that E/G is Hausdorff. (a) Let K be a compact subset of E and let U be an open neighborhood of K. Show that there exists a continuous mapping of E into [0,1] which takes the value 1 on w - ’ ( r ( K ) ) and the value 0 on the complement of a-‘(a(U)). (b) Deduce from (a) that there exists a continuous mapping of E/G into [0, 11, taking the value 1 on T ( K ) and the value 0 on the complement of w(U). (Show first that there exists a relatively compact open neighborhood U, of K in E such that 0, c U. Deduce that there exist two continuous mappings fi,fi of E into [0, t ] such that f i takes the value 4 on T-’(T(K)) and the value 0 on E - w-’(T(U,)),and fi takes the value 4 on T - ~ ( T ( ~ ~and ) ) the value 0 on E - T-~(T(U)). Consider the functionf, f2 Iterate this “interpolation” indefinitely and pass to the limit.) (c) If E is separable, show that there exists a sequence (U,) of relatively compact open sets in E such that the n(UJ form a basis for the topology of E/G. (d) Suppose that E is separable. Show that E/G is metrizable. (For each pair of indices m, n such that 0, c U., consider a continuous mappingf,. of E/G into [0, 11 which takes the value 1 on T(U,) and the value 0 on the complement of n(U,). Consider the continuous mapping x ~ ( f , . ( x ) ) of E/G into the product space RN N).

+ .

6. If M is a monoid with neutral element e , we define an action of M on a set E as at the beginning of (12.10). Suppose that E is a compact metric space and that, for each s E M, the mapping XHS . x is continuous. The closed orbit of x with respect to M is defined to be the set M * x ; it is stable under the action of M. Show that for each x E E there exists in M . x a minimal closed orbit Z (i.e., such that M . z = Z for all z E Z). (For each y E M . x let h(y) be the least upper bound, as t runs through M .y, of the distances from t to a closed orbit contained in M . t . Show that the greatest lower bound of the numbers h(y) with y E M .xis 0, by showing that otherwise E would not be precompact. -Deduce that there exists a sequence (y.) of points of M * x such that y , , , E M * y . and the

-

sequence (h(y.)) tends to 0. Now use the compactness of E to complete the proof.)

11. HOMOGENEOUS SPACES

Let G be a group, H a subgroup of G.Recall that the set of left cosets xH (resp. right cosets Hx)of H in G is denoted by G/H(resp. H\G). For each

11 HOMOGENEOUS SPACES

53

coset J = xH (resp. J = Hx) in G/H (resp. H\G) and each s E G, write s 3 = (sx)H (resp. 1 s = H(xs)). Then it is clear that G acts on the left (resp. on the right) on G/H (resp. H\G), and that this action is transitive. The set G/H (resp. H\G) together with this action of G is called the homogeneous space of left (resp. right) cosets of H in G. We shall work throughout with G/H. Everything we shall do can be transposed in an obvious way to apply to H\G. Let n denote the canonical mapping XHXH of G onto G/H. Clearly G/H is the set of orbits of elements of G for the right action (h, x ) w x h (12.10.13) of H on G. If G is a topological group, we can therefore topologize G/H with the topology defined in (12.10), which is called the quotient by H of the topology of G. If x, E G and if 3, = x,H is its image in G/H, we obtain a fundamental system of neighborhoods of 3 , in G/H by considering the canonical images in G/H of the neighborhoods V of x, in G (i.e., for each V, the set of cosets x H of the elements x E V, or equivalently the image of V (or of VH) under the canonical mapping n : G + G/H). Whenever we consider G/H as a topological space, it will always be this topology that is meant, unless the contrary is expressly stated. (12.11.1)

The group G acts continuously on G/H.

Let (so, a,) be a point of G x (G/H) and let xo be an element of the coset 3,, . Every neighborhood of so 3 , is of the form n(V), where V is a neighborhood of soxo in G . There exists a neighborhood U of so and a neighborhood W of xo such that the relations s E U and x E W imply sx E V; hence the relations s E U and 3 E n(W) imply s J E n(V).

-

-

(12.11.2) Let G be a topological group, H a subgroup of G. (i) G/H is Hausdorfl if and only if H is closed. (ii) G/H is discrete ifand only i f H is open. (iii) If H is discrete, then every x E G has a neighborhood V such that the restriction of n to V is a homeomorphism of V onto the neighborhood n(V) of n(x) = 3 in G/H.

(i) If G/H is Hausdorff, then { n ( e ) }is closed in G/H (12.3.4) and hence H = n-'(n(e)) is closed in G. Conversely, if H is closed in G, the set of pairs (x, y ) E G x G belonging to the same orbit under the action of H on the right is the set of (x, y ) such that x - ' y E H, and is therefore closed, being the inverse image of H under the continuous mapping (x, y ) w x - ' y . Hence G/H is Hausdorff (12.10.8). (ii) If G/H is discrete, the set (n(e)) is open in G/H and therefore H = n-'(n(e)) is open in G. Conversely, if H is open in G, then so is each

54

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

coset x H , and therefore { n ( x ) }= 7c(xH)is open in G/H (12.10.6). Hence G/H is discrete. (iii) Let U, be a neighborhood of e in G such that U, n H = { e } , and let V , be a symmetric open neighborhood of e such that V$ c.Uo (12.8.3). Then, for each x E G, the restriction of 7c to V = x V , is injective, because if h, h’ in H are such that xzh = xz’h’ with z , z’ in V,, we have h’h-’ = z ’ - l z E Vg c U, , so that h’ = h and z’ = z. Since the image under 71 of any open set in V is open in n(V) (12.10.6), and 7c is continuous, it follows that the restriction of I[ to V is a homeomorphism onto n(V). (12.11.3) Let G be a metrizable group, H a closed subgroup of G, and let d be a right-invariant distance dejning the topology of G (12.9.1). For any two points 1, j in G/H, put do(i,j ) = d(xH, y H ) (3.4). Then do is a distance

defining the topology of G/H. If G is complete (12.9.2), then G/H is complete with respect ro d o .

We remark first that if x E f and y

Ej

, we have do(f,j ) = d(x, yH). For

d(x,y H ) = inf d(x, yh) and therefore, for every h’ E H, we have d(xh’, yH)

=

hEH

d(x,y H ) by virtue of the right-invariance of d. Hence, for each z E G, we have ldo(f, i) - dO(3,ill

=

14%zH) - dcv, zH)I 6 4x9 Y )

by (3.4.2). Since this is true for all x , y in the respective cosets f, j , it follows that Id,(& 2 ) - do(j, i)l6 do(& j ) , which shows that do is a pseudo-distance on G/H. The relation do(f, 3 ) < r is equivalent to the existence of a point y E 3 such that d ( x , y ) c r . Since H is closed in G, this proves that do is a distance, and that if B(x; r ) is the open ball with center x and radius r (with respect to d ) , then its image under the canonical mapping n : G + G/H is the open ball with center n(x) and radius r with respect to do. Hence do defines the topology of G/H. Now suppose that G is complete. Let (1,)be a Cauchy sequence with respect to do. We shall show that, by passing to a subsequence of (1,)if necessary, we can reduce to the case where there exists a right Cauchy sequence (x,) in G such that n(x,) = 1,. Since by hypothesis the sequence (x,) converges, this will show that ( i nhas ) a cluster value and therefore converges (3.14.2), and hence that G/H is complete. There exists in G a denumerable fundamental system (V,) of neighborhoods of e such that V;,, c V, for all n (12.9.1 and 12.8.3). Let (&), be a sequence of real numbers > O such that the relation d(e, z ) < E, implies that z E V, . The hypothesis implies that we can define inductively a subsequence (Ank) of (a,) such that d o ( i n pin4) , < E~ for all p 2 k and q 2 k. Replacing the original sequence (1,)by (ink), we can therefore assume that d O ( i pkq) , < E, for

12 QUOTIENT GROUPS

55

all p 2 n and q >= n. Since d is right-invariant, this means that for all y E Rp the intersection of R, and the neighborhood V ,y is not empty. We can then define, by induction on n, a sequence (x,) in G such that x , E R , and x,, E V ,x, for all n. Hence it follows, by induction on p , that for all p > 0 we n ~ V n - V,2+1 1 ~ n c V , for have x,+*EV,+ p - 1 V n + p - 2 . . . V n t l V n ~ (because all n). Hence the sequence (x,) is a right Cauchy sequence in G. Q.E.D. Let G be a topological group, and let E be a topological space on which G acts continuously and transitively. Let x E E and let S, be the stabilizer of x in G; then (12.10.1) the continuous mapping h, : SHS * x of G into E, which by hypothesis is surjective, factorizes canonically as (12.1 1.4)

h, : G 2 G/S, 2 E,

where n, is the canonical mapping of G onto the homogeneous space G/S,, and f, is the bijection SS,H s x . It follows from (12.10.6) that f, is a continuous bijection, but it is not necessarily a homeomorphism (Section 12.12, Problem 2). (12.1 1.5) With the notation above, the bijection f, : G/S, -,E is a homeomorphism for each x E E if and only if there exists a point xo E E such that the mapping h,, : SHS * xo transforms each neighborhood of e in G into a neighborhood of xo in E.

In view of (12.10.6) we need only prove that the condition is sufficient. Notice first that each x E E can by hypothesis be written in the form x = t xo for some t E G. I f V is a neighborhood of e, it follows from the hypothesis that V . x = ( V t ) xo is a neighborhood of x, because we can write ( V t ) x o = t ( ( t - l v t ) . xo), and t -'Vt is a neighborhood of e in G (12.8.3), and ~ H Sy is a homeomorphism of E onto E, for each s E G (12.10.3). Since each open set in G/S, is the image under n, of some open set in G, it is enough to show that for each open set U in G, and each x E E, the set h,(U) =f,(n,(U)) = U * x is open in E. Now, for each t E U, the set t -'U is a neighborhood of e, and therefore ( t -'U) x is a neighborhood of x in E, from the first part of the proof. Hence U * x = t . ( ( t- ' U ) x ) is a neighborhood of t * x ; by (3.6.4), the proof is complete.

-

-

12. QUOTIENT GROUPS

If H is a normal subgroup of a group G, then xH = H x for each x E G and therefore the sets G/H and H\G are identical; and G/H has a group structure

56

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

(the quotient of G by H) for which the canonical mapping n : G -+ G/H is a homomorphism. (12.12.1) Let G be a topologicalgroup and H a normalsubgroup of G. Then the quotient by H of the topology of G is compatible with the group structure of G/H.

Let (,to,j o ) be a point of (G/H) x (G/H), and let xo (resp. yo) be an element of the coset f, (resp. jo).Every neighborhood of i o j 0is of the form n(U), where U is a neighborhood of xo y o . There exists a neighborhood V of xo and a neighborhood W of yo in G such that the relations X E V and y e W imply x y U. ~ Hence the relations , t ~ n ( V )and j ~ n ( W imply ) i j E n(U), and therefore the mapping ( i j,) ~ + , t jis continuous at the point (a,, yo). The proof of continuity of the map 2 w i - l is similar. Whenever we speak of the quotient G/H o f a topological group G by a subgroup H, it is to be understood that G/H carries the quotient topology. (12.12.2) Let G be a topologicalgroup and let H , K be two normal subgroups of G such that H 3 K. Then the canonical bijection cp : G/K -+ (G/H)/(K/H) is an isomorphism of topological groups.

We know that cp is the unique mapping which makes the following diagram commutative : G

A G/K

where n, d ,and n" are the canonical homomorphisms. The definition of a quotient topology (12.10) shows that a subset U of (G/H)/(K/H) is open if and only if n'-'(n''--'(U>)is open in G, that is, if and only if n-'(cp-'(U)) is open in G ; but this last condition signifies that cp-'(U) is open in G/K. Since cp is bijective, the result follows. (12.12.3) Let G be a topologicalgroup, H a normal subgroup of G , 71: G + G / H the canonical mapping, and A a subgroup of G. Then the canonical bijection of n(A) onto the quotient group AH/H is an isomorphism of topological groups.

This is a particular case of (12.10.7), applied to the right action (h, x ) ~ x of h H on G.

12 QUOTIENT GROUPS

57

There is also a canonical bijection II/ of the quotient group A/(A n H) onto the subgroup n(A) of G/H ;this bijection II/ is the unique mapping which makes the following diagram commutative: A

(1 2.12.4)

/n

b n o j

A/(A n H)+

n(A)

where n' is the canonical mapping and j : A + G the canonical injection. It follows from (12.10.6) that is continuous, but it is not necessarily bicontinuous (Problem 2). However:

(12.12.5) Let G be a metrizable group and H a compact subgroup of G . For each closed subgroup A of G , the canonical bijection II/ : A/(A n H) -+ n(A) is an isomorphism of topological groups. It is enough to show that the image under II/ of an arbitrary closed subset F of A/(A n H) is closed in n(A). Now n'-'(F) = B is a closed subset of A, hence closed in G , and $(F) = n(j(B)) = n(BH). But since H is compact, the set BH is closed in G (12.10.5) and saturated, hence n(BH) is closed in G/H (12.10) and therefore also closed in n(A).

(12.12.6) Let G,, G2 be two topological groups and HI (resp. H,) a normal subgroup of GI (resp. G 2 ) .Then the canonical bijection 0 : (GI

x G2)/(H, x H2)

-+

(G,/H,) x (G2/H2)

is an isomorphism of topological groups. This is an immediate consequence of (12.10.11 ) .

(12.12.7) Let G, G' be two topological groups and u : G -+ G' a continuous homorphism. Then u factorizes canonically as G 5 G/N 1;u ( ~2) G! where N is the kernel of u, p is the canonical homomorphism, and j the canonical injection. The mapping u is a continuous bijection (12.10.6(iii)), but is not necessarily bicontinuous. When u is bicontinuous, the homomorphism u is said to be a strict morphism of G into G ' ; for this to be the case, it is necessary and sufficient that for each neighborhood V of e in G, the set u(V) should be a neighborhood of e' = u(e) in u(G) (12.11.5).

58

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

PROBLEMS

1. (a) Let G be a topological group, K its identity component, H a subgroup of G contained i n K. Show that the connected components of the space G/H are the images of the connected components of G under the canonical mapping T : G + G / H . Show that K is the smallest of the subgroups L of G such that G/L is totally disconnected. (b) Let G be the additive subgroup of the Banach space d , ( N ) (7.1.3) consisting of the mappings nt-+f(n) such that f(n) E Q for all n E N and lim f ( n ) exists in R. The

"- uo

distance induced on G by that on d R ( N ) defines a topology compatible with the group structure of G , and with respect to this topology G is totally disconnected. Let H be the subgroup of G consisting of all f~G such that lim f(n) = 0. Show that H is closed n- m

in G and that G / H is isomorphic to R,and therefore connected. (c) Let G be a locally compact metrizable group, H a closed subgroup of G, and rr : G + G/H the canonical mapping. Show that the connected components of G/H are the closures of the images under T of the connected components of G. (Reduce to the case where G is totally disconnected, and use Problem 3 of Section 12.9.) 2.

In the additive group R,let H be the subgroup Z and let A be the subgroup 0Z, where 0 is an irrational number. Show that the canonical bijection A/(A n H) +(A H)/H is not an isomorphism of topological groups.

+

3. Let p be a prime number and let (GJ be an infinite sequence of topological groups all equal to the discrete group Z / p 2 Z .Let H. be the subgroup p Z / p 2 Z of G, . Let G be the subgroup of the product g r o u p n G. consisting of all x = (x.) such that xn E H. for all n

but a finite set of values of n. Let 23 be the set of neighborhoods of 0 in the product group H, = H c G. Show that W is a fundamental system of neighborhoods of 0

n

for a topology on G compatible with the group structure, and that in this topology G is metrizable and locally compact and G/H is discrete. Show that the homomorphism u : X H ~ Xof G into G is not a strict morphism of G into G, and that u(G)is not closed in G. 4.

Let G be a metrizable group, K a closed normal subgroup of G. If K and G/K are complete, show that G is complete.

5. (a) Let G be a topological group, K a normal subgroup of G. Show that if K and G/K have no small subgroups (Section 12.9, Problem 6), then G has no small subgroups. (b) Deduce from (a) that if H I , H 1are two normal subgrouos of a topological group G, such that G/H1 and G/H, have no small subgroups, then G/(H, n H,) has no small subgroups. 6. Let G be a connected, locally compact, metrizable group, K a compact normal subgroup of G , and N a closed normal subgroup of K, such that K/N has no small subgroups (Section 12.9, Problem 6). Show that N is a normal subgroup of G. (Observe that the hypothesis on K / N implies the existence of a neighborhood U of e in G such that xNx-' c N for all x E U.)

7. Let G be a connected metrizable group and H a compact normal commutative subgroup of G, with no small subgroups. Show that H is contained in the center of G .

13 TOPOLOGICAL VECTOR SPACES

59

(Observe that, if s is close to e in G, then the image of H under the mapping is close to e.)

xt-+sxs-'x-'

8. Let G be a locally compact metrizable group and H a closed normal subgroup of G , such that G/H is not discrete and has no small subgroups. Show that there exists a f continuous homomorphism f:R + G such that the composition R + G +G/H is nontrivial (see Section 12.9, Problem 7).

13. T O P O L O G I C A L V E C T O R SPACES

We shall adhere to the conventions of (5.1), so that all vector spaces under consideration have as field of scalars either the real field R or the complex field C. If E is a vector space over R (resp. C),a topology on E is said to be compatible with the vector space structure if the mappings ( x ,y ) - x + y of E x E into E, and (A, X ) H A X of R x E (resp. C x E) into E are continuous. A vector space endowed with a topology compatible with its vector space structure is called a topological vector space. Since -x = (- l)x, the conditions above imply that the topology is afortiori compatible with the additive group structure of E, and all the notions and properties established for topological groups in the preceding sections will apply in particular to topological vector spaces (provided of course that these properties are transcribed into additive notation). An isomorphism of a topological vector space E onto a topological vector space F is a linear bijection of E onto F which is a homeomorphism. If llxll is a norm on a vector space E, the topology on E defined by the distance IIx - yll is compatible with the vector space structure of E (5.1.5) and therefore makes E a topological vector space. Two equivalent norms (5.6) define the same topology. In a topological vector space E, the translations XHX a and the homotheties XH Ax ( A # 0 ) are homeomorphisms of E onto E, the inverse mappings being, respectively, the translation X H x - a and the homothety xwa-lx. In a vector space E over R (resp. C) a set M is said to be balanced if, for each x E M and each scalar A such that 1A1 =< I , we have Ax E M (in other words, if AM c M whenever 111 5 1). The set M is said to be absorbing if, for each X E E , there exists a real number a > 0 such that A X E M whenever (A16 a. Every absorbing subset of E generates the vector space E. For a balanced subset M of E to be absorbing, it is enough that for each x E E there should exist a scalar A # 0 such that Ax E M.

+

60

XI1 TOPOLOGY A N D TOPOLOGICAL ALGEBRA

(1 2.13.1 ) In a topological vector space E, the absorbing balanced neighborhoods of 0 form a fundamental system of neighborhoods of 0.

For each xo E E, the mapping AH AxO of the scalar field into E is continuous at the point 0; hence, for each neighborhood V of 0 in E, there exists ct > 0 such that the relation 1 11 S CI implies Ax, E V. This shows that every neighborhood of 0 is absorbing. On the other hand, the continuity of the mapping (A, x) I--, Ax at the point (0,O) implies that, for each neighborhood V of 0 in E, there exists ct > 0 and a neighborhood W of 0 in E such that the relations 111 6 ct and x E W imply Ax E V. The union U of the sets 1W, where 1A1 6 CI, is clearly a balanced neighborhood of 0 in E, and is contained in V. Hence the result. Let F be a vector subspace of a topological vector space E. Then the topology induced on F by the topology on E is compatible with the vector space structure of F. Whenever we consider a vector subspace F as a topological vector space, it is always the induced topology that is meant, unless the contrary is expressly stated. The proof of (5.4.1) applies without any change and shows that F is a vector subspace of E. The definition of a total subset of E is the same as in (5.4). On the quotient vector space E/F, the quotient topology (12.11) is compatible with the vector space structure. For if n : E + E/F is the canonical mapping, let ( A o , 2,) be any point of R x (E/F) (resp. C x (E/F)) and let xo be any point of 2,. Then every neighborhood of 1, i o contains a neighborhood of the form n(V),where V is a neighborhood of 1, *yo.There exists a neighborhood U of A, in R (resp. C ) and a neighborhood W of x, in E such that the relations 1 E U and x E W imply Ax E V; consequently the relations 1 E U and R E n(W) imply 1i E n(V), and the assertion is proved. Whenever we consider E/F as a topological vector space, it is always with the quotient topology, unless the contrary is expressly stated. The criteria for continuity in a product ((3.20.15) and (12.5)) show immediately that if El, E, are two topological vector spaces, the product topology on El x E2 is compatible with the vector space structure of El x E, . The vector space El x E 2 , endowed with this topology, is called the product of the topological vector spaces El and E 2 . Proposition (5.4.2) and the definition of the topological direct sum of two subspaces are valid without modification for arbitrary topological vector spaces. If E is the direct sum of two subspaces F,, F, , and if p 1 : E + F, is the corresponding projection, then the kernel of p 1 is F, , and p1 therefore has a

13 TOPOLOGICAL VECTOR SPACES

61

canonical factorization E + E/F2% F,, where jl is a bijection. To say that E is the topological direct sum of F1 and F2 signifies that .jl is continuous (12.10.6), or equivalently that the inverse bijection, which is the restriction to F, of the canonical mapping n2 : E -+ E/F2 (and which is always continuous) is bicontinuous. (12.1 3.2) (i) Let E be a topological vector space and let H be a hyperp1ane.h E , with equation f ( x ) = 0. Then H is closed in E ifand only if the linear form f is continuous on E. (ii) Let E be a HausdorJf topological vector space over R (resp. C ) of finite dimension n. I f ( a i ) 1 6 i s nis a basis of E, the mapping

of R" (resp. C") onto E is an isomorphism of topological vector spaces. (iii) Let E be a topological vector space, M a closed vector subspace of E , and F a finite-dimensional vector subspace of E. Then M F is closed in E. In particular, if E is Hausdorf, every finite-dimensional vector subspace of E is closed in E. (iv) Let E be a topological vector space and let M be a closed vector subspace of finite codimension in E. Then every algebraic supplement N of M in E is a topological supplement.

+

We shall prove (ii) first. Since u is continuous and bijective, we have to show that u is bicontinuous, or equivalently (considering the inverse images under u of open sets in E) that if D is a Hausdorf topology compatible with the vector space structure of R" (resp. C"), and coarser than the product topology Do, then 0 = Do. Put JIxIJ= ~ u p l 0 such that p'(x) _I c . sup pi(x) i for aN x E E.

Apply (12.14.11) to the identity map u

=

l E (12.2.1).

(12.14.13) Finally, we remark that the notions of series and convergent series are defined in a topological vector space E just as in a normed space (5.2); propositions (5.2.2), (5.2.3), (5.2.4) and the fact that the general term of a convergent series tends to 0 remain valid without change. If E is locally convex and its topology is defined by a family ( p J of seminorms, then for a series with general term x, to converge in E it is necessary that, for each E > 0 and each scalar I , there should exist an integer no such that

for all n 2 no and q > 0. If E is a FrCchet space, this condition is also sufficient, as follows from Cauchy's criterion (12.9).

PROBLEMS

1. Let E be a vector space over R (resp. C) and let 8 be a set of subsets of E satisfying conditions (V,) and (V,,) of Section 12.3, Problem 3, together with the following: (EV,) Every set V E 8 is balanced and absorbing.

14 LOCALLY CONVEX SPACES

69

(EV,,) For all V E 23 and all h # 0 in R (resp. C), we have XV E B. (EVIII) For each V E B, there exists W E B such that W W c V. Show that there exists a unique topology on E, compatible with the vector space structure, for which 8 is a fundamental system of neighborhoods of 0.

+

2.

Let E be a vector space over R with a denumerable infinite basis (en),and let 5 be the set of aN balanced absorbing sets in E. Show that 5 does not satisfy axiom (EVIII) of Problem 1. (For each n > 0, let A. be the set of points

n

1=1

he, such that [tIl6 l/n for

1 5 i 5 n, and let A be the union of the sets A,, . Show that there exists no set M € 5 such that M M C A.)

+

3. Let E be the vector space VR(R) of continuous mappings of R into R. For each continuous function m such that m(f) > 0 for all r E R,let V, denote the set offunctions YE E such that If(r)l 5 m(f) for all t E R. (a) Show that the V, form a fundamental system of neighborhoods of 0 in a topology .Toon E which is compatible with the additive group structure of E, but not with the vector space structure. (b) Let D be the vector subspace of E consisting of functions with compact support (12.6). Show that the topology Y induced by T oon D is compatible with the vector space structure of D, and is not metrizable. 4.

Let (EJaG1be any family of topological vector spaces, and E =

n E. the product a.1

vector space. Show that the product of the topologies of the Em(12.5) is compatible with the vector space structure of E. If each E. is locally convex and its topology is defined by a family of seminorms (p& Lm, then the topology of E is locally convex and is defined by the family of seminorms par pr. (for all a E I and all A E Le).In particular, every (finite or) denumerable product of Frkhet spaces is a Frkchet space. 0

5. Let p be a seminorm on a (real or complex) vector space E. Show that the set N of x E E such that p(x) = 0 is a vector subspace of E. The seminorm defined by (12.14.8.1) on E/N is a norm. Let E be a Hausdorff locally convex space whose topology is defined by a family (pa) of seminorms, and for each a let N. be the subspace of E consisting of those x E E for which p.(x) = 0. Show that E is isomorphic to a subspace of the product of the normed spaces E/N, (the norm on E/N. being i.). In particular. every Frkchet space is isomorphic to a closed subspace of a denumerable product of normed spaces. 6. A subset A of a vector space E over R is said to absorb a subset B of E if there exists a > 0 such that hB c A whenever 5 a. If E is a topological vector space, a subset

1x1

B of E is said to be bounded(with respect to the topological vector space structure of E) if every neighborhood of 0 absorbs B. (a) The closure of a bounded set is bounded. Every finite set is bounded. If A, B are bounded, then so are A u B and A B. If E is metrizable, every precompact set in E is bounded. (b) Suppose that E is Hausdorff. Then a subset B of E is bounded if and only if, for every sequence (x,,) in B and every sequence (A,) of real numbers tending to zero, the sequence (h.~.) tends to 0 in E. (c) Let (E.) be a family of topological vector spaces and E E. their product

+

=n 8

(Problem 4). Show that a subset B of E is bounded if and only if pra(B) is bounded in E. for each a.

70

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA (d) Let E, F be two topological vector spaces and f:E + F a continuous linear mapping. Then for each bounded subset B of E, the image set f(B) is bounded in F.

7. (a) Let E be a (real or complex) Hausdorff topological vector space. Show that, if there exists a bounded neighborhood V of 0 in E (Problem 6), the sets n-'V (n an integer > 0 ) form a fundamental system of neighborhoods of 0 in E, and consequently E is metrizable. If furthermore E is locally convex, then its topology can be defined by a single norm. (b) Let E be a metrizable locally convex space whose topology is defined by an increasing sequence ( p , ) of seminorms. Show that the topology of E can be defined by a single norm if and only if there exists an integer N such that, for all n 2 N, there exists a real number k. 2 0 for which pn 2 k n p N . (c) Let E be the real vector space of indefinitely differentiable real-valued functions on the interval I = [0, I ] in R.For each n 2 0 and eachfe E, put

(where Dof=f). Show that each p . is a norm on E and that the topology defined by the sequence of norms (p,) cannot be defined by a single norm (cf. (17.1)). (d) Let E be a Hausdorff locally convex space whose topology is defined by a sequence of seminorms but cannot be defined by a single norm. If d is a translationinvariant distance defining the topology of E (1 2.9.2) show that every set in E which is bounded in the sense of the topological vector space structure of E is bounded with respect to the distance d , but there exist subsets of E which are bounded with respect to d but are unbounded with respect to the topological vector space structure of E. 8. Let E be a metrizable topological vector space. (a) Show that every balanced subset of E which absorbs all sequences in E which converge to 0 is a neighborhood of 0 in E. Deduce that, if u is a linear mappingof E into a topological vector space F, and u transforms every sequence converging to 0 in E into a bounded sequence in F, then u is continuous. (b) Let (B,) be any sequence of bounded subsets of E. Show that there exists a sequence (A,) of nonzero scalars such that the union of the sets A,B. is bounded. 9. Let E be a topological vector space. A set b of bounded subsets of E is called a fundamental system of bounded subsets if every bounded set in E is contained in a set belonging to b. Show that, if E is a metrizable locally convex space whose topology

cannot be defined by a single norm, no fundamental system of bounded subsets of E is denumerable (use Problem 8).

10. Let (En)nslbe a sequence of normed spaces such that Elis not separable. Let F be a E,, such that pr.(F) = E, for each n. Show that there exists vector subspace of E

=n n

a real number 6 > 0 and a bounded sequence (x,,Jmal in F such that llprlxl- pr1xAl

28

whenever i # j . (Remark that, for each index n and each nondenumerable subset A of F, there exists a nondenumerable subset B of A such that pr.(B) is bounded in En.) Deduce that a Frkchet space is separable if every bounded subset is relariuely compact. 11. A subset A of a real or complex vector space E is convex if, whenever x and y belong to A and 0 5 A 6 1, we have kr + (1 - A)y E A (Section 8.5, Problem 9). It follows

71

14 LOCALLY CONVEX SPACES

that, if A is convex and ( x , ) ~ , < ,is, 0, show that there exists a continuous mapping s : L -+ E such that p ( z - f ( s ( z ) ) ) E for all z E L. (Consider a locally finite denumerable open covering (U,) of L such that p(x' - x) 2 &e whenever x and x' lie in the same U,, a continuous partition of unity (h,) subordinate to (U"),and for each n a point a,,E U, nf(E). If b, ~ f - ' ( a . )show , that the function s defined by s(z)

=Cb,,h,(z) satisfies the required conditions.) I

15 WEAK TOPOLOGIES

73

15. WEAK TOPOLOGIES

(12.15.1) Let (E,),e I be a family of locally convex spaces, and f o r each a let Lm be a family of seminorms dejining the topology of E, . Let E = E,

n

be the product vector space, and for each a E I and each I (12.1 5.1 . l )

&(x)

= pal(prorx)

E L,

, put

UEE

for all x E E.

Then the pad are seminorms on E and dejine the product topology (12.5) on E. It is straightforward to check that the pLl are seminorms, and then the second assertion follows from the definition (12.5) of a product of uniformizable spaces. In particular, when all the E, are equal to the same locally convex space F, then the product topology on the vector space F' of all mappings of I into F is defined by the seminorms

f

HP ( f

(4)

where a E I and p belongs to a set r of seminorms defining the topology of F. This topology is called the topology of simple (or pointwise) convergence on the vector space F', or on any of its vector subspaces (cf. (12.5)). With respect to this topoiogy, each mapping f w f (a) (a E I) is a continuous linear mapping on F', and maps each open set in F' to an open set in F (12.5.2). The topology of simple convergence is Hausdorf if F is Hausdorff (12.5.7). A mapping t wft of a topological space T into F' is continuous with respect to the topology of simple convergence if and only if the mapping t w f , ( a ) of T into F is continuous, for each a E I (12.5.5). When F is a Banach space over K (= R or C) and T is an open subset of K", a mapping t wft of T into F' is said to be p times diferentiable (resp. indefinitely diferentiable, resp. analytic) with respect to the topology of simple convergence if, for each index a E I, the mapping t w f t ( a ) of T into F has the corresponding property. We shall be particularly concerned with the case in which F is the field of scalars K, and I is a vector space E over K. Let V be a subspace of KEwhose elements are linear forms on E. The topology of simple convergence on such a subspace V is called the weak topology; it is defined by seminorms (12.1 5.2)

f-

If(X>l,

where x runs through E. For each x E E, the mapping f ~ f l x is) a continuous linearform on V with respect to the weak topology. A sequence (f.)of elements

74

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

of V which converges to f E V with respect to the weak topology (i.e., is such that lim f,(x) =f ( x ) for all x E E) is also said to be weakly conuergent. On n-) m

the same principle, we shall speak of subsets of V being weakly closed (in V), weakly compact, etc. A subset B of KE is said to be weakly bounded if all the seminorms (12.15.2) are bounded on B, that is if sup I f ( x ) l < 00 for

+

JEB

all x E E. We shall also use the phrases weakly continuous (resp. weakly differentiable, weakly analytic) or scalarly continuous (resp. scalarly differentiable, scalarly analytic) in place of “continuous (resp. differentiable, analytic) with respect to the weak topology.”

If E is a locally conuex space (over K = R or C), the vector space of continuous linear forms on E is called the dual of E, and is often denoted by E’. If x’ E E is a continuous linear form on E, its value at a point x E E is often denoted by the expression (x, x’) or (x’, x) instead of x’(x). Since E‘ is a vector subspace of KE, it is endowed with the weak topology defined by the seminorms X ’ H I(x, x’)l. For each x E E, the function X ’ H ( x ,x‘) is a weakly continuous linear form on E’. If E is a FrCchet space and E, a dense vector subspace of E, then it follows from (12.9.4) and the principle of extension of identities (3.15.2) that the mapping X ’ H X ’ I E, is an (algebraic) isomorphism of the dual E‘ of E onto the dual Eb of E, . (12.15.3) Let E, F be two locally convex spaces, E‘ and F’ their respective duals, and u : E -+ F a continuous linear mapping. Then Y ‘ H Y ‘ u is a linear mapping of F’ into E’ which is continuous with respect to the weak topologies. 0

If y’ is a continuous linear form on F, then y’ u is a continuous linear form on E. Given x, E E, there exists yo E F such that (x,, , y’ u ) = ( y o , y’) for all y’ E F’, namely yo = u(x,); the continuity of the mapping y ’ y’ ~ u is therefore an immediate consequence of (12.14.11) and the definition (12.15.2) of the seminorms which define the weak topology. 0

0

0

, the linear mapping ‘u : F’ -+ E’ is called the We write y’ u = ‘ ~ ( y ’ )and transpose of the continuous linear mapping u. Thus we have 0

(12.15.4)

(u(x), Y‘> = (x, ‘4Y’))

for all x E E and y’ E F’. Clearly (12.15.5)

yu1

+ u2) = +

tU2,

‘(Au)= A . ‘u

for any scalar 1,and for any continuous linear mappings u l , u2 of E into F. If

15 WEAK TOPOLOGIES

75

v is a continuous linear mapping of F into a locally convex space G, then (12.15.6)

‘(0

0

u ) = 5.4 0 %.

The weak topology on the dual of a locally convex space is not metrizable in general (Problem 2). However, there is the following result: (12.15.7) Let E be a separable metrizable locally convex space. If H is any equicontinuous (7.5) subset of the dual E’ of E, then the weak closure of H in E‘ is a compact metrizable space in the weak topology.

We shall first prove the following lemma: (12.15.7.1 ) Let E be a metrizable vector space and F a normed space. In order that a set H of linear mappings of E into F should be eguicontinuous, it is necessary and suficient that there should exist a neighborhood V of 0 in E and a real number c > 0 such that IIu(x)I/I c for all x E V and all u E H.

(If E is normed, this condition is also equivalent to sup llull < +a UEH

by (5.7.1).)

The condition expresses that H is equicontinuous at the point 0. If this is the case, then for each xo E E and each x E xo EV,we have

+

IIU(4

for all u

E

- u(xo>ll= l l 4 X

- X0)Il

I EC

H, and therefore H is equicontinuous at xo .

In proving (12.15.7), we may first of all restrict ourselves to the case where H is weakly closed in E’. For if R is the weak closure of H in E’, the continuity of the mapping X’H ( x , x’) on E’ implies that if I(x, x ’ ) ] 5 c for all x E V and all x’ E H, then also I(x, x’)l 5 c for all x E V and all x’E R (3.15.4), whence the result by (12.15.7.1). Now let (a,,) be a sequence which is dense in E. We shall show that the weak topology on H is defined by the pseudo-distances

d,,(x’, y’) = I & , ,

x’ - Y Y .

Clearly the topology defined by these pseudo-distances is coarser than the weak topology; hence it is enough to show that it is alsofiner. By hypothesis, every neighborhood of a point xb E H in the weak topology contains a subset W of H consisting of elements x’ E H satisfying ajinite number of inequalities (12.15.7.2)

[ ( x i ,x’- xb)l < r

(1

5 i 5 m)

76

XI1 TOPOLOGY A N D TOPOLOGICAL ALGEBRA

where the x i are points of E, and r > 0. Suppose that H satisfies the condition (12.15.7.1). Choose E > 0 such that ~ E < C +r. For each index i, there exists an a,,(i)such that x i - a,,(i)E EV. By virtue of (12.15.7.1), it follows that for each xf E H we have [ ( x i - a,,(i),xf - xb)l

5~

< +r,

E C

and the relations x’ - xb)l < f r therefore imply the inequalities (12.15.7.2). Since the weak topology is Hausdorff, we have shown that H is metrizable (12.4.6). More precisely, the above argument shows that the map~ H into the product space KN (where K = R or C) ping x’t-+((a,, x ’ ) ) , , ~of is a homeomorphism of H onto a subspace L of KN.So it remains to be shown that L is compact. In the first place, the projections of L are boundedsubsets of K (and therefore relatively compact ((3.17.6) and (3.20.16))): for each index n there exists A,, > 0 such that AnanE V and therefore [(a,,, x’)l 5 c/A,, for all x’ E H. By virtue of (12.5.9), (12.5.4), and (3.17.3), it remains to be shown that L is closed in KN.Now if (xh) is a sequence of points in H such that for each n the sequence ((a,,, ~ k ) ) , , , > converges ~ in K, then it follows from (7.5.5) that the sequence (xk) converges simply in V. Since for each Z E E there exists a scalar y such that yz E V (so that (z, x i ) = y-l(yz, xh)), the sequence (xh) converges simply in E: its limit y’ is a linear mapping, by the principle of extension of identities, and is continuous by (7.5.5), and is therefore a point Q.E.D. of H. We recall that if E is a normed space, its dual E’ = 2 ( E ; K) is a Banach space with respect to the norm IIx’II = sup I(x, x’)l (5.7.3); since IIXII 5

1

I ( x , x’)I 5 ((x(( I(xf((, the topology defined by this norm (sometimes called the strong topology on E‘) isjner than the weak topology on E’.

(12.15.8) Let E be a normed space. The mapping X’H I(x’I( of E’ into R is lower semicontinuous with respect to the weak topology.

For it is the upper envelope of the weakly continuous mappings

x’H((x, x’)], where x runs through the ball (1x115 1 in E. Hence the result follows from (12.7.7).

If (x;) is a sequence in E’ which converges weakly to a’, then by (12.7.13) we have (12.15.8.1)

lim inf (lxLll 2 Ila’ll. n-tm

15 WEAK TOPOLOGIES

77

(12.15.9) Let E be a separable normed space. Then every closed ball B': IIx'II

5 r in E' is metrizable and compact with respect to the weak topology.

By (12.15.7) and (12.15.7.1) it is enough to show that B' is weakly closed, and this follows from the fact that the function x ' w llx'II is lower semicontinuous on E' (12.7.2). Now let E be a Hilbert space (6.2). For each x E E, let j ( x ) denote the continuous linear form y ~ ( I x) y on E. It follows from (6.3.2) that X H ~ ( X ) is a semilinear isometry (i.e., we have Axl

+ x2) = j(xl> +Ax2),

j(Ax) = JAx)

for all scalars A) of E onto its dual E'. We can therefore " transport" to E the weak topology on E': the weak topology on E is therefore the topology defined by the seminorms x ~ l ( a l x ) as l a runs through E. This topology is coarser than the topology defined by the norm on E (which is called the strong topology of the Hilbert space E). From (12.15.9) we have: (12.15.10) In a separable Hilbert space, every closed ball is metrizable and compact with respect to the weak topology.

If u is a continuous endomorphism (with respect to the strong topology) of a Hilbert space E, then by virtue of (12.15.4) and (11.5.1) MX)

I Y ) = (u(x), iW> = ( x , ' 4 A Y ) ) ) = (x l~-1(r4AY)N) = (x I U*(Y))

for all x, y in E , and therefore (6.3.2) (12.15.11)

u*

=j-1

o'uoj.

This relation shows immediately that 'u is strongly continuous and that = IIu*II = (lull.Replacing u by u* in (12.15.11), we see that every strongly continuous endomorphism of E is also weakly continuous (cf. (12.16.7)).

II'uII

(12.1 5.12) In a Hilbert space E , a sequence (x,) converges strongly to apoint a

if and o n b if it converges weakly to a and lim llxnll = Ilall. n+ w

The conditions are clearly necessary. To see that they are sufficient, consider the formula IIX,

- allZ = IIXnII' - 29W, I a) + IIaI12.

Since by hypothesis the sequence ((x, 1 a))converges to (a I a ) = 11a1I2,it follows that IIx, - all tends to 0.

78

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

PROBLEMS 1. Let E be a vector space over K (= R or C) and let F be a vector subspace of KEwhose elements are linear forms on E. We write ( x , x ’ ) in place of x’(x) for x E E and x’ E F. Given a finite sequence ( x i ) l 6 l a n of elements of F, show that for this sequence to be free (A. 4.l)it is necessary and sufficient that there should exist n vectors x1 (1 i 5 n) in E such that ( x , , x ; ) = (Kronecker delta). The vector space E is then the direct sum of the subspace V of dimension n generated by the X I , and the subspace W of all x E E such that ( x , x i ) = 0 for 1 5 n. A linear form x’ E F is such that ( x , x’> = 0 for all x E W if and only if x’ is a linear combination of the x { . (Prove these assertions simultaneously by induction on n.)

s

sr,

is

2.

The notation and hypotheses are the same as in Problem 1. Suppose also that there exists no vector x # 0 in E such that ( x , x ‘ ) = 0 for all x’ E F. (a) Show that the weak topology on F is metrizable if and only if E has an at most and ( z , ) ~ ~are , ~ two ~ finite denumerable basis (A. 4.1). (Observe that if (yf)161sm sequences of vectors in E such that the relation sup I ( y l , x’)l 5 1 implies the relation

sup I (z,,

16J6n

x’)l

51

ISl6rn

in F, then the z, are linear combinations of the y , .

Show that if we have ( y , , x’) = 0 for 1 5 i 5 m,then we must also have (z,, x ’ ) = 0 for 1 6j 5 n, and apply the result of Problem 1 by identifying E with the set of linear forms X‘H ( x , x’) on F.) (b) Deduce from (a) that, in an infinite-dimensional separable Hilbert space, the weak topology is not metrizable (cf. Section 5.9, Problem 2). (c) Show that every linear form on F which is continuous with respect to the weak topology can be written x ’ w ( x , x ’ ) for a uniquely determined x E E (argue as in (a)).

3. (a) In a Hilbert space E, let A be a closed convex set (Section 12.14, Problem 11) and a a point of E. Show that there exists a unique point b E A such that d(a, A) = d(a, b) (the projection of a on A). For each z E A show that (z - b I b - a) 20 (argue as in (6.3.1)). (b) Deduce from (a) that every closed convex set A in E is the intersection of closed half-spaces whose frontiers are hyperplanes of support of A (Section 5.8, Problem 3). Deduce that A is weakly closed, and that if C is any convex subset of E, the strong and weak closures of C are the same. Every bounded closed convex subset of E is weakly compact. (c) Show that the weak closure in E of the sphere S : ((xi(= 1 is the ball ( / X I / 0 for all n.) (e) Suppose that E is separable, and let (en)nblbe basis (6.5.2) of E. Let A be the closed convex hull (Section 12.14, Problem 13) of the set of points e./n. Show that A is compact and that the interior of A is empty, and that A has no closed hyperplane of support containing the (frontier) point 0, although there exist lines D passing through 0 and such that D n A = {O}.

15 WEAK TOPOLOGIES

4.

79

Let E be a separable real Frechet space a n d p a continuous seminorm on E. (a) Let (an)n2O be a total (12.13) free sequence in E, and let E. be the subspace of dimension n 1 generated by a o , a ] , . . . , a,. Show by induction on n that for each n there exists a linear formf, on E, such that f. is the restriction of fn+] to En, fo(ao)= p ( a o ) and fn(x) s p ( x ) for all x E E.. (Consider the hyperplane H in given by the equation f n - l ( x )=fn -l(ao), and use Problem 3(d) in the plane which is the quotient of E, by the hyperplane Ho : j m - I ( x= ) 0 in En-l.) (b) Deduce from (a) that there exists a continuous linear form f o n E such that f ( a 0 ) =p(ao) and If(x)i s . p ( x ) for all x E E (Hahn-Banach theorem for a separable Frechet space). In particular, if E is a separable Banach space, then for each x E E we have

+

IIX II = S U P I I IlY’ll s I (the norm on the dual E being that defined in (5.7.3)). Deduce that if E and F are separable Banach spaces, E and F’ their duals, and u : E -+ F a continuous linear mapping, then llfu/1= IIu/l. (c) Deduce from (b) the geometrical form of the Hahn-Banach theorem: given any nonempty convex open set A in E, and a point x o $ A, there exists a closed hyperplane with equation g ( x ) - a = 0 such that g(x) > a for all x E A, and g ( x o ) 5 a. (Reduce to the case where 0 E A and x o is a frontier point of A. We can then assume that A is symmetrical about 0 and therefore is the set of all x such that p ( x ) < 1 , where p is a continuous seminorm on E.) (d) Let A, B be two disjoint closed convex sets in E, one of which (say, A) is compact. Show that there exists a closed hyperplane H, with equation g(x) - a = 0, which strictly separates A and B: that is to say, such that g ( x ) > a for all x f A and g ( x ) < a for all x E B. (e) Deduce from (d) that every closed convex set in E is an intersection of closed half-spaces, and that every closed linear variety in E is an intersection of closed hyperplanes. (f) Deduce that for a vector subspace F not to be dense in E it is necessary and sufficient that there should exist a continuous linear formff 0 on E such that f ( x ) = 0 for all x E F.

5. Let A be a convex set in a real vector space. A point a E A is said to be an extremal point of A if there exists no line-segment containing more than one point, which is contained in A and has a as an interior point: in other words, if the relations b E A, ~ E Aa ,= X b $ ( l - & c , O < h < l imply b = c = a .

(a) Let E be a Hilbert space, A a bounded closed convex set in E, 6 > 0 the diameter of A, and a, b two points of A such that Ilb - all 2 &dE. 6 . Show that there exists a supporting hyperplane H of A orthogonal to the vector b - a and such that the set H n A has diameter 548. (b) Deduce from (a) that A has at least one extremal point (proof by induction). (c) Show that A is the closed convex hull of the set of its extremal points (KreinMilman fheorem for Hilbert space). (First deduce from (b) that every closed hyperplane of support of A contains at least one extremal point. Then argue by contradiction, assuming that the closed convex hull B of the set of extremal points of A is distinct from A; by using Problem 4(e), show that there would then exist a hyperplane of support of A which did not meet B.) (Cf. Section 13.10, Problem 8.)

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XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

(d) If C is a convex set in E, the intersection of C with a closed (resp. open) halfspace is called a closed (resp. open) cap in C . If a E A is an extremal point, show that the open caps in A which contain a form a fundamental system of neighborhoods of a in A. (Remark that the convex hull of the union of two closed caps in A is weakly compact and does not contain a if neither of the two closed caps contains a ; then use Problem 4(e), and Section 12.3, Problem 6(e).) (e) Suppose that there exists a weakly compact set K c A such that the closed convex hull of K is A. Show that every extremal point of A belongs to K (use (d)). (f) In the Banach space (co) (Section 5.3, Problem 5) show that the closed ball llxll 1 has no extremal points. 6. Let E, F be two locally convex spaces. Let (p.) be a family of seminorms defining the topology of F, and let 6 be a set of bounded subsets of E (Section 12.14, Problem 6). For each continuous linear mapping u of E into F, and each B E (5, put p., .(u) = supp.(u(x)). Show that p., is a seminorm on the vector space P ( E ; F) of conZ€B

tinuous linear mappings of E into F. If the set of homothetic images of the sets of 6 covers E, then the topology defined by the seminorms p a , on 9 ( E ; F)is Hausdorff. If E and F are normed spaces and if 6 consists only of the closed unit ball /\XI\ 5 1 in E, then the corresponding topology on 9 ( E ; F) is that defined by the norm (5.7.1).

7. Let E be an infinite-dimensional separable Hilbert space. Show that, with respect to the topology defined by the norm (5.7.1) on P ( E ; E) (this topology is called the norm topology), the unit ball V consisting of the operators U with norm l\Ul\ 1 (the "contractions" of E) is not separable. (Let (en),>,, be an orthonormal basis of E. For each point z = ({.).20 of the Banach space 1" (Section 5.7, Problem l), let U,E 9 ( E ; E) be the mapping defined by U,. en= [.en for all n 2 0. Show that zt-+ Uzis an isometric linear mapping of I" onto a subspace of 9 ( E ; E), and use the Problem of Section 5.10.)

8. Let E be a separable Hilbert space of infinite dimension. The strong topology on P(E; E) is that obtained by the procedure of Problem 6 by taking (5 to be the set of all finite subsets of E, so that this topology is defined by the seminorms p,(U> = llU. x(I for all x E E. The strong topology is coarser than the norm topology.

(a) Show that the strong topology is not metrizable. (Observe that 9 ( E ; E) contains vector subspaces which, when given the topology induced by the strong topology, are isomorphic to E endowed with the weak topology.) (b) The unit ball V of 9 ( E ; E), endowed with the strong topology, is separable and metrizable. (Proceed as in (12.5.7), by embedding V in EN.) (c) Show that the mapping (U,V ) w UV is continuous in the strong topology on Q x 9;p(E;E) and not continuous on 9(E: F,)X Q. (d) Show that the mapping UHU' of V into V is not continuous with respect to the strong topology. (Let (en)nzObe an orthonormal basis of E, and consider the sequence of operators uk such that uk * e, = 0 if n < k and U k. e . = en-k if n 2 k . ) (e) Let U(E) c V be the unirury group of E, consisting of all linear isometries of E onto E. Show that the topology induced on U(E) by the strong topology is compatible with the group structure. Give an example in this group of a right Cauchy sequence which is not a left Cauchy sequence. Show that U(E) is closed in V. 9.

Let E be a separable Hilbert space of infinite dimension. The weak ropology on Y(E; E) is that obtained by the procedure of Problem 6 by takingG to be the set of

16 BAIRE'S THEOREM A N D ITS CONSEQUENCES

81

finite subsets of E, but taking E with the weak topology, so that this topology is defined by the seminorms p x ,,(U)= I (U . x I y ) 1 for all x , y in E. This topology is coarser than the strong topology. (a) Show that the weak topology is not metrizable (same methodas in Problem 8fa)). Ib) The unit ball Q in Y(E; E) is metrizable and compact in the weak topology (method of (12.5.7)). (c) The bounded sets in 9 ( E ; E) are the same for the weak topology as for the strong topology. (d) The mapping U H U * of Y(E; E) into itself is continuous with respect to the weak topology. (e) Show that the mapping (U, V ) 4 U V of V x V into V is not continuous for the weak topology. (IF is a Hilbert basis of E indexed by Z, consider the "shift operator" U defined by U .en = en+l for all n E Z, and its powers U" and U+.) The topology induced on the unitary group U(E) by the weak topology is identical with that induced by the strong topology, and hence is compatible with the group structure. Also U(E) is not closed in G9 for the weak topology. (f) Show that, for each Uo f Y(E; E), the linear mappings VH Uo V and V++ VUO are continuous on Y(E; E) for the weak topology. (9) Let uly be a submonoid of V (i.e., uly is stable under multiplication). Show that the closure of uly with respect to the weak topology on 9 ( E ; E) is a compact monoid, which is commutative if uly is commutative. 10. The notation is the same as in Problem 9.

(a) Let U E Q. Show that the relations U * x = x , (x [ (I.x ) = (x [ x), and U* . x = x are equivalent. (b) Deduce that the idempotent operators in are the orthogonal projections (Sections 6.3 and 11.5.)

11. Let E be a separable Hilbert space, A a submonoid containing the identity l E which is weakly closed (and therefore weakly compact) in V (notation of Problem 8). For each x E E, the orbit of x under A is the set .I. n of vectors U * x where U E 4; it is weakly compact in E. We say that x is a flight vector with respect to uly if 0 E uly * x, and that x is a reversibfe vector with respect to &if for each U EX the re exists V E X such that VU . x = x . Let F(uly) (resp. R(uly)) denote the set of flight (resp. reversible) vectors with respect to uly. If x is reversible, then IIU .xII = llxll

for all U E A. (a) Every orbit uly * x contains a minimal orbit N (Section 12.10, Problem 6), and every y E N is reversible with respect to uly. Let U E uly be such that U . x = y E N, and let d be the weakly closed submonoid generated by l E and U : this submonoid d is Commutative. Let P C d . y c N be a minimal orbit with respect to d , and let V E d be such that V .y = z E P. Show that there exists W E d such that VU W . z = I;deduce that the vector x - W . z is a flight vector, and hence that every x E E is the sum of a flight vector and a reversible vector belonging to A * x . (b) Show that if x E E is reversible with respect to X then x is also reversible with respect to every weakly closed submonoid M . (Split up x into the sum of a vector y E R( N)belonging to N . x and a vector z E F(N);use the fact that if a , b are two b)ll lim infl/x.-all.

”-.

0)

16. BAIRE’S THEOREM A N D ITS CONSEQUENCES

(Baire’s theorem) Let E be a topological space in which eoery (12.16.1) point has a neighborhood homeomorphic to a complete metric space. If(U,) is a sequence of dense open sets in E, then the intersection of the U, is dense in E. It is enough to prove that, for each x E E and each neighborhood V of x, the intersection of V and the U, is not empty. Hence we may assume that E itself is a complete metric space and (bearing in mind (3.14.5)) prove that the intersection G of the U,, is nonempty. Let d be a distance defining the topology of E, with respect to which E is complete. We shall define by induction on n a sequence (x,) of points of E and a sequence (r,) of real numbers > O , as follows: x1 E U1; I , < l / n for each n >= 1 ; the closed ball B’(x,,; r,) is contained in U, n B’(xnWl;r,,-l). This is possible because U, n B(xn-l; r,,-l) is a nonempty open set in E, since U, is dense in E. Clearly we have d(x,, x,+,,) r, < l / n for each n 2 1 and each p > 0, and therefore (x,) is a Cauchy sequence in E. By hypothesis it converges to a point a E E, and since x,+,, E B’(x,; r,) for each p > 0, we have a E B’(x,; I,,) c U, for each n since B‘(x,; r,) is closed in E. The point a therefore belongs to G . Q.E.D. We shall apply Baire’s theorem mainly when E is an open subspace of a complete metric space or a locally compact rnetrizable space (by virtue of (3.16.1)). In a topological space E, a subset A is said to be nowhere dense if the open set E - i% is dense (or, equivalently, if A contains no nonempty open set, or if has empty interior). For example, in a Hausdorff space E, a set { a } consisting

84

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

of a single point is nowhere dense unless it is open, i.e., unless a is isolaied in E (3.10.10). In a topological group G, every closed subgroup H which is not open is nowhere dense (12.8.7). In a topological space E, a subset B is said to be meager if it is the union of at most denumerably many nowhere-dense sets. Since the closure of a nowheredense set is nowhere dense by definition, Baire's theorem can be restated in the form that, in a space E satisfying the conditions of (12.16.1), the complement of a meager set is dense in E. For example, in a Frtchet space E, a denumerable union of closed linear varieties, each distinct from E, i.e., sets of the form a, + V,, where V, # E is a closed vector subspace of E, has a dense complement in E. For it follows from (12.13.1) that a vector subspace V of a topological vector space E cannot be open in E unless V = E. In particular, a denumerable set in a Frtchet space is meager, but it should be noted that such a set may also be dense (if the space is separable). (12.16.2) Let E be a topological space in which every point has a neighborhood homeomorphic to a complete metric space, and let u be a lower semicontinuous function on E. Ifu(x) < + 00 for each x E E , then given any nonempty open set U in E there exists a nonempty open set V c U such that sup u(x) < 00. xov

+

It is enough to prove the result when U = E. For each integer n > 0, let F, be the set of all x E E such that u(x) 5 n. By hypothesis, F, is a closed set (12.7.2), and E is the union of the F,; hence at least one of the F, is not nowhere dense (12.16.1), and therefore contains a nonempty open set. Q.E.D. Notice that under the hypotheses of (12.16.2) it can happen that sup u(x) = xsE

+ 00.

Consider for example the real-valued function on R which is equal to 0 at x = 0, and equal to 1/x2when x # 0. In particular: (12.1 6.3) In a Fre'chet space E , every lower semicontinuous seminorm is continuous.

By definition, a seminorm p on E is finite at all points of E. If p is lower semicontinuous, it follows from (12.16.2) that there exists a point x,, E E, a

16 BAIRE’S THEOREM AND ITS CONSEQUENCES

85

neighborhood V of 0 in E, and a real number c > 0 such that p(x) 5 c for all

x E xo + V. Hence

P(Z)

for all z

E V,

5P b O )

+ P(X0 + z ) 2 c + P ( X 0 )

which proves the result (12.14.2).

(Banach-Steinhaus theorem) Let E be a Frkchet space, F a (12.16.4) normed space. Let H be a set of continuous linear mappings of E into F. Suppose that sup IIu(x)II < + co for all x E E . Then H is equicontinuous. ueH

The function p(x) = sup IIu(x)II, being finite for all x E E, is a seminorm usH

on E (12.14.1). Since each of the functions X H IIu(x)I[ is continuous, p is lower semicontinuous (12.7.7). The result therefore follows from (12.1 6.3) and (12.15.7.1).

As a consequence, we have: (12.16.5) (i) Under the hypotheses of (12.16.4), let (u,) be a sequence of continuous linear mappings of E into F which converges simply in E to a mapping u of E into F. Then u is a continuous linear mapping of E into F. (ii) More generally, let Z be a metric space, A a subset of Z , and z w u, a mappie3 of A into the space of continuous linear mappings of E into F. Let zo E Z be a point in the closure of A, and suppose that the limit

lim u,(x) = v(x)

z-+zo,zeA

exists in F, for each x E E. Then v is a continuous linear mapping of E into F .

(i) From the hypotheses we have sup IIu,(x)II < +co for all x n

E

E, hence

the Banach-Steinhaus theorem shows that the sequence (u,) is equicontinuous, and the continuity of u follows from (7.5.5). The fact that u is linear is an immediate consequence of the principle of extension of identities. (ii) The point zo is the limit of a sequence of points (z,) in A (3.13.13), hence u(x) = lim U J X ) for all x E E. Now apply (i). n-tm

(12.16.6) Let E be a Frkchet space, F a Banach space, I an open interval in R. Let Y ( E ; F ) denote the space of continuous linear mappings of E into F , endowed with the topology of simple convergence (12.15). I f a mapping t t - f , of I into Y(E; F ) is diferentiable with respect to the topology of simple convergence (12.15), then there exists a mapping t H f j of I into 2 ( E ; F ) such that f ; ( x ) = D(f,(x))for all x E E.

86

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

For D(f,(x)) is the limit, as h # 0 tends to 0, of the mapping

h + + ( f r + h -fr)lh, and the result therefore follows from (12.16.5). The mapping f iois said to be the derivative at the point to of the mapping t w f , (with respect to the topology of simple convergence), or the weak derivative when F is the field of scalars (and therefore Y(E; F) = E’). Remark

(12.16.6.1) Let A be an open set in C and let Z H ~ , be a weakly analytic mapping (12.15) of A into the dual E’ of a complex FrCchet space F. Then the same argument as in (12.16.6) shows that there exists a weak derivative 2-f: of Z H ~ ,, with fL(x) = D(f,(x)) for all x E E, and this weak derivative is weakly analytic. Moreover, for each a E A and each circuit y contained in A - { a } , we have the Cauchy formula (12.16.6.2) Conversely, let y be a road in C defined in an interval I = [b, c ] in R,and let z H g , be a weakly continuous mapping of y(1) into E’. Then, for z # y(I), the mapping

(12.16.6.3) is a linear formf, on E, belonging to E’. Indeed, for each x E E the right-hand side of (12.16.6.3) is the limit of a sequence

where

((3.16.5) and (8.7.8)),hence our assertion is a consequence of the BanachSteinhaus theorem. By virtue of (9.9.2),f, is weakly analytic in C - y(I), and for a 4 y(1) we can write (12.16.6.4)

m

f Z ( 4

=

1 cn(x)(z - a)”,

n=O

the series being convergent in any disk with center a not meeting y(1). The

16 BAIRE’S THEOREM AND ITS CONSEQUENCES

87

coefficients c,, E E’ are given by

Applying this result to the situation where y is a circuit t H a + reit (0 5 t 5 2n) in A - { a } we obtain, for any weakly analytic function ZI+L on A, the Taylor expansion (12.16.6.5)

convergent in the disk ) z - a1 < r (independent of z E E), and the derivatives are given by (12.16.6.6)

More generally, suppose that in A - { a } we have (12.16.6.7)

where the x: are linear forms on E (a priori, not necessarily continuous), and zHg, is a weakly analytic function on A - { a } which is weakly bounded in a neighborhood of a. Then, by (9.15), for each x E E, the function zt-tg,(x) can be extended by continuity to the point a, and the extended function is analytic in A. Hence there exists a weakly analytic function z H h, in A whose restriction to A - ( a } is zI+g,. Furthermore, the x: are continuous linear forms on E (i.e., they are elements of E’). Indeed, for each x E E we have ( Z - u ) ~ - ’ ~ = ( dz x)

where y is a circuit tt+a + rei‘ (0 5 t 2n) with r sufficiently small and independent of x (9.14); our assertion therefore follows from the considerations above (12.16.6.3). Finally, we remark that the principle of analytic continuation (9.4.2) remains valid for weakly analytic functions. This is an immediate consequence of the definitions. (12.16.7) Let E be a Hilbert space and u an endomorphism of the (nontopological) vector space E. Then the following three conditions are equivalent: (a) u is continuous; (b) u is weakly continuous; (c) u has an adjoint (11S).

88

XI1 TOPOLOGY A N D TOPOLOGICAL ALGEBRA

We have already seen (12.15.11) that (a) implies (b). If u is weakly continuous, then for all y E E the linear form X H (u(x) I y ) is weakly continuous, and therefore a fortiori continuous with respect to the strong topology, which is finer than the weak topology. Consequently (6.3.2), for each y E E there exists a unique point u*(y) such that (u(x) I y) = (x I u*(y)) for all x E E: in other words (11.5), u has an adjoint, and therefore (b) implies (c). Finally, if u has an adjoint, then we have IIU*(Y)ll = SUP

llxll 5 1

IbIU*(Y))l

= SUP

llxll 6 1

I(u(x)lv)l,

and each of the linear forms y ~ ( I u(x)) y = (u(x) l y ) is continuous, hence

YH IIu*(y)(l is a lower semicontinuous seminorm on E, and is therefore continuous in the strong topology (12.1 6.3). But this is equivalent to saying that

u* is (strongly) continuous, and therefore so is u = u** (11.5.2).

We remark that it is essential for the truth of (1 2.16.4), (12.16.5), (12.1 6.6), and (12.16.7) that the space E should be complete (cf. Problems 21 and 22). (12.1 6.8) (Banach's theorem) Let E, F be two Fre'chet spaces and let u be a continuous linear mapping of E into F. Then either u(E) is meager in F, or else u(E) = F. In the latter case, if N is the kernel of u and E --+ E/N & F the canonical factorization of u, the mapping v is an isomorphism of the Frtchet space E/N (12.14.9) onto F (in other words (12.12.7), u is a strict morphism of E onto F).

Suppose that u(E) is not meager in F. By virtue of (12.13.1) and (12.12.7), the theorem will be proved if we can show that, for each neighborhood V of 0 in E, u(V) is a neighborhood of 0 in F. There are two steps to the proof. (12.16.8.1) Let E, F be two topological vector spaces, and u a linear mapping of E into F such that u(E) is not meager in F. Thenfor each neighborhood V of 0 in E, the closure u(V) of the image of V is a neighborhood of 0 in F.

Let W be a balanced neighborhood of 0 in E such that W + W c V ((12.13.1) and (12.8.3)). Then for each x E E there exists an integer n 2 1 such that x E nW. Consequently u(E) is the union of the sets u(nW) = n u(W). Since u(E) is not meager, at least one of the sets n * u(W) has an interior point, and therefore so does u(W). But since __ -u(W) = u(W), we have -~ - u(W) = u(W). So if yo is an interior point of u(W), so is - y o , and -then therefore 0 = yo ( y o ) is an interior point of u(W) u(W) (12.8.2). But _ _ u(W) u(W) is contained in theclosure of u(W) + u(W) = u(W W ) c u(V), and so (12.16.8.1) is proved.

+

+

+

+

16

BAIRE'S THEOREM A N D ITS CONSEQUENCES

89

Now suppose that E, F are equipped with translation-invariant distances d, d , respectively, compatible with their topologies (12.9.1). By (12.16.8.1), the hypothesis that u(E) is not meager in F implies that, for each r > 0, there exists p = p(r) > 0 such that B'(0; p) c u(B'(0; r)). By translation it follows that for each x E E we have B'(u(x); p ) c u(B'(x; r)). Hence it is enough to prove the following lemma: (12.16.8.2) Let E be a complete metric space, F a metric space, and u a continuous mapping of E into F with the following property: for each r > 0 there exists p = p(r) > 0 such that B'(u(x);p ) c u(B'(x;r ) ) for all x E E. Then B'(u(x);p ) c u(B'(x; 2r))for all x E E.

For each integer n 2 1, there exists by hypothesis a number p, > 0 such that B'(u(x); p,) c u(B'(x; 2-"+'r)) for all x E E. We may take p1 = p and (replacing p, if necessary by inf(p, , 2 - 7 ) assume that lirn p, = 0. Let xo be n+m

any point of E, and let y E B'(u(x,); p). We shall prove that y E u(B'(x,; 2r)). For this we define inductively a sequence (x,),,, of points of E such that, for each n 2 1, we have x, E B'(x,,-,; 2 - " + l r ) and u(x,) E B'(y; pn+,). Suppose that xl, . . . , x,-, have been chosen to satisfy these conditions. Then we have y E B'(u(x,,-,);p,), and since B'(u(x,,-,); p,) c u(B'(x,-,; 2-"+'r)), there exists a point x, E B ' ( X , , - ~2; - " + ' r ) such that u(x,,) E B'(y; p,+J. So the inductive construction can continue. Now the sequence (x,) is a Cauchy sequence in E, because for each n 2 0 we have + 2-"-p+' r 5- 2-"+'r d(x,, x,,~) 2-"r 2 - " - l r +

s

+

-

for all p > 0. Since E is complete, the sequence converges in E to a point

x such that d(x,, x ) 5 2r. Since u is continuous, we have u(x) = lirn u(x,,);

and since d'(u(x,), y )

s pnil,

it follows that lim u(x,) = y . n+m

n-+m

Q.E.D.

This theorem has the following corollaries : (12.1 6.9) Let E, F be two Frkchet spaces. Then every continuous bijective linear mapping of E onto F is an isomorphism.

In this proposition it is essential to assume that both E and F are complete. For example, if we take E = I; (6.5) and F to be the canonical image of E under the identity mapping E-tBR(N) (7.1), endowed with the topology induced by that of gR(N), then this mapping is continuous because

90

XI1 TOPOLOGY AND TOPOLOGICAL ALGEBRA

C 4; 2 sup (,2 n

n

(5.5.1),but the inverse bijection of F onto E is not continuous:

otherwise F would be complete, and therefore closed in g,(N) (3.14.4),which is absurd, because its closure in gIR(N)is the set of all sequences (5,) such that lim 4, = 0. n-, m

(1 2.16.10) Let E, F be two Banach spaces, and u a continuous surjective linear mapping of E to F. Then there exists a number m > 0 such that for each x E E there exists x' E E,for which u(x) = u(x') and IIu(x)II 2 m * IIx'II. Bearing in mind (12.14.8)and (5.5.1),this expresses that the bijection F .+ E/u-'(O), the inverse of the bijection E/u-'(O) F induced by u, is continuous on F. -+

(12.16.11) (Closed graph theorem) Let E, F be two Frtchet spaces. For a linear mapping u of E into F to be continuous, it is necessary and sufficient that the graph (1.4)of u should be closed in the product space E x F. In general, iff is a continuous mapping of a topological space X into a Hausdorff topological space Y, the graph offis closed in X x Y, because it is the set of all z E X x Y satisfying the relation pr, z =f(prlz), and theassertion follows from (12.3.5).To show that the condition stated in (12.16.11)is sufficient, remark that it implies that the graph G of u, being a closed vector subspace of the FrCchet space E x F (3.20.16(iv)),is a FrCchet space (3.14.5). The projection z w p r l z of G onto E is therefore a continuous bijective linear mapping, hence an isomorphism (12.16.9). Since the inverse mapping is u : X H ( X , u(x)), it follows that xt-,u(x) = pr,(u(x)) is continuous on E. The condition of (12.16.11)may also be expressed by saying that if a sequence (x,, u(x,)) in E x F tends to a point (x, y ) , then y = u(x). Replacing x, by x, - x, and using the linearity of u, an equivalent formulation is that i f a sequence (x,) in E tends to 0 and is such that the sequence ( ~ ( x , )tends ) to a limit y , then y = 0. It is this criterion that we shall apply in practice to verify the continuity of u. Finally, the following consequence of Baire's theorem allows us to use the criterion (12.11.5):

(12.16.12) Let G be a separable, metrizable, locally compact group, acting continuously and transitively on a Hausdorff topological space E in which every point has a neighborhood homeomorphic to a complete metric space. For each x E E, let S, be the stabilizer of x. Then the canonical bijectionf, : G/S, -+ E is a homeomorphism.

16 BAIRE’S THEOREM A N D ITS CONSEQUENCES

91

Let xo E E. We have to show that, for each neighborhood V of e in G, the set V xo is a neighborhood of x, in E (12.11.5). Let W be a compact symmetric neighborhood of e in G such that W2 c V, and let (s,) be a sequence which is dense in G, such that (s,,W) is a covering of G. Each of the sets s,,W . x, is closed in E, because st+s x, is continuous (12.3.61, and E is the union of the denumerable sequence of closed sets s, W xo . By Baire’s theorem (12.16.1), there exists an index n such that s,,W * xo has an interior point s,s xo, with s E W. It follows (12.10.3) that x, is an interior point of the set a

s-’s,’(s,W.

xo) = s-’w* xo

in other words, V * x, is a neighborhood of xo .

c

v . x,; Q.E.D.

(12.16.13) Let G be a separable, metrizable, locally compact group, let G’ be a metrizable group, and let f : G -+ G’ be a continuous surjective homomorphism. Then f is a sfrict morphism (12.12.7) (in other words, if H is the kernel off; then the canonical bijection g : G/H G’ is an isomorphism of topological groups). --f

For we may consider G’ as a space on which G acts continuously and transitively by the rule (s, t ’ )f (~s ) f ’ and , the stabilizer of the neutral element e‘ of G’ is H. The result therefore follows from (12.16.12).

PROBLEMS 1. Let E, F be two metric spaces, A a dense subspace of E, and f a continuous mapping of A into F. If F is complete, show that the set of points in E at whichfhas no limit relative to A (3.1 3 ) is meager in E. (For each n , consider the set of points x E E at which the oscillation offwith respect to A (3.14) is > l/n.) 2.

Let E, F be two complete metric spaces, and f a homeomorphism of a dense subspace A of E onto a dense subspace B of F. Show that there exists a subspace C 3 A in E (resp. D 3 B in F) which is a denumerable intersection of open sets, and an extension offto a homeomorphism g of C onto D. (Apply Problem 1 to f and its inverse.)

3. Let E be a complete metric space, F a metric space, and (fn) a sequence of continuous mappings of E into F which converges simply (pointwise) in E to a mapping f. Show that the set of points x E E at which f is not continuous is meager in E. (Let G p ,,, be the set of x E E for which the distance between f,(x) and &(x) is 5 1/2n for all q z p . Show that the union of the interiors of the sets G,, for p 2 1 is a dense open set, by using Baire’s theorem. Deduce that the set of points at which the oscillation of f’is 0, let A. be the set of points x E [a, b[ such that thereexist pointsy, z E [a, h[ satisfyingx < y < z < x l / n and

+

Show that A. is a dense open subset of [a, b[.)

16 BAIRE'S THEOREM AND ITS CONSEQUENCES

97

28. Let f b e an indefinitely differentiable real-valued function defined on an open interval ]u,b[ of R.Suppose that for each point x E Ju,b[ there exists an integer N(x) such that DN("'f(x)= 0. Show that f i s a polynomial. (One may proceed as follows: (a) Let G be the open set of points x E ]a, b[ such that in some neighborhood of x the function fcoincides with a polynomial, and let F be the complement of G in ]u,b[. Show that F has no isolated points (3.10.10). (b) For each integer n, let En be the closed subset of F consisting of the points x E F such that D"f(x)= 0. If F is not empty, show that there exists a nonempty open interval 1 C ]a, b[ and an integer N such that F n I is nonempty and is contained in EN (use Baire's theorem). Then deduce from (a) that F n I c E, for all n > N. (c) Deduce from (b) that F n I is nowhere dense in I, and then that DNf(x)= 0 on every component interval of G n I. Hence derive a contradiction of the hypothesis F f 121.) 29. Let E, F be two separable Banach spaces, F' the dual of F. Suppose that F is contained

in a Hausdorff locally convex space G , and that the topology on F induced by that of G is coarser than the weak topology of F . Let u : E -+ G be a continuous linear mapping. (a) Show that, for every ball B in the Banach space F', the inverse image u-'(B) is a closed subset of E. (Use (12.15.9) and (12.15.8.1).) (b) Suppose that there exists a nonmeager subset A of E such that u(x) E F' for all x E A. Show that u(E) c F' and that u is continuous for the topology defined by the norm on F'.

30. Let E be a Frechet space, and let (u.) be a family of continuous linear mappings of E into a normed space F. Suppose that there exists a nonmeager subset A of E such that for each x E A the set of points tl.(x) is bounded in F. Show that the family (u,) is equicontinuous. (For each integer n 1, consider the set of points x E E such that l/u,(x)lI5 n for each index a,)

CHAPTER XI11

INTEGRATION

The theory of integration which we shall develop in this chapter is restricted to separable metrizable locally compact spaces, this being sufficient for our purposes in later chapters. We have followed fairly closely the exposition of N. Bourbaki [22], with the simplifications afforded by our more restricted hypotheses. The key results of the theory of integration are Lebesgue’s convergence theorems (1 3.8), the Fischer-Riesz theorem (13.11.4), the Lebesgue-Nikodym theorem (13.1 5.5) and the Lebesgue-Fubini theorem (1 3.21.7). Unfortunately it is necessary to include rather a lot of material on upper integrals, measurable functions and negligible functions, which are indispensable technical tools. The important properties of certain particular measures on locally compact groups or on differential manifolds will be examined in Chapters XIV and XVI. We have also included, amongst the problems, applications of integration which are not dealt with in the text, especially to ergodic theory and orthogonal systems. The reader who wishes to go further in these directions should consult [21], [26a], [28], [30], and [31]. Nowadays the purposes of a theory of integration are very different from what they were at the beginning of this century. If the aim was only to be able to integrate “ very discontinuous ” functions, integration would hardly have gone beyond the rather narrow confines of the “ fine ” theory of functions of one or more real variables. The reasons for the importance that Lebesgue’s concept of integral has acquired in modern analysis are of quite a different nature. One is that it leads naturally to the consideration of various new complete function spaces, which can be conveniently handled precisely because they are spaces offunctions (or of classes of “equivalent ” functions) and not just abstract objects, as is usually the case when one constructs the completion of a space. Another is that the theorem of Lebesgue-Nikodym 98

1 DEFINITION OF A MEASURE

99

and the properties of measures defined by densities (13.15) give us a method for dealing with denumerable families of measures on a locally compact space, by fixing a basic measure and working with the densities relative to this basic measure (hence again withfunctions); this again proves to be extremely convenient. Here the modern point of view emerges: given a p-integrable function5 what is important is not the values taken byfso much as the way in whichf operates on the space of bounded continuous functions by means of the linear mapping g w fg dp (this mapping depends only on the equivalence class off and therefore does not change when we modifyfat the points of a set of measure zero). The development of this point of view will lead in Chapter XVII to the theory of distributions, which is a natural generalization of the notion of measure on differential manifolds. Throughout this chapter, the phrase “ locally compact space ” will always mean ‘‘ separable metrizable locally compact space.” 1. DEFINITION OF A MEASURE

To begin with, let X be a compact (metrizable) space. A measure (or complex measure) on X is by definition an element of the dual of the Banach space gc(X) of complex-vaiued continuous functions on X (7.2),that is to say ((12.1 5) and (5.5.1 )) it is a linear form f tt p(f) on VC(X)which satisfies an inequality of the form (13.1.1)

for a l l y € %,(X) (recall that

Icl(f)l 2 a I l f

llfll

= sup

II

1f ( x ) l ) .

XEX

Now let X be a locally compact space (metrizable and separable, in accordance with our conventions). For every compact subset K of X, let X ( X ; K) (or X,(X; K)) denote the vector subspace of g,-(X) consisting of the functions whose support (12.6) is contained in K (and is thereforecompact). We shall denote by X,(X) (or X ( X ) )the union of the X c ( X ;K ) as K runs through all compact subsets of X. In other words, X,(X) is the vector space of (complex-valued) continuous functions with compact support. Clearly .X,(X) = V:cm(X). A measure (or complex measure) on X is by definition a linear form p on X,(X) with the following property: for each compact subset K of X, there exists a real number aK >= 0 (in general depending on K) such that (13.1.2)

for allfE X ( X ;K).

Ip(f)I

5 OK l l f l l

XI11 INTEGRATION

100

This definition agrees with the preceding one when X is compact. It expresses that the restriction of p to X ( X ;K) is continuous with respect to the topology induced by that of %?,?(X).We remark that X ( X ; K) is closed in %?F(X),and therefore a Banach space (3.14.5). I n general, a measure is not necessarily continuous on X,(X) with respect to the topology induced by that of %?F(X)(i.e., the topology defined by the norm 11 f 11). We shall examine this question later (13.20). Examples of Measures (13.1.3) Let X be a locally compact space, and let X E X .The mapping f t +f ( x ) of X ( X )into C is a measure, for it is linear and we have If(x)l 5 i l f l l for each compact subset K of X such that f E X ( X ; K). This measure is called the Dirac measure at the point x , or the measure defined by the unit mass at thepoint x , and is denoted by E,. More generally, let (a,) be a sequence of distinct points in X , and (t,,) a sequence of complex numbers, such that for each compact K c X the subsequence formed by the t,, f o r which a, E K is absolutely summable (5.3). Let cK = ItJ. Then, for each function f E X ( X ;K), the series t,,f(a,) is a,

E

1

K

n

absolutely convergent, because the only nonzero terms are those for which a, E K, and we have

C ItnSta,JI S IfII C = CK . IISIIThis also shows that f H C t,,f (a,) is a measure on X . This measure is said to *

U"EK

a,eK

tn

n

be defined by the masses t, at the points a,, for all n (cf. (13.18.8)). (13.1.4) Let f~ X c ( R ) . For each interval [a, b ] containing the support off, the value of the integral Jabf(t)dt (8.7) is the same, and we denote it by

s'^

f ( t ) dt. The mappingfwJ+OOf(t) -00 dt is a linear form on XX,(R),and it is a measure because, for each compact interval K = [a, b ] in R and each functionfE X ( R ; K) we have -m

by the mean value theorem (8.7.7). This measure is called Lebesgue measure on the real line R. (1 3.1.5) Let p be a measure on X and let g E %',(X). Then for each function f E X ( X ) it is clear that g f E , X ( X ) , and the mapping f w p ( g f ) is therefore

1 DEFINITION OF A MEASURE

101

a h e a r form on X(X). It is a measure, for if K is any compact subset of X and i f f E X ( X ;K), then IIgfII g 11 f 11 sup Ig(x)l, and consequently

-

XEK

Ip(gf)l 5 bK (1 f 11, where bK = aK sup Ig(x)l. This measure is denoted by g * p, X€K

and it is called the measure with density g relative to p (cf. (1 3.13)). (13.1.6) Let n : X + X' be a homeomorphism of X onto a locally compact space X'. For each functionfE X(X'),the function f 0 n belongs to X(X), and we have Supp(f0 n) = n-'(Supp f). It follows immediately that, if p is any measure on X, then f Hp( f 0 n) is a measure on X', called the image of p under n and denoted by n(p). (13.1.7) Let Y be a closed subset of X (and therefore a locally compact subspace of X (3.18.4)) and v a measure on Y. For each f E X ( X ; K), the restriction f I Y belongs to X ( Y ; K n Y), and hence there exists a constant CK such that If(y)l 5 c K l l f l l Iv(flY>I5 cK ' YE

Y

for all f E X ( X ;K). The mapping f Hv( f I Y) is therefore a measure on X, called the image of v under the canonical injection Y + X, or the canonical extension of v to X. (1 3.1.8) Let U be an open subset of X (and therefore again a locally compact subspace (3.18.4)). For each compact subset K of U, it is clear that the mappingft+ f I U is an isometry of X ( X ;K) onto X ( U ; K). The image under the inverse isometry of a function g E X ( U ; K) is the function g' which agrees with g on U and is zero on X - U. (By abuse of notation, we shall often write f in place off/ U when Supp(f) c U, and g in place of g'). The mapping g w g U of Xc(U) into X,(X) is therefore injective. If p is any measure on X , the mapping g H p(g') is a measure on U, said to be induced by p on U, or the restriction of p to U, and denoted by pu or p I U. It should be noted that a measure v on U is not necessarily the restriction of a measure on X (Section (13.4), Problem l), and that an " extension " of v to X, if it exists, is not in general unique. However, there is the following result: (13.1.9) Let (U,),€, be an open covering of X. For each a E I let pa be a measure on U, such that, for each pair of indices a, p, the restrictions of pa and p, to U, n U, (1 3.1.8) are equal. 'Then there exists a unique measure p on X whose restriction to U, is p a , for each a E 1.

We shall first show that each function , f X,(X) ~ can be written in the n

form f = cfiwhere, for each index i, there exists ai E I such that f i E Xc(X) i= 1

102

Xlll INTEGRATION

and Supp(fi)c U,, . For this purpose we observe that if K is the support of f, then there exist finitely many indices cli E I (1 5 i 5 n ) such that the U,, cover K. Hence ((3.18.2) and (12.6.4)) there exist n continuous mappings hi : X -+ [0, I] such that Supp(hi) is compact and is contained in U,, for 1 5 i 5 n, and such that

n

h,(x) = 1 for all x E K. Then the functions f i =f hi

i=1

satisfy the required conditions. This already proves the uniqueness of p: for by definition we must have

~ ( f=)i= C1pU(fhi)= i2= 1pa,Uhi). n

To prove the existence of a linear form p on X,(X) whose restriction to X,.(U,) is p, for each a E I, it is enough to establish the following assertion: given two finite sequences (gJl s i j m and j g n of functions belonging to X,(X), such that Supp(g,) c U,, for 1 5 i 5 m and Supp(hj) c U,, for 1 S j S n, and such that

for all x E Supp(f),then we have

Now, we have

and therefore

Similarly,

But since Supp(fgihj) is contained in U,, n U,,, it follows from the hypotheses that pu,(f g i hi) = ppj(fgihj),and our assertion follows. It remains to be shown that the linear form p so defined is a measure. Let K be a compact subset of X, and define the U,, and the hi as at the beginning of the proof. If H i = Supp(hi), then by hypothesis there exists a

2 REAL MEASURES

number a, 2 0 such that lpai(g)l 5 a i llgll for all g Hence, for each function f E X ( X ;K), we have Ipai(.fhi>l5 aiiifhiil

so that

I ~ W5I

E

.X(X; Hi) (13.1.2).

5 aiilfil,

(,faijli/ii* I=

103

Q.E.D.

1

(13.1.10) If I and p are two measures on X, then so are A + p and a l for any scalar a E C. The set of all measures on X is therefore a uecior subspace of CXC('), which we denote by M,(X) or M(X). By analogy with the example (13.1.4), if p is a measure on a locally com-

s

s

pact space X, we write f dp or f ( x ) dp(x) (or also (f, p ) or ( p , f ) ) in place of p[f), for any f E X ( X ) , and we call this number the integral o f f with respect to p. 2. REAL MEASURES

Let X be a locally compact space. Let X , ( X ) denote the set of all realualued continuous functions on X with compact support, and X,(X; K) the set of those whose support is contained in K. Clearly XR(X)is a real vector subspace of X,(X), and we can write YC(x)=

x,(x) @ iXR(x)

(direct sum). For every (complex) measure p , the restriction of p to XR(X) is an R-linear mapping p o of X R ( X ) into C; moreover p,, determines p uniquely, for i f f = f l ifz with f i t f z in .X,(X),then p ( f ) = p o ( f i ) ipo(fz). Conversely, if an R-linear mapping p o : X,(X) + C is such that, for each compact subset K of X, there exists aK > 0 with the property that Ipo(f)l 5 uK llfll for allfE X , ( X ; K), then it is immediately obvious that the mapping

+

+

fi + $2

HP O ( f 1 )

+ iPO(f2)

is a (complex) measure on X. Hence we may identify each measure on X with its restriction to .X,(X).

Let p be a (complex) measure on X . It follows immediately from (13.1.2) that the m a p p i n g f t + X ) is also a measure on X, called the conjugate of p and denoted by ji. We have p = / i , and if I , .iiare measures o n X and a, b are any two complex numbers, then aI = bp = ZX hii. More generally, if g is any function belonging to %,(X) and p is any measure on X, then we have F p = S ji (13.1.5).

+

-

104

Xlll INTEGRATION

A measure p on X is said to be real if ji = p, or equivalently if p ( f ) is real for everyfe X,(X). We may therefore identify the set of real measures on X with a vector space of linear forms on the real vector space X,(X). This vector space is denoted by MR(X). Lebesgue measure and all Dirac measures are real. If p is any complex measure, then the measures p1 = ( p + ji)/2 and p 2 = ( p - F)/2i are real. They are called respectively the real and imaginary parts of p, and are denoted by 92p and Yp respectively. For each function f E X,(X), we have

and by definition (13.2.2)

p

= 92p

+ iYp,

ji

= 9?p - iYp.

3. POSITIVE MEASURES: T H E ABSOLUTE V A L U E O F A MEASURE

A measure p on a locally compact space X is said to be positive if, for each function f~ ,X,(X) such that f 2 0, we have p(f) 2 0. Consequently, i f f and g are two functions belonging to X,(X) such that f 5 g, we have p ( f ) 5 p(g). Since each f E X,(X) can be written in the form f = f + -f (where f + ( x ) = ( f (x))' and f - ( x ) = ( f ( x ) ) - (2.2)), it follows that a positive measure is a real measure. We denote by M+(X) the set of all positive measures on X. Surprisingly, the property of positivity alone implies the defining property (13.1.2) of a measure: (13.3.1) p(f)

Let p be a linear form on the real vector space X,(X) such that is a (positioe) measlire on X.

2 0 whenever f 2 0 . Then p

We have to show that (13.1.2) is satisfied. There exists a function g E X,(X) with values in [0, I ] and equal to I throughout K ((3.18.2) and (4.5.2)). Hence for all f E X,(X; K), we have

0 S . f + 5 ll.fll* g ,

0 5 . r 5 llfll . g

and therefore 0

s A f + )5 llf'll . A g > ,

so that finally

Icc(f'>I S 2 llfll . A g ) .

05 K

- )s I l f II .

3 POSITIVE MEASURES: THE ABSOLUTE VALUE OF A MEASURE

105

The notion of a positive measure enables us to define an order relation on the vector space MR(X) of real measures on X. We write p S v if the measure v - p is positive. Since the relations p 2 0 and p S 0 imply that p ( f ) = p ( f + ) - p ( f - ) = 0 for allfc X,(X), and therefore that p = 0, it follows that p v is indeed an order relation on MR(X)(not, in general, a total ordering). It is clear that p 5 v implies that I + p 5 I + v for all real measures I , and ap =< av for all real scalars a 2 0. (For a study of this order relation, see (13.15).) (13.3.2) Let p be a (complex) measure on X . Then there exists a smallest positive measure p on X such that Ip(f)l 5 p(1fl)for a l l f c X,(X).

For every positive measure v such that Ip(f)l S v(lfl) for all f~ .X,(X), the relations g 2 0 and Ihl 2 g (9,h E ,X,(X)) imply Ip(h)l Iv(lhl) 5 v(g). We shall show that there exists a positive measure p on X such that, for each functionfz 0 in .X,(X), we have (13.3.2.1)

This p will then clearly satisfy the conditions of (13.3.2). To begin with, we remark that the right-hand side of (13.3.2.1) isJinite; for if K = Supp(f), then Supp(g) c K, and Il*(g)l 5 a K ' llgll

5 aK llfll

whenever 1g(s A by virtue of (13.1.2). Also it is clear that, for any real scalar a 2 0, we have p(af) = ap(f). We shall show next that, if f , and f 2 are two functions 2 0 belonging to ,XR(X),then

For each E > 0, there exists g i E X,(X) such that lgil 5 fi and

IAgi>lZ p ( f i ) - E (i = 192).

Multiplying gi by a complex number with absolute value 1, we may assume that p ( g i ) = Ip(gi)l, and then we have P(91

+ g2) = IP(Sl>l + 1AS;)l 2 P(f1) + p(f2)

- 2%

+

and since lgl + g215 f l + f 2 we have p(f, + f 2 ) 2 p(f,) p(f2) - 2 ~ Since . is arbitrary, it follows that p(f,) + p ( f 2 ) 2 p ( f i + f 2 ) . On the other hand, let h E .X,(X) be such that Ihl 5fl + f 2 . Let hi be the function which is equal to hf,/(fi + f 2 ) at the points x where f i ( x ) + f 2 ( x ) # 0 and is zero E

106

Xlll INTEGRATION

elsewhere (i = 1, 2). Then hi is continuous on X, because fi/(fl + f 2 ) is continuous at the points x wheref,(x) +fz(x) > 0, and also Ihi(x)l 2 Ih(x)l for all x E X, which proves that hi is continuous at the points x where f i ( x ) +fz(x)= 0, because h(x) also vanishes at these points. It is clear that / h i /Sfi (i = 1, 2) and that h = h, + h,; hence lP(h>l I lP(M

+ lP(h2)l I P ( f , ) + P(.fz)*

Since Ip(h)l may be taken to be arbitrarily close to p ( f , +f2),it follows that p(fl +fz) I p ( , f l ) p(f,). Hence (13.3.2.2)is proved. We shall now extend the definition of p ( f ) to all functionsf€ X,(X). To do this we write p ( f ) = p(f’) - p ( f ” ) , wheref=f’ -f ” is any decomposition off as the difference of two functions f ’ , f “ 2 0 belonging to X,(X). The value of p ( f ) so obtained is independent of the decomposition, because if f = f ; - f ; = f ; - f i , then f ; +f;’ =f;’ + f ; and therefore p(f;) p(f;) = p(f;’) + p(f;) by virtue of (1 3.3.2.2). With this definition, the formula (13.3.2.2) is valid for allfl,fz in X,(X). For we can writef, =f ; -f ; , f 2 =f ; - f i wherefi’,fi” (i = 1,2) are 20 and belong to X,(X); sincef, + f z = ( f ; + f ; ) - ( f : +f;’), our assertion follows from the definition above and from (13.3.2.2)for functions 2 0 . Finally, the above definition shows that, for each scalar a 2 0, we have p(uf) = up(f ) ; and if a < 0, then we have

+

+

P ( d > = p(af’

- a ! ” )= d - a f ” )

+ p(af’>

- p(-af‘) = ( - M f ”) (- 4 P C f ’ ) = (-a)p(f”)

= ap(f).

Hence the relation p(uf) = ap(f) is valid for all real scalars a, and therefore we have proved that p is a (real) linear form on X,(X). Hence by (13.3.1)it is a positive measure. Q.E.D. The measure p defined by (13.3.2)is called the absolute value of the complex measure p, and is denoted by IpI. Hence by definition we have

It is immediately seen that if a E C and p is a measure on X, we have

(13.3.4)

laPl = I4 - IPl.

If p is a positive measure on X, then

(1 3.3.5)

IPI = P.

3

POSITIVE MEASURES: THE ABSOLUTE VALUE OF A MEASURE

107

By virtue of (13.3.2.1) it is enough to show that Iy(f)l S y(lf1) for all = 1 and Iy(f)l = [y(f) = p ( ( f ) ; since p is real, we have y(Tf) = y(9(cf)); and since y 2 0 and 9(Cf) 5 Ifl, it follows that y(B([f)) 5 y(lfl), which establishes the assertion.

f~ X,-(X).Givenf, there exists [ E C such that l[l

If y is any real measure on X, it follows from (13.3.3) that y 5 lyl, and therefore :

(13.3.6) Every real measure on X is the diference of two positive measures (for a more precise result, see (13.15)). If y is any (complex) measure on X, it follows from (13.3.3) and (13.3.2) that we have

Also if y, v are any two measures on X we have

(13.3.8)

IP +

4 5 IPI + I4

by virtue of (1 3.3.2). Finally, it follows immediately from the definitions that, if n : X -+ X’ is a homeomorphism and y is any measure on X, then

PROBLEMS

1. Let X be a locally compact space, E a vector subspace of eR(X)and P a conuex cone in U,(X) (i.e., a subset of this space such that the relations f~ P and g E P imply f+ g E P and a f P~ for all scalars a > 0). Suppose that for each function h E S R ( X ) there exists f~ E such that f- h E P. Let u be a real linear form on E such that the relation f~ E n P implies u ( f ) 2 0. Let h E XR(X)and let PA be the set of all f~ E such that h - f P, ~ and Pi the set of a l l f s E such thatf- h E P. Show that these two sets PA, Pi are nonempty, and that if a’ is the least upper bound of the u ( f ) with f~ P A and a“ the greatest lower bound of the u ( f ) with f s Pl , then a’ and a” are finite and a’ 5 a”. Deduce that there exists a linear form u1 on the subspace El = E + Rh of VR(X) which extends u, such that the relationf, E El n P implies ul(fi)2 0. Show that a’ 2 u,(h) 6 a# for any such extension u1 of u, and that the extension is unique if and only if a’ = a”.

108 2.

XI11 INTEGRATION

Let X be a compact space and let p be a real-valued function on VR(X) satisfying the following two conditions: (i) p ( f + g ) g p ( f ) + p ( g ) ; (ii) p(uf) = u p ( f ) for all u > 0. Then the set P of functions f~ VR(X) such that p ( f ) 5 0 is a convex cone. Suppose furthermore that (iii) inff(x) s p ( f ) 5 supf(x) for all f~ V,(X), so that we have X E X

X E X

p(l) = 1. Show that there exists a positive measure p of mass 1 on X such that p ( f ) sp ( f ) for allfc VR(X).If moreover we are given a linear form u on a vector subspace E of VR(X) such that u ( f ) ( p ( f ) for all f~ E, then there exists a measure (L of the above type which extends the form u. (Consider a denumerable total set ( g n ) n ) Oin WR(X), with go = 1 ; use the result of Problem 1 inductively to obtain, on the subspace G of 'e,(X) generated by E and thegn, a linear form u such that -p( -f) 2 u (f)5 p c f )

for allfg G, and deduce that u extends by continuity to a measure on X.) Show that the measure p i s unique if and only ifp(f) p ( - f ) = 0 for allfe E.

+

4. T H E V A G U E T O P O L O G Y

Since the space M,(X) is a subspace of CXC('), we can define the weak topology, i.e., the topology of simple convergence in X c ( X ) (12.15). This topology on M,(X) is called the vague topology. To say that a sequence (p,) of measures on X converges cyaguely to a measure p therefore means that, for each function f E X,(X), the sequence ( p , ( f ) ) converges to p ( f ) in C. (13.4.1 ) Let (p,,)be a sequence of measures on X such that,for each f E X,(X), the sequence (pn(f )) tends to a limit p( f ) in C . Thenf Hp ( f ) is a measure on X and is the vague limit of the sequence (p,). If the pn are all positive, then so is p.

We have already remarked (13.1) that, for each compact subset K of X, X ( X ;K) is a Banach space and the restrictions of the p, to X ( X ;K) are continuous linear forms. Hence it follows from the Banach-Steinhaus theorem (12.16.5) that the restriction of p to X ( X ;K) is continuous, and therefore that p is a measure on X (and clearly a positive measure if the ,u, are positive). We recall (12.15) that a subset H of M,(X) is said to be vaguely bounded (or just bounded, if there is no risk of ambiguity) if, for each f E X,(X), we have sup I p ( f ) l < +a.Every vaguely convergent sequence is vaguely bounded.

PEH

(13.4.2) Let H be a bounded subset of M,(X). (i) For each conipact subset K of X, there exists a real number cK > 0 such that, for each p E H and each,f E .X,(X), we have Ip(.f>I 5

cK

(and therefore (13.3.2.1), 1p1(1 f I) 5 cK 11 f 11).

llfil

4 THE VAGUE TOPOLOGY

109

(ii) The vague closure of H in M,(X) is a compact metrizable space with respect to the vague topology. (i) This is an immediate consequence of the Banach-Steinhaus theorem (12.1 6.4). (ii) Let (U,) be a sequence of relatively compact open sets in X which cover X and are such that 0, c U,,+l(3.18.3). Since every compact subset K of X is contained in some U, (3.16), each space X ( X ;K) can be identified hence with a closed with a closed subspace of one of the Banach spaces W(u,,), subspace of X ( X ;0,).Now we know (7.4.4) that %(On)is separable, hence the same is true of X ( X ;0,) (3.10.9). Let (fmn)mL1 be a dense sequence in X ( X ;On).To show that H is metrizable it is enough to show that the vague topology on H can be defined by the pseudo-distances I(f, p - v ) l (12.4.6). This means that if g i (1 5 i 5 p ) is a finite sequence of functions belonging to X,(X), if p o is an element of H and r a real number >O, there exists a finite number of functions f,,,, (1 =< k =< q ) such that the relations p E H and I(fmknk P - ~ o > 5 l ! ~ r(1 5 k _I 4 ) imply I(gi P - PO)I 5 r (1 5 i 5 P ) . But the g i all belong to X ( X ;0,)for some fixed n, and the set of restrictions is equicontinuous, by (i) above and (12.15.7.1). of the p E H to X ( X ;D,,) Hence the assertion follows from (12.15.7). It remains to show (by the same reasoning as in (12.15.7)) that, when we identify H with its image L in the product space C N x Nby means of the mapping PI--+(( f,, , p)), L is closed in CNx .' If ( p k ) is a sequence of points of H such that each of the sequences ((fmn, p k ) ) k L 1 is convergent, then it follows from (12.15.7) that, in each X ( X ;On),the restrictions of the pk converge to a continuous linear form. Hence the sequence (pk) converges vaguely to a measure on X. Q.E.D. 3

9

We shall see later (13.20) that condition (i) in (13.4.2) can also be written in the form IpI(K) 5 cK for each measure p E H. Notice also that (13.4.2(ii)) implies (13.4.1) as a particular case. In particular: (1 3.4.3) Let v be a positive measure on X . Then the set of complex measures p such that lpl 5 v is metrizable and compact with respect to the vague topology.

For this set is evidently bounded and closed in M,.(X) in the vague topology. It should be noted that a bounded set of measures does not necessarily satisfy the hypothesis of (13.4.3).

110

Xlll INTEGRATION

(13.4.4) Let (pn) be an increasing sequence of real measures on X such that, for each function f 2 0 belonging to XR(X), the sequence ( p n ( f ) )is bounded above in R. Then the sequence ( p n ) has a vague limit in MR(X) which is also its least upper bound f o r the order-relation on MR(X).

For each function f 2 0 in XR(X), the sequence (pn(f)) is increasing and bounded above in R, hence (4.2.1) has a limit p ( f ) in R which is equal to sup p n ( f ) .Since every function belonging to X,(X) is a linear combination n

of four functions 2 0 belonging to X,(X), it follows that the sequence (p") is vaguely convergent (13.4.1), and it is clear that p is its least upper bound, by the definition of the order-relation on MR(X). (13.4.5) I f a series of positive measures on X , with general term p n , is such that, for each f 2 0 in X,(X), the series with general term p,,(f) 2 0 is convergent in R, then the series with general term p n is vaguely convergent in MR(X), and its sum p = p,,is such that p( f)= 2 p n ( f ) f o rall f E X,(X).

1 n

n

Apply (13.4.4) to the partial sums of the series.

PROBLEMS

+

1. Let be Lebesgue measure on R, let ,u be its restriction to R t =lo, m [, and let g be the function X H l/x on Rf Show that the measure g . ,u cannot be extended to a measure on R (cf. (17.9)).

.

2. On the real line R,show that the sequence of Dirac measures E. (unit mass at the point +n) converges vaguely to 0. Give an example of a sequence (f.) in Xc(R) which converges to 0 in the Frkchet space Qc(R) (12.14.6) but is such that the sequence ( where ak, ,, = (1 - (k - l ) / n ) ~ ~ -and ~ , uo, = 1, and Px. is a polynomial of degree s k - 1 whose coefficients are less than Ck/n in absolute value, where ck is a constant independent of n. (Differentiate the equation ( * k ) with respect to t , then multiply through by t . ) (c) Deduce from (b) and Weierstrass' theorem that, for every continuous function f c %'(I), the sequence (B".,) converges uniformly t o f o n 1. 10.

Let X be a metrizable compact space, let (f.).,,, be a sequence of complex-valued continuous functions on X, and let ( c . ) . , ~ be a sequence of complex numbers. (a) In order that there should exist a complex measure p on X such that p(f.) = c, for all n , it is necessary and sufficient that there should exist a number A > 0 with the following property: for every finite sequence of complex numbers, we have

(Use the Hahn-Banach theorem.) (b) Suppose that the fk are real-valued, the numbers ck real, and fo = I . Then there exists a positiue measure p on X such that p(fn)= c. for all n if and only if, for each sequence (A,),,,

2

I=O

11.

/\k

ck

of real numbers such that

"

1X,fk(x) 2 0 for all x E X, we have

k=O

2 o (cf. Section I3.3, Problem 2).

In Problem 10, take X = [0, I ] and For each sequence (c,) of scalars, put

f.(r)

= 1"

("Hausdorff's moment problem").

(a) Show that there exists a complex measure p on X such that p(fn)= c, for all n if and only if there exists a number A > 0 such that

(Observe that Akc. must be the value of p for the polynomial t"(1 - t ) k , and use Problem 9(b) by remarking that, for each real polynomial P, there exists a constant Cp such that I 1 Bn,P - - CPS P 5 B",P - CP

+

for all n.) (b) There exists a posiriue measure p on X such that p(f,)= c. for all n if and only if Akc. 2 0 for all k 2 0 and all n >= 0 (same method).

5 UPPER A N D LOWER INTEGRALS

113

12. Let X be a compact space. Show that in M(X) the set of positive measures with finite support and total mass 1 is dense (with respect to the vague topology) in the set P of all positive measures with total mass 1. (Let U be a neighborhood of p E P with respect to the vague topology, consisting of measures v E P such that Ip(fi)- v(fi)l 5 6, with fi E %‘,(X). Consider a continuous partition (g,) of unity and points U, E X such that Ifi(x) -Cfi(u,)g,(x)l 5 6 for all i . ) J

13. Let X be the unit interval [0, 11 in R.Show that the set K of Dirac measures E , (x E X) is vaguely compact, and that in M(X) the Lebesgue measure lies in the vague closure of the convex hull of K, but does not lie in the convex hull of K.

5. UPPER A N D LOWER INTEGRALS WITH RESPECT TO A POSITIVE MEASURE

In Sections (13.5) to (13.14)(including the problems), p denotes a positive measure on a locally compact space X. We shall show (13.7.3)that p can be extended from .XR(X) to a vector subspace YA(X, p) of RX,depending on p and containing (and in general distinct from) .XR(X),in such a way that this extension (also denoted by p) is apositive linear form on 2’A(X, p) (i.e., p takes values 2 0 on functions 20 belonging to Y ; ( X , p)) and possesses the fundamental property of passage to the limit for increasing sequences: that is to say, if (f,) is an increasing sequence of functions in 9A(X, p) whose upper envelopef (12.7.5)also belongs to Y A ( X , p), then we have lim p(fn) = &-). n+m

Let 9 (or 4(X)) be the set of all functions f : X + which are lower semicontinuous on X and bounded below by a function belonging to X,(X) (this implies that f ( x ) > - cx) for all x E X ; but we can have f ( x ) = + 00 at some points x E X : indeed, the constant function equal to + 00 belongs to 9). Every function f 2 0 which is lower semicontinuous on X belongs to 4 . For every f E 9,we put

this is a real number, or +a. Clearly iff E .X,(X) we have p * ( f ) = p ( f ) ; iff, g E 9 a n d f s g , we have p * ( f ) 5 p*(g); and for any real number a > 0, we have p*(af) = a p * ( f ) for all f E 4 .

(13.5.2) Let (f,) be an increasing sequence of functions belonging to 4 , and = supf, (so that f E 4 (12.7.6)).Then

let f

n

Xlll INTEGRATION

114

Suppose first of all that f E .X,(X) and that f,E X,(X) for all n. It is clear that the supports of all thef, are contained in the same compact set K = Supp(f) u Supp(f,). By Dini's theorem (7.2.2), the sequence (f.)converges uniformly t o f o n X. The formula (13.5.2.1) then follows from the fact that the restriction of p to X ( X ;K) is a continuous linear form on this Banach space. Now pass to the general case. Clearly we have p*(f,) 5 p * ( f ) for all n. Hence it is enough to show that, for each function u E X,(X) such that u 55 we have p*(u) S sup p*(f,). Now by (12.7.8) we know that, for each function,f, , there exists an increasir.5 sequence (gm,JnlZlof functions belonging to X,(X) such that f, = sup gmn. We have f = sup gmn, hence also m

m, n

f = sup h, , where h, = sup g p q. The functions h, clearly belong to X,(X) n

p l n , qsn

and form an increasing sequence. Since u SJ, we have u = sup(inf(u, h,,)); the n

sequence of functions inf(u, h,) is increasing and belongs to X,(X);since also u E X,(X), the first part of the proof shows that p*(u) = sup p*(inf(u, h,,)); n

but since h, S.d, we have p*(inf(rr, h,)) 5 p*(f,), and therefore finally P*@) 5 SUP P*(f,). Q.E.D. n

We remark that because the functions in 9 never take the value -03, the sumfl +fi of two such functions is defined at every point of X, and belongs to 9 (12.7.5).

We may write . f l

=

lim g, and f2 = lim h,, where (g,) and (h,) are two

n-. w

n+ a,

increasing sequences of functions belonging to X,(X) (12.7.8). Hence we have fl +f2= lim (g, h,) (4.1.8). Since p ( g , h,) = p(g.) + p(h,), the ti+

m

+

+

result follows from (13.5.2) and (4.1.8). Let (t,) be any sequence whose terms are real numbers 20, or +a. Since the partial sums s, = t , + -.-+ r, are defined (4.1.8) and form an increasing sequence, this sequence has a limit in R (4.2.1), denoted by

m

C t,

n= 1

and called the sum of the series with general term t,. For every sequence

(f,)of positiue functions belonging to I, the function

m

XH

c f , ( x ) is therefore

n= 1

5 UPPER A N D LOWER INTEGRALS

defined, and is denoted by

m

115

f n ; also it belongs to 9 (12.7.6). If we apply

n= 1

(13.5.2) to the sequence of partial sums

cfn,and (13.5.3) to each of the N

n= 1

terms in this partial sum, we obtain the following corollary: (13.5.4)

Zf(fn)ntl is a sequence of functions 20 belonging to 9,then

Now consider an arbitrary mapping f of X into R. There always exist functions h E 9 such that h 2 J for example the constant function equal to +co. Put (13.5.5)

p*(f) =

inf

htj,he/

p*(h);

this number p * ( f ) is called the upper integral o f f with respect to the measure p. Iff E 9 it is clear that this definition agrees with the preceding one. Here the value of p * ( f ) can be any element of R. The relation f 6 g implies p * ( f ) 5 p*(g). For any scalar a > 0 we have p*(af) = ap*(f).

+

(13.5.6) Ifthe sum f l f , of two mappings of X into R is definedat allpoints of X , and if p*( f l ) > - 00 and p*( f,) > - 03, then

+ f J 5 P*(fl) + P*(f,).

P*(fl

This is obvious if one of the numbers p * ( f i ) , p * ( f 2 ) is +a.If not, given any a > p * ( f i ) and b > p*(f2), there exist h,, h, in 9 such that h,, f , 5 h, and p*(hl) 5 a, p*(h,) b. It follows that h, + h, 2 f l +f , fi and p*(h, + h,) 5 a + b by (13.5.3). Hence the result. (I3.5.7) Zf (f.) is any increasing sequence of mappings of X into p*( f n ) > - 03 for all suficiently large n, then

1

P* SUPfn (

n

such that

= SUP P*(fn) = lim p*(fn)* n+

n

m

2 sup p*(fn) is clear. Let us prove the reverse

The inequality p* supf ( n

W

n)

n

inequality. We may assume that sup p*(fn) < n

+

03,

otherwise there is

nothing to prove. By hypothesis, we may therefore assume that sup p * ( f . ) n

and all the p * ( f . ) arefinite. For each E > 0, we shall show that there exists an

116

Xlll INTEGRATION

increasing sequence (g,) of functions belonging to 9 such that for each n we have f , 5 gn and p*(gn) 5 p*(f,) + E . Then, if we put f = supf, and n

g = sup g, , we shall have g E 9,f 5 g, p*(g) = sup p*(gn) 5 sup p*(f,) n

n

by virtue of (13.5.2),and finally p * ( f ) 5 p * ( g ) 5 sup p * ( f , ) + E . Since n

E

+E

was

arbitrary, this will complete the proof. By definition, for each n there exists h, E Y such that f n 5 h, and p*( f , ) 5 p*(hn)5 p*(f,) 2 - k Put g, = sup(h,, . . . , I?,,), so that gnE 9 (12.7.5).Clearly the sequence (g,) is increasing, and f,5 gn. We shall show by induction on n that

+

This is evident from the definitions when n = 1. Suppose that (13.5.7.1)is true, and remark that gn+l= sup(hn+lrg,) and f , 5 inf(h,+,, gn). Since the functions h,+l and g, do not take the value -co,we have SUp(hn+ 1,

gn>

+ inf(hn+

13

Sn) = h n +

1

+ gn

and therefore, by virtue of (13.5.3)and the fact that p*(g,,) and p*(h,+l) are finite,

by the inductive hypothesis (1 3.5.7.1).

Q.E.D.

It should be noted that for a decreasing sequence (fn), it is no.t necessarily true that p* infS, = inf p*(f,), even if all the numbers p*(fn) are $finite

L1

(Section 13.8,Problem 13). (13.5.8) Z f ( f n ) is any sequence of functions 20,then

5

UPPER AND LOWER INTEGRALS

117

For each integer N 2 1 it follows from (13.5.7) that

now apply (12.5.7) to the increasing sequence of partial sums

n= 1

R, the mapping

If (f n )is any sequence of mappings of X into

x H l i m inff,(x)

N

Cfn.

(resp. x H l i m supfn(x)) n-t m

n+w

is defined for all x E X. It is denoted by lirn inff, (resp. lirn supf,). n- m

n+m

(13.5.9)

(Fatou’s lemma) r f ( f n )is any sequence ofjiunctions 2 0 , then p* lirn inff,,

(

n-tm

1 s lirn inf p * ( f , ) . n+m

For each n 2 1 , let gn = inf (f,+,). Clearly we have p*(gn) 5 inf p * ( f , + & , P20

and gn 2 0; since the sequence (gn) is increasing and lirn inf f , n+ m

foIlows from (13.5.7)that p* lirn inff,

(

n+m

= sup p*(g,) 1

= sup g n , it

5 sup (inf p*(fn+,)) n

n

P20

pZ0

= lirn inf p*(f,). n+ w

Iff: X + R is any mapping, we define p * ( f ) = - p * ( - f ) ; this number is called the lower integral off with respect to the measure p. All the properties above which were proved for the upper integral can be immediately translated into properties of the lower integral. In particular, if we put -9 = 9 (or Y ( X ) ) , then Y is the set of all upper semicontinuous functions on X which are bounded above by a function belonging to X,(X) (which implies that they do not take the value + 00). For all functions f E 9, we have

CL*U)=

inf

9 b f 9 9 E XR(X)

P(S>

and for all f:X + R we have P*(f) =

(13.5.10)

SUP P*(h).

hsf,heY

I f f is any mapping of X into R, then p * ( f ) 5 p * ( f ) .

118

Xlll INTEGRATION

By virtue of the definitions, it is enough to show that if u E 9, u E 9 and u 6 u, then p*(u) p*(u). Since - u E 9 it follows that u - u = u ( - u ) is defined throughout X,belongs to Y and is 20;hence

+

0 5 p*(u - u) = p*(u

+ (- u)) = p*(u) + p*( =p*(d

- u)

- P*b)

by virtue of (13.5.3). We shall oftenwrite j * f d p or s * f ( x ) dp(x) (resp. S*fdpor f t f ( x ) dp(x)) instead of p * ( f ) (resp. p*(f)). For any subset A of X, we put p*(A) = p*(qA) and p,(A) = p*(qA), where qA is the characteristic function of A (12.7). These numbers p*(A) and p,(A) are 20 (possibly +a);they are called, respectively, the outer measure and inner measure of A. Example (13.5.11) Let I be Lebesgue measure on R and let I = ] a , b[ be an open interval in R. We shall show that I*(I) = b - a (which is equal to + 03 if b = + 03 or a = - a).If a < a' < b' < b, then there exists a continuous mappingfof R into the unit interval [0, 11, with support contained in [a, b ] , and equal to 1 throughout [a', b'] (4.5.2). We have J -00 + m f ( t )dt 2 b' - a'. Conversely, for each function g E XX,(R) such that 0 5 g 5 q r ,we have / +- mm g ( t )dt 5 b - a. Hence it follows that I*(I) = b - a. Now let U be any open set in R. Then its connected components (3.19) are open intervals ((3.19.1) and (3.19.5)), and the set of connected components is at most denumerable, because each component contains a point of the denumerable set Q, and the components are pairwise disjoint. Hence if they are r k = ] & , bk[, then q u = C qlk,and therefore ((12.7.4) and (13.5.4)) k

n*(u)= 1 ( b k - ak). k

(13.5.11.1)

6. NEGLIGIBLE F U N C T I O N S A N D SETS

A mapping f : X --t R is said to be negligible (with respect to the measure

p), or p-negligible, if p * ( l f l ) = 0. Then ufis also negligible for all a # 0 in

and if 191 5

If[,

then g is negligible.

R;

6 NEGLIGIBLE FUNCTIONS AND SETS

(13.6.1) I f ( f . ) is a sequence of negligible functions 20, then

m

119

fn is negligible.

n=l

This follows immediately from (13.5.8).

A subset N of X is said to be negligible (with respect to p) or p-negligible if its characteristic function cp, (12.7) is p-negligible. Clearly any subset of a negligible set is negligible. (1 3.6.2)

A denumerable union of negligible sets is negligible.

For if ( N k )is a sequence of negligible sets and N (P,

= sup cpNk k

=

5 1 q N kand , the result follows from (13.6.1).

u k

N k , we have

k

For example, with respect to Lebesgue measure A, every set { t } consisting of a single real number is negligible. For if E > 0 there exists a function f E XX,(R)with values in [0, I] which is equal to 1 at the point t and vanishes on the complement of the interval [t - E , t + E ] , and therefore L*((P{f,)

I 4f1 I 28.

Hence, by (13.6.2), it follows that every denumerable subset of R (for example, the set Q of rational numbers) is negligible with respect to Lebesgue measure. One can also give examples of nondenumerable sets in R which are negligible with respect to Lebesgue measure (Section 13.8, Problem 4).

A property P(x) of the points of X is said to be true almost everywhere (with respect to p) if the complement of the set of points for which P(x) is true is pnegligibie. (13.6.3) A mapping f :X -+ everywhere.

(P,

is negligible if and only

if it is zero almost

Let N be the set of all x E X such that If(x)l > 0. Then we have 5 sup nl f 1, and If1 5 sup ncp,, hence the result follows from (13.5.7). n

n

(1 3.6.4) I f a mapping f : X -+ R issuch that p*( f)< + cx) (resp. p * ( f ) > - a), then f ( x ) < co (resp. f ( x ) > - a)almost everywhere.

+

It is enough to prove the assertion relating to the upper integral. By hypothesis, there exists a function h E 9 such that f 5 h and p*(h) < + 00. We may therefore limit ourselves to the case where f E 9,and since there is

120

Xlll

INTEGRATION

then a function u E .X,(X) such thatf - u >= 0, we may also assume t h a t f z 0. Now let N be the set of points x E X such thatf(x) = + 03 ; we have ncp, 5 f for all integers n > 0, hence np*(cp,) 5 p*(f), and the hypothesis therefore implies that p*((pN) = 0. The relation “f(x) = g(x) almost everywhere in X ” is an equivalence relation between mappings of X into a set E, because the union of two negligible sets is negligible. We say then that f and g are equivalent (with respect to p) or p-equivalent, and we shall denote by f the equivalence class of a mapping f :X --+ E. A mapping f :A -,E, where A is a subset of X, is said (by abuse of language) to be defined almost everywhere in X if X - A is negligible. The equivalence class of such a mapping f is then defined to be the equivalence class of any mapping of X into E which extendsf; as before, we denote the equivalence class by f. Two functions f, g defined almost everywhere are said to be equivalent i f f = 8; this means that the set of points x E X at which f and g are both defined andf(x) = g(x) has a negligible set for its complement. (1 3.6.5)

Let f,g be two equivalent mappings of X into 8. Then p*( f) = p*(g).

Let N be the negligible set of points x E X such that f (x) # g(x). Since the functions f, g and sup(f, g ) are equal on X - N, we may assume that f 5 g. Let h be the negligible function which is equal to co at the points of N, and to 0 elsewhere. If v E 9 is such t h a t f s v, the function v h is defined at all points of X, and we have g 5 v + h, so that

+

+

by virtue of (13.5.6), because p*(v) > - co. From the definition of p * ( f ) , it follows that p*(g) 5 p * ( f ) and hence that p*(g) = p*(f). Iffis a mapping into 8 which is defined and finite almost everywhere in X, then f is equivalent to afinite function defined on the whole of X. Iff, g are functions defined almost everywhere in X, with values in R, and finite almost everywhere, then the same is true o f f + g andfg, and the equivalence classes of these functions depend only on f and 8. They are denoted by f + and fg, respectively. Iff(x) 5 g(x) almost everywhere, thenfl(x) 5 gl(x) almost everywhere for any functions fl, g1 equivalent to f, g, respectively. In this case we write f ij , and this defines an order relation on the set of equivalence classes (with respect to p ) of mappings of X into W.

7 INTEGRABLE FUNCTIONS A N D SETS

121

7. INTEGRABLE F U N C T I O N S A N D SETS

Wehaveseen (13.5.10) that every function f:X + R satisfies p * ( f ) S p * ( f ) . The function f is said to be integrable (with respect to p) or p-integrable if p * ( f ) and p * ( f ) are finite and equal. Their common value is then called the integral off with respect to p, and is written p ( f ) , or (S, p), or j f dp, or

1f ( x ) dp(x). Clearly every function f

.X,(X) is integrable and its integral is the value of p atf, so that our notation is consistent. An integrable function is therefore finite almost everywhere (13.6.4), but a bounded function is not necessarily integrable. For example, a constant nonIxl), is not integrable zero function on R, or the continuous function 1/(1 (with respect to Lebesgue measure) (cf. (13.20)). In view of the definitions in (13.5), we have the following criterion for integrability : E

+

(13.7.1) (i) For a function f :X -+ R to be integrable, it is necessary and sufficient that, given any E > 0, there should exist g E 9’and h E J such that g S f h and p*(h) - p*(g) 5 E (or, equivalently (13.5.3), p*(h - g ) 6 e). (ii) I f f is integrable, there exists a decreasing sequence (h,) of functions belonging to 9 and an increasing sequence (g,) of functions belonging to Y such that g, 5 f 5 h, for all n and lim p*(hn) = lim p*(gn) = Af).

n+ w

n+

m

(i) The condition implies that p*(h) and p J g ) are necessarily finite (and therefore so are p*( f ) and p*( f ) ) , and that p*( f ) = p*( f ) . (ii) For each n there exists h; E J and g; E 9’ such that g; S f S hi and p ( f ) - n-l S p*(gA) 5 p ( f ) 5 p*(h:) 5 p ( f ) + n-’. The required conditions are then satisfied by taking h, = inf(h;, . . . , h@ and g, = sup(g;, .. .,g;). Iff is integrable, then clearly so is any function f l equivalent to f (13.6), and we have f i dp = f dp (which we denote also by p ( f ) ) . It is now clear how integrability should be defined for functions defined almost everywhere in X: such a function f is integrable if the functions equivalent to f and defined on the whole of X are integrable. The number p(f) is also denoted by f dp,

s

s

f

or j f W dP(X), or A f ) ,or (S, P>. (13.7.2) For a mapping f : X + to be integrable it is necessary and suficient that, given any E > 0, there exists a function u E .X,(X) such that cc*(lf - 4) S E.

Xlll INTEGRATION

122

Necessity. Suppose that f is integrable, then there exist functions g E 9’and h E 9 such that g 5 f 5 h and p*(h - g ) 5 +E. Also ((13.5,l) and (13.5.3)) there exists u E X,(X) such that u 5 h and p*(h - u) +&. Since If - uI 5 Ih - uI + Ih - 91, it follows by (13.5.3) that

s

s

p*(lf - 4) p*(lh - 4)+ p*(lh - g l ) S

6.

Suficiency. If u E .X,(X) is such that p * ( l f - ul) 5 E , then by definition (13.5.5) there exists a function v E 9 such that If - uj v and p*(v) 5 2 ~ . But the relation - u 5 f - u 5 v can be written in the form

s

+ u5f5 v + u have - u + u E 9’and v + u E .f (12.7.5), -v

(u being finite); also we p*(v u - (- u u ) ) = 2p*(v)

+

+

5 4 ~ hence ; f is integrable.

and

s

The set L?i(X, p) (also denoted by L?A(p) or 2’:) offinite p-integrable functions on X is a vector space over R,and the mapping f H f dp is a positive ( I 3.7.3)

s

linear form on 9; (i.e.) the relation f 2 0 implies f dp 2 0).

Iff is finite and integrable, then clearly so is af for every scalar a E R, and we have af dp = a f dp. Iff and g are finite and integrable, it follows from (13.5.6) applied to f and g and to -f and -9 that

I

s

s s s* f&+ B 4 - I

(f+B)dPS

which completes the proof.

s*

(f+B)dPS

s s f4+

94.4

It follows that iff and g are integrable functions (finite or not) on X with values in R, then f + g (which is defined almost everywhere) is integrable, and that

J ( f -k 9 ) dP = / f

+ 1 9 4.

(13.7.4) I f f is integrable, then so are If I, f

+

and f -, and we haue

(13.7.4.1)

I f f and g are integrable, then so are sup(f, g ) and inf(A 9).

If

uE

X,(X) is such that p * ( l f - ul) 5 E , then since

I If1 - I4 I S I f - 4,

7 INTEGRABLE FUNCTIONS AND SETS

)I

it follows that p*(I If1 - ( u J 5 E , which by (13.7.2) shows that rable. Moreover, we have -If[ 5 f 5 If1 and hence

If1

123

is integ-

Ifl) = -PL(ISI) 5 P*(f) = C lm 5 P*(lfl> = A l f I)? which establishes (13.7.4.1). Since f = f ( f + If[) and f - = +(If1 -f), it /A*(-

+

follows thatf' and f - are integrable (13.7.3). Iff and g are integrable, then f - g is defined almost everywhere and integrable, and the functions (everywhere defined) sup(f, g ) and inf(f, g ) are equivalent respectively to the functions (defined almost everywhere) + ( f g Ifgl) and +(f g - I f - 91). Hence they are integrable.

+ +

+

(13.7.5) For a function h E 9 (resp. g E 9) to be integrable, it is necessary and suflcient that p*(h) < + 00 (resp. p*(g) > - 00).

For if p*(h)
0 there exists u E XR(X)such that

by (13.5.1) and (13.5.3). The result now follows from the definition of integrable functions. E,

A subset A of X is said to be integrable if its characteristic function cpA is integrable, or equivalently if p*(A) and p*(A) are jinite and equal. Their common value cp d is then denoted by p(A) and is called the measure of A. We have p ( 0 ) = 0. The negligible sets are the same as the integrable sets with measure zero. If A is integrable, then so is every set B for which A n eB and B n EA are negligible, and p(B) = p(A).

1.p

(1 3.7.6) I f A and B are integrable sets, then A v B, A n B and A n CB are integrable.

This follows from the formulas (12.7.3). (13.7.7) Every compact set is integrable. For an open set U to be integrable it is necessary and suficient that p*(U) < + 00. In particular, every relatively compact open set is integrable.

This follows immediately from (1 3.7.5) and (12.7.4).

Example (13.7.8) With respect to Lebesgue measure on R, every bounded interval with endpoints a, b is integrable and its measure is Ib - a / , by virtue of

124

Xlll

INTEGRATION

(13.5.11) and the fact that all finite sets are negligible (for Lebesgue measure) (13.6). If I = [a, b] is a bounded interval, the “Dirichlet function” (3.11) (P, - ( P , ( P ~ (which is 0 at the rational points of I and on the complement of I, and 1 at the irrational points of I) is integrable, and its integral is b - a, since the set Q is negligible (13.6). (13.7.9) A subset A of X is integrable i f and only i f , given any E > 0, there exists a compact set K and an open set G such that K c A c G and p*(G - K) E.

The condition is sufficient by virtue of the definition of integrable functions and (12.7.4). Suppose conversely that A is integrable, and let E be such that 0 < E < 1. By hypothesis, there exists h E 9 such that ( P 5 ~ h and

s

- (PA) dp 5

Let G be the set of points x E X such that h(x) > 1 - E ; then G is open (12.7.2) and contains A. Clearly h 2 (1 - E ) ( P ~ ,hence

I

Also there exists g E Y such that g 5 ( P and ~ ( q A- g ) dp 5 E , and by the definition of Y the set of points x E X such that g ( x ) > 0 is relativelycompact. Choose 6 > 0 so that Sp(A) 5 E , and let K be the set of all x E X such that g ( x ) 2 S; then K is closed in X (12.7.2) and therefcre compact (3.17.3), since it is contained in a relatively compact set. Clearly we have K c A and g 5 ( P ~ Sq,, where B = A - K ; hence

+

j g dp

S p(K) + M B ) S p(K) + M A ) S

and finally p(A) 5 j g dp

p(K)

+ E,

+ I p(K) + 2 ~ .

Q.E.D.

E

There exist bounded subsets of R which are not integrable with respect to Lebesgue measure (Section 13.21, Problem 6). (13.7.10) Let n : X -+X‘ be a homeomorphism. It follows directly from the definitions that iff’ is any function on X with values in R, we have

j*Yd(nW) = J>ft

.

O

dP.

125

8 LEBESGUE’S CONVERGENCE THEOREMS

The function f ’ is n(p)-integrable if and only if ,f’ that case we have J f ‘ d(n(p))= f ’ o 71) dp.

I(

0

71

is p-integrable, and in

8. LEBESGUE’S C O N V E R G E N C E T H E O R E M S

(13.8.1) Let (f,) be an increasing sequence of integrable functions. For supf, to be integrable, it is necessary and sufficient that n

and if this condition is satisfied we have

(13.8.1.1) Since J*f,dp > - co for all n, we have

j*(sup f.) n

dp

= sup n

/fndp

(13.5.7).This already shows that the condition is necessary. Conversely, if the condition is satisfied, let f = supf , . Then, given any E > 0, there exists n n

such that the functi0n.f -f, (which is defined almost everywhere because f and f n are finite almost everywhere (13.6.4))satisfies

(13.5.6). But there exists a function (13.7.2),hence by (13.5.6)we have

uE

X,(X) such that J Ifn - uI dp 5 E

and the result follows by (13.7.2). There is of course a corresponding theorem for decreasing sequences of integrable functions, obtained by replacing f , by -A in (13.8.1). (13.8.2) Let (f,) be any sequence of integrablefunctions. In order thatf = sup fn

should be integrable, it is necessary and suficient that there should exist a g dp < + 00 and f n 5 g almost everywhere. function g 2 0 such that

I*

The condition is obviously necessary (take g =f+).Conversely, if it is satisfied, let gn = sup fk . Then g, is integrable (13.7.4) and f = sup gn . 1S k S n

n

Xlll INTEGRATION

126

0. Show that the set B of points of X which n

belong to infinitely many of the sets A. is integrable, and that p(B) 2 rn. 8.

Let A be Lebesgue measure on the interval 1 = [0, 11 and let A. be a sequence of integrable sets in I such that inf h(A.) = rn > 0. For each integer k , let %k denote the

"

+

set of 2k intervals of the form [ j . 2-,, ( j 1) .2-,] in I(0 5 j < 2'9, and CSk the set of all the unions of sets of a,. Show that there exists a decreasing sequence ( I k ) k r O of subsets of I such that 1, = I , I, E Ek for all k , and a subsequence (Ank) of the sequence (A,) with the following properties: (i) &A,, n (I - I k ) ) 5 &rn;\(l - I t ) for all r >= k ) ; (ii) &Anvn J) 2 trnA(J) for all r 2 k and all J E Dk contained in I,. (Proceed by induction on k , using the diagonal trick.) Deduce that there exists a subsequence (B.) of (A,) such that, for all k and all J E ' D k contained in I, , the intersection of J with the sequence (B,) is not empty. (Use Problem 7 and the diagonal trick.) Deduce that B, contains a non-empty compact set with no isolated points (and therefore n

nondenumerable (Section 4.2, Problem 3(c))).

9. Letfbe a real-valued function 2 0 on X. Show that the mapping p ~ p * ( f 'of) M+(X) into R is continuous with respect to the vague topology (13.4) if and only iff is continuous and compactly supported. The mapping p ~ p * ( fis) lower semicontinuous with respect to the vague topology if and only iffis lower semicontinuous. 10. Let (pn)be an increasing sequence of positive measures on X. Suppose that the sequence is bounded above in M+(X), and let t~ be its least upper bound (13.4.4). Show that, for every functionfz 0 on X, we have p * ( f ) = lim pcL:(/). = "-1

Let (pn)be a decreasing sequence of positive measures on X, and let p be the greatest lower bound of the sequence in M+(X) (13.4.4). Iff is a function 2 0 on X such that pz(f)< +to for all sufficiently large n, show that p*(f)= limp,*(f).

11. (a)

+

"+ m

(If g 2 0 is lower semicontinuous and p*(g) < a,observe that there exists a sequence (h,) of continuous functions 2 0 with compact supports, such that h,, s g and p.*(g)

=?p.*(h,)

x m

for all n.)

(b) On the discrete space N, let p n be the measure such that p n ( { m }= ) 0 for rn < n and 1 for rn 2 n. Show that the greatest lower bound of the decreasing sequence (p.) in M+(N) is 0, but that p,*(N)= to for all n.

+

8 LEBESGUE'S CONVERGENCE THEOREMS

133

12. Let X, Y be two locally compact spaces and T : X -+Y a proper continuous mapping (Section 12.7, Problem 2). Let p be a positive measure on X, and let v = ~ ( p(Section ) 13.4, Problem 8). Show that v * ( g ) = p*(g T)for all functions g 2 0 on Y. (Consider first the case in which g has compact support, and then use Section 12.7, Problem 0

2(b).) Show that N c Y is v-negligible if and only if n-l(N) is p-negligible.

13. Assume that there exists in I = [0, I ] an increasing sequence (H.) of nonmeasurable H. = I and h,(H,) = 0 for all n. sets (with respect to Lebesgue measure h) such that

u"

Show that h* inf (1 - yHn)) = 0, but that h*(l - vHn) = 1 for all n.

("

14. (a)

Let F be a convex function defined on a convex open set A c R" (Section 8.5, Problem 8) and let fk (1 6 k 6 n) be n integrable functions on X, such that the mapping x ~ ( f * ( x ) takes ) ~ ~its~ values ~ ~ in A, and such that the composite function x ~ F ( f i ( x .).,. , f . ( x ) )is integrable. Show that if u , is bounded (13.9.16) and p(X) = 1, then

(Use the Hahn-Banach theorem applied to the convex set of points (ti, ..., ~ , , , u ) E A X R

such that u 2 F(r,, .. . , t,,).) Consider in particular the case n = 1, F ( t ) = e', and deduce the inequality of the means uyu;2

. . . u:m

s

U1Ul

where the uk are >O, the uIiare 2 0 and

1-u2 u2

m k= t

+ ..+ '

umum,

uk= 1.

(b) Suppose that p is bounded and p ( X ) = l . Show that if two functions .f2_0, g 2 0 are integrable and such thatfg 2 1, then ( l f d p ) ( s g d p ) 2 1 (use (a)). 15. For each finite sequence (A,)

,5 , d m of p-integrable sets, put

if none of the A, is pnegligible. (Remark that a point of D(AI, . . ., An) belongs to a t least n - 1 of the sets D(A,, A,) (i 1. Then the intersection N of the sets A -

(13.8.7).

u Ki n

i= 1

is negligible, by

A subset of A is said to be measurable (with respect to p) or p-measurable if there exists a partition of A consisting of a sequence of compact sets and a negligible set. Jf K' and K are compact sets such that K' c K, then K - K' is integrable ((13.7.6) and (13.7.7)), and therefore, by virtue of (13.9.1), an equivalent statement is that A is the union of a sequence of compact sets and a negligible set. The same reasoning shows also that another equivalent statement is that A is the union of a sequence of integrable sets. A set A c X is said to be universally measurable if it is measurable with respect to every positive measure on X.

(13.9.2) For a subset A of X to be measurable it is necessary and suficient that A n K should be integrable,for each compact set K in X . The condition is necessary, by virtue of (13.8.7(ii)). It is sufficient, because X is the union of an increasing sequence (K,) of compact sets (3.18.3), and therefore A I S the union of the sequence (A n K,) of integrable sets. In particular, we see that the space X itself is measurable.

(13.9.3) (i) The complement of a measurable set is measurable. (ii) Denumerable unions and denumerable intersections of measurable sets are measurable. (iii) All open set,r and all closed sets are universally measurable. Assertion (i) follows from (1 3.9.2) and (1 3.7.6); assertion (ii) from (13.9.2) and (13.8.7); assertion (iii) from (13.9.2) and (13.7.7) for closed sets, and then by (i) for open sets.

9

MEASURABLE FUNCTIONS

135

The sets obtained from the open (or closed) sets of X by iterated application of the operations (i) and (ii) above are therefore also universally measurable. The definition of a p-measurable set A also shows that there exists a universally measurable set B contained in A such that A - B is p-negligible. By applying this result to X - A, we see that there also exists a universully measurable set C containing A such that C - A is p-negligible. A mapping u of X into a topological space Y is said to be measurable (with respect to p) or p-measurable if there exists a partition of X into a sequence of compact sets K, and a p-negligible set N such that each of the restrictions u I Kn is continuous. The mapping u is said to be universally measurable if it is measurable for all measures on X. A continuous function is universally measurable. For a subset A of X to be measurable (resp. universally measurable) it is necessary and sufficient that its characteristic function q A should be measurable (resp. universally measurable). For if we have a partition of X consisting of a sequence of compact sets K, and a negligible set N, such that qAI K, is continuous for all n, then K, is the union of two disjoint compact subsets, namely A n K, and K, - (A n K,)((3.11.4) and (3.17.3)), and A is the union of the sets A n K, and A n N, and is therefore measurable. Conversely, if A is measurable, then so is X - A (13.9.3), hence there exists a partition of X consisting of a sequence of compact sets K, and a negligible set N, such that either K, c A or K, c X - A for all n. Consequently qAI K, is continuous for all n. (13.9.4) Let u be a mapping of X into a topological space Y . Then thefollowing three properties are equivalent: (a) u is p-measurable. (b) For each compact subset K of X and each E > 0, there exists u compact subset K' of K such that p(K - K') 5 E and such that the restriction of u to K' is continuous. (c) For each compact subset K of X, there exists apartition of K consisting of a sequence (L,) of compact sets and a p-negligible set M, such that each of the restrictions u I L, is continuous. To show that (a) implies (c), we remark that the hypothesis (a) implies the existence of a partition of X consisting of a sequence (K,) of compact sets and a negligible set N, such that u I K, is continuous for all n ; hence (c) is satisfied by taking L, = K n K, and M = K n N. To show that (c) implies (b), we remark that it follows from (c) that p(K)

a,

=

1 p(L,)

fl=

1

(13.8.7), and hence that there is an integer n such that

p(K - K') 5 E , where K' =

u Li .

icn

Clearly K' satisfies the conditions of (b).

136

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INTEGRATION

Finally, we have to show that (b) implies (a). The space X is the union of an increasing sequence (H,) of compact sets (3.18.3). For each n we use the hypothesis (b) together with (13.7.9)to define a sequence (Km,),21 of compact sets such that K,, c H, - H,-,, K,, c (H, K i n , and

u

i 0, there exists a compact subset K' of K such that p(K - K') 5 E and such that u, I K' is continuous for all n. Using (13.9.4)we define inductively a decreasing sequence (K,) of compact subsets of K such that p(K - K,) 5 f~ and p(K, - K,+l) 5 e/2"+', and such that u, 1 K, is continuous. Then by (13.8.7)it is clear that K' = K, satisfies

0 n

the required conditions.

(1 3.9.6) Let (Yi)l be afinite sequence of topological spaces, Z a topological space, and v a continuous mapping of Y i into Z. Let :ii : X -+ Y i (1 2 i 5 n)

n

be measurable. Then the mapping

i

XF-+ v(u,(x),

... , u,,(x)) is measurable.

This follows immediately from (1 3.9.5) and (1 3.9.4).

(13.9.7) V J g are two measurable functions with values in 8, then sup(f, g ) and inf(f, g ) are measurable. r f u, v are measurable mappings of X into a (real or complex) vector space, then u + v and au (where a is any scalar) are measurable.

9 MEASURABLE FUNCTIONS

137

(13.9.8) A function equivalent to a measurable function is measurable.

This follows from (13.9.4)and the fact that if K is compact and N c K is negligible, then there exists a compact set K' c K - N with measure arbitrarily close to the measure of K (1 3.7.9). A mapping f defined almost everywhere in X , with values in a topological space Y, is said to be measurable if every mapping of X into Y which is equivalent to f (13.6)is measurable. Clearly (13.9.8)it is enough for one such mapping to be measurable.

(13.9.8.1) In particular, it follows from (13.9.6)and (13.9.8)that iff and g (with values in R) are defined andjinite almost everywherein X and measurable, then f g and f g (which are defined almost everywhere) are measurable.

+

(13.9.9) Let u be a measurable mapping of X into a topologicalspace Y . I f M is any open or closed subset of Y , then u-'(M) is measurable. Consider a partition of X consisting of a sequence of compact sets Kn and a negligible set N, such that the restrictions u I K, are continuous. If M c Y is closed, then u-'(M) n K, is compact (3.11.4),hence u-'(M) is measurable. If M is open, then X - u-'(M) = u-'(Y - M) is measurable, and therefore so is u-'(M) (13.9.3).

(13.9.10) (Egoroffs theorem) Let Y be a metric space and let (f,)be a sequence of measurable mappings of X into Y such that, for almost all x E X , the sequence (f,(x)) converges to a limit f (x). Then ( 1 ) f is measurable; ( 2 ) f o r each compact subset K of X and each E > 0, there exists a compact subset K' of K such that p(K - K ' ) S E and such that the restrictions f,I K' are continuous and converge uniformly in K' to f I K'. Clearly it is sufficient to prove the second assertion ((1 3.9.4)and (7.2.1)). Let KObe a compact subset of K such that p(K - KO)5 +E and such that the restrictionsf, I KO are continuous for all n (13.9.5). If d denotes the distance fmction on Y, let B,,, be the set of all x E KO such that d(fp(x),fq(x))2 l / r for ar least one pair of integers ( p , q ) such that p 2 n and q 2 n. If A p,q,r is the set of points x E KOsuch that d(f,(x),f,(x)) 2 l/r, then we have Bn,r

=

U

phn. s t n

Ap.q,r*

138

Xlll INTEGRATION

Now the sets A p , q ,are r closed (3.11.4),hence B,,,r is integrable (13.8.7). Moreover, by hypothesis, for each integer r 2 l we have p( B,,, = 0, and the

() .)

sequence (B,,,,),,? is decreasing, so that by (13.8.7) we have lim p(BnPr)= 0. n-m

Hence there exists an integer n, such that P ( B , , , ~ 2 ) 142"'~.Let B the set B is integrable, and we have p(B) 5

m

C142'''

=&

=

u r

B,,,.,,;

by (13.8.7). The

I=

set C = KO- B is also integrable, and by definition the sequence ( f , I C ) converges uniformly to f l C . Hence we may take K' to be any compact subset of C such that p(C - K') 5 4 6 (13.7.9). (1 3.9.1 1) Let (f,)be any sequence of measurable functions on X with values in R. Then the functions inff, , sup f,, , lim sup f, and lim inff, are measurable. n

n

n-+ w

n-m

For supf, is the limit of the increasing sequence of functions g,

=

sup f;.,

l 0, there exists a measurable set A, an integrable function g 2 0, and an integer no such that, for all n 2 n o , we have If.(x)l

s

5 Ig(x)l for all x E A. Show that f i s integrable and that lim If-f.1 n-m

dp= 0.

Consider the converse. (c) Show by examples that the conditions of (a) are not sufficient and that the con-

I

I

ditions of (b) are not necessary for f to be integrable and f d p = lim f. dp. n-m

If A is measurable and B is any subset of X, show that

+

p*(B) = p*(B n A)

+ p*(B n (X -A)).

(If p*(B)< m, consider an integrable set B1 3 B such that p*(B) = p(BI).) Conversely, show that if A satisfies this condition (for all B c X), then A is measurable (cf. Section 13.8, Problem 3). Suppose that X is compact. A bounded real-valued function f on X is said to be continuous almost everywhere (with respect to p) on X if the set of points of discontinuity off is negligible. (a) Give an example of a functionfwhich is continuous almost everywhere and such that there exists no continuous function g which is equal to falmost everywhere. (b) Suppose that the support of p is equal to X. Show that a bounded real-valued function fdefined on X is equal almost everywhere to an almost everywhere continuous function on X if and only if there exists a subset A of X such that X - A is negligible and f l A is continuous. (To show that the condition is sufficient, observe that A is dense in X and that the lower semicontinuous extension of f JA to X is continuous at every point of A.) Deduce that f is measurable. Show that there exists a sequence (A) of continuous functions on X which converges at every point of X and whose limit is almost everywhere equal tof(cf. Section 13.11, Problem 3). (c) Show that a real-valued function f on R which is continuous on the right (i.e., such thatf(x+) = f ( x ) for all x E R) is continuous except at the points of an at most denumerable. set, and is therefore continuous almost everywhere with respect to Lebesgue measure. (Apply Section 3.9, Problem 3 to the set A, of points x E R at which the oscillation offis > l/n.) A partition w = (A.) of a set E is said to be finer than a partition m’ = (A;) of E if, for each index a , there exists B such that A, c A;. The partitions of E form an ordered set with respect to this relation. (a) Let X be a compact metric space. For each finite partition zu = (Ak)of X consisting of integrable sets and each bounded real-valued function f on X, put

(“Riemann sums” relative tofand the partition

and that if

w

w ) . Show that

is finer than m’, then s , # ( f ) s , ( f ) and S , ( f )

6 S,,(f).

9

MEASURABLE FUNCTIONS

145

(b) A sequence (m,)of finite partitions of X is said to be fundamental if ZU,,+~ is finer than w, for all n and if the maximum of the diameters of the sets of w, tends to 0 as n tends to m. Show that iffis a bounded integrable function on X, then there exists at least one fundamental sequence (m,) of finite partitions of X, consisting of

+

integrable sets, and such that the sequences ( s W n ( f and ) ) [S,"(f)) both tend to

s

f dp.

(c) Letfbe bounded and continuous almost everywhere on X (Problem 6). Show that, for all fundamental sequences (m,) of finite partitions of X consisting of integrable

f dp. (Observe that if A. is the J X at which the oscillation off (Section is Ll/n, then

sets, the sequences (s,Jf)) and ( S m n ( f ) ) both tend to

closed set of points x E 3.14) p(A,,) tends to 0 as n tends to m ; each A,, has an open neighborhood V. such that the points of V,,have a distance < l/n from A,, and such that p(V,) tends to 0. For a partition m k whose sets all have diameter < l/n, consider separately the sets of the partition which meet V. and those which do not.) Iffis bounded and lower semi-

+

s

continuous on X, show that s,,(f) tends to f d p for every fundamental sequence (m,) of finite partitions of X consisting of integrable sets. (d) A subset A of X is said to be quadrable (with respect to p) if its characteristic function va is continuous almost everywhere, or equivalently if the frontier of A is p-negligible. Show that every point xo in X has a fundamental system of quadrable open neighborhoods. (For each neighborhood V of xo , let f :X + [0, I ] be a continuous mapping such that f ( x o ) = 1 and f ( x ) = 0 for all x E X - V. For each a E 30, 1 [, consider the set of points x E X such that f ( x ) > E . ) Deduce that there exists a fundamental sequence of finite partitions of X consisting of sets which are either open or negligible. Give an example of a nonquadrable closed set (Section 13.8, Problem 4(a); cf. Section 13.21, Problem 2). (e) Let (w,) be a fundamental sequence of finite partitions of X consisting of sets which are either open or negligible. Iffis a bounded function on X, let g be the largest lower semicontinuous function sfi show that s&) = s,(g). Deduce that the sequences ( s m n ( f ) and ) (S,"(f)) tend to the same limit if and only iff is continuous almost everywhere. (f) Use (e) to give an example of a negligible functionfand a fundamental sequence (w") of finite partitions of X consisting of integrable sets, such that the sequences (s,Jf)) and ( S m n ( f ) )do not tend to the same limit. 8.

Let f be a measurable mapping of X into a complete metric space E, and let K be a compact subset of X. Show that f l K can be approximated uniformly by measurable step-functions if and only iff(K) is relatively compact in E.

9. Suppose that X is compact. Let (f.) be a sequence of p-measurable finite real-valued functions on X. Show that the following properties are equivalent: (1) There exists a subsequence of (A) which tends to 0 almost everywhere in X. (2) There exists a sequence (I,) of finite real numbers, such that lim sup 1t.l > 0 n-m

and such that the series with general term f,f.(x) converges almost everywhere in X. (3) There exists a sequence (t.) of finite real numbers such that C 1r.l = m n

+

and such that the series with general term t.f.(x) converges absolutely almost everywhere in X.

146

Xlll

INTEGRATION

(To show that (I) implies (2) and (3), use Egoroff's theorem. To show that (3) implies (1). show that (3) implies that there exists a n increasing sequence of integrable subsets A, of X and a subsequence ( f n k ) of the sequence (fn), such that p(Ak)tends to

p ( X ) and

s*, Ifn,.[

d p tends to 0, as k tends to

+

03.)

10. Let X, Y be two locally compact spaces and 7r : X + Y a proper continuous mapping. ) 13.4, Problem 8). Let p be a positive measure on X, and let v = ~ ( p(Section (a) A mapping g of Y into a topological space is v-measurable if and only if g 0 T

is p-measurable. (b) A mapping g : Y -+ R is v-integrable if and only if g we have s g dv = / ( g

0

7r)

0

7r

is p-integrable, and then

dp. (Use Problem 12 of Section 13.8.)

(c) Show that the support of v is the closure of

7r

(Supp(p)).

11. Let X be a locally compact space, u : X + X a continuous mapping, and p a positive

measure on X. Suppose that for each p-negligible set N, the inverse image u - ' ( N ) is p-negligible (cf. Problem 17). We shall write u-"(A)for (u")-'(A),for each integer n 1 and each subset A of X, and uo = lx. If A is any p-measurable subset of X, let A,,,

u u-"(A) be the set of points m

=

"=O

which "enter A at least once," and let A,,, = A n u-'(A,,,) be the set of points of A which "return at least once to A". Also put A,,,

inf =

An

n u-"(A,,,), the set of m

"=O

points of A which "return to A infinitely often;' The set A is said to be wandering (relative to u) if the sets rc-"(A)( n 2 0) are pairwise disjoint. The mapping u is said to be incompressible if, for each p-measurable subset A of X such that A c u.-'(A),the set u-'(A) - A is negligible; or equivalently if, for each p-measurable subset B of X such that u-'(B) c B, the set B - u-'(B) is negligible. If u is not incompressible it is said to be compressible. (a) Show that the following four properties are equivalent: ( a ) u is incompressible. (p) The wandering sets (relative to u) are negligible. ( y ) For each measurable set A, the set A - A,,, is negligible. is negligible. (6) For each measurable set A, the set A - A,,, (Observe that u-'(Ae0,) c A,,, and that A - Are, is wandering.) (b) Show that u is compressible if and only if there exists a measurable real-valued function f on X such that f ( u ( x ) )>f(x) for all x E X and such that the set of points x E X at whichf(u(x)) > f ( x ) has measure >O. (To show that the condition is sufficient, argue by contradiction to show that it implies that there exists a rational number r for which the set of points x E X such thatf(x) < r < f ( u ( x ) ) is not negligible.) Deduce that if u" is incompressible for some integer n > 1, then u is incompressible. (c) Suppose that X is compact and that the measure p is inoariant with respect t o u (cf. Section 13.4, Problem 8). Show that u is incompressible (" Poincarb's recurrence theorem "). (d) Show that if u is incompressible, then there exists a negligible set N in X such that, for every x $ N and every neighborhood V of x in X, there are infinitely many integers n such that u"(x) E V. (Consider the sets U - U,,, i n f , as U runs through a denumerable basis of open sets of X.)

9

MEASURABLE FUNCTIONS

147

Let (t,)oarsnbe a sequence of real numbers, and let m be an integer ( n . Let L, be the set of indices i with the following property: there exists an integer p such t12_0.(Observethat t h a t O _ I p s m a n d t f + t i + l + ~ ~ ~ + t i +2p 0- .- Show l that

12. (a)

lELm

if i E L,, and if p is the smallest of the integers with the above property, then i + 1, ..., i + p - 1 belong to L,,,.) (b) Let X be a locally compact space and p a positive measure on X. Let u : X + X be a proper continuous mapping such that u ( p ) = p. Iff’is any p-integrable function on X, putf, = f a n d f x = f o up ( k 2 1). For each integer m , let A, be the measurable set of points x E X such that one of the sumsfo(x) +fi(x) ...+f,(x), wherep 5 m,

+

is

2 0. Show that

JAM

f(x) dp(x) 2 0. (For each integer n > 0 and each x E X, consider

the sequence ( f r ( ~ ) ), and ~ ~apply ~ ~ (a) ~ to + this ~ sequence, denoting the corresponding set of indices by L,(x). For each k 5 n m, let Bkbe the set of x E X such that k

E

Lm(x), and deduce from (a) that

+

n+m

C

k=O

B!,

fx(x) dp(x) 2 0. Now use the fact that

Bk= Kk(Am)for 0 5 k 5 n to deduce that

for all n.) If A is the union of the Am, conclude that

/A

f ( x ) dp(x) 2 0 (“maximal

ergodic theorem ”). (c) Let a be a real number and C a n integrable set such that

x

1”-1 a < Iim sup fx(x) n-+m

for all x E C . Show that ap(C) 6

nk=o

1

If(x)l dp(x). (Apply (b) to the function f- avc .)

(d) Let a , b be two real numbers such that a < b. If E is the set of points x E X such that 1”-1

Iim infn-m

1”-1

1fk(x) < a < b < Iimn + msup -n kC= o /i(x),

k=O

show that E is p-negligible. (Deduce first from (c) that E is integrable, and then apply (b) to each of the functions (f- b)vEand (a -f)yE, using the fact that u(E) C E.) Hence show that,for almost aNx E X, the sequence

converges to a limit f*(x), that f* is integrable and that f*(u(x)) = f * ( x ) almost everywhere (G. D. Birkhoff’s ergodic theorem). (Take a , b to be all pairs of rational numbers such that a < b.)

s

s

f* dp = f dp. (Reduce to the case f 2_ 0, and consider first the case wherefis bounded; then pass to the general case by observing

(e) If X is compact, show that that

/ If’l

dp 5

If1 dp.1

148

Xlll INTEGRATION (f) Suppose that X i s compact and p(X) = 1. Consider a sequence (gn)of pintegrable functions such that lim g,(x) = f ( x ) almost everywhere. Suppose moreover that there nr'"

exists a p-integrable function G 2 0 such that /gnI 5 G almost everywhere, for each integer n. Show that under these conditions the sequence

converges almost everywhere to f * ( x ) . (Reduce to the case f = 0. For each e E 10, 1 [, there exists 6 > 0 such that, for each p-integrable subset B of X satisfying p(B) 5 6, we have of points x

G d p e2 (13.15.5). Next, there exists an integer m such that the set A, X satisfying

E

SUP Ig,(x)l

nbm

Ie2

has measure p(A,) 2 1 - 6. Let G , be the function which is equal to G on X - A,; then for n 2 m and all x E X, we may write

Deduce from (c) that the set of points x

1:

lim sup "'rn

has measure 13.

5 26.)

E

6'

on A, and to

X such that g,,(uk(x))

2e

Let X be a locally compact space, p a positive measure on X, and u : X +X a continuous mapping such that, for each p-negligible set N, the set u-'(N) is p-negligible. A measurable real-valued function f on X is said to be pinvariant with respect to u if f a u and J' are equal almost everywhere (relative to p). A p-measurable subset A of X is said to be p-inuarianr with respect to u if its characteristic function is pinvariant with respect to u. (a) For a measurable function f to be pinvariant with respect t o u, it is necessary and sufficient that, for each rational number r , the set of points X E X such that f ( x ) 2 r should bc p-invariant with respect to u. (To show that the condition is sufficient, consider the set of points x E X such thatf(u(x)) # f ( x ) . ) (b) From now on, suppose that u is proper and the measure p-invariant with respect to u (i.e., u ( p ) = p). The mapping u is said to be ergodic with respect to p (or p is ergodic with respect to u) if the only p-measurable functions which are pinvariant with respect to u are constant almost everywhere. Then, for every p-integrable function f on X, the function

(sf(.)

is constant almost everywhere. If X is compact, the value of the constant is

dp(x))/p(X). Conversely, if X is compact and i f f * is constant almost every-

where, for every p-integrable function A then u is ergodic.

9

MEASURABLE FUNCTIONS

149

(c) Take X to be the circle U : IzJ= 1 in C , and p to be the image of Lebesgue measure under the mapping t w e Z n i rof [0, 11 onto U. If 6 is irrational, show that the mapping u : z w e Z n i e zleaves p invariant and is ergodic with respect to p. (Use Bohl's theorem, Section 13.4, Problem 7.) (d) Suppose that X is compact and u ergodic with respect to p. If A and B are p-measurable subsets of X, show that

Conversely, if this formula is satisfied for all pairs of measurable subsets A, B in X, then u is ergodic with respect to p. (e) Suppose that X is compact and u ergodic with respect to p. If a measurable 0 is such that function

fz

exists almost everywhere, thenfis integrable. (Consider f a s the limit of an increasing sequence of bounded functions.) (f) Suppose that X is compact, that u is ergodic with respect to p, and that Supp(p) = X. Show that, for almost all x E X, the set of points u"(x) (n 2 0) is dense in X. 14.

Let X be a compact space, u : X + X a ccntinuous mapping, and p a positive measure on X, invariant with respect to u and of total mass 1. Let A be a p-measurable subset of X. For each x E A, let n(x) denote the smallest integer n 2 1 such that u"(x) E A, If we put A, = u - " ( A ) and B. = u-"(X - A), then the set of points x E A for which n(x) = rn is A,, n B, n Bz n ... n B,-, n A,,,, and hence the function x ~ n ( x is) measurable. (a) Put a0 = 1 and a,

= p(Bo

n B1 n . . . n B",-,) (rn 2 1). Show that

p(Ao n B1 n B, n ... n B,,-l n B, n A,,,,)

=

+

a,,, - 2a,,,+, a m f 2

for all m 2 0. (b) Show that the series whose general term is a,,, - a,+1 is convergent and that its terms form a decreasing sequence (interpret a,,, - urn+,as the measure of a set). Deduce that lim rn(tL, = 0.

n B..

m-m

(c) Let B,

=

Show that IAnLL(x)dp(x) = 1 - pCL(Bm)

("Kac's theorem": use (a) and (b)). Consider the case where u is ergodic with respect to p, and p(A) > 0. (d) Suppose that u is bijective and ergodic with respect to p, and that p(A) > 0. Let En, be the set of points x t A such that n(x) = rn. Show that the complement in

A of

u Emis negligible (cf. Problem Il(a)). Show that the sets up(E,) for rn 2 1 m

m=l

and 0 ' p < rn are mutually disjoint, and that the complement of their union in X is negligible (" Kakutani's skyscraper "1.

150

Xlll

INTEGRATION

15.

Let f be a lower (or upper) semicontinuous real-valued function on R". Let X be a locallycompact space and ul , . . . , universally measurablefinite real-valued functions. Show thatf(ul, . . . , u,) is universally measurable (cf. Problem 17).

16.

With the notation of Section 12.7, Problem 3, show that there exist n universally measurable mappings f w z k ( t )of C" into C (1 5 k 5 n) such that P,(X)=X"+ r,X"-'+

...+ t . - n ( x - z & ) ) . k=O

(By induction on the degree n , using Problem 15.) 17. Let f be the continuous function on I = [0, I ] which was defined in Section 4.2, Problem 2(d), such that the restriction o f f t o the Cantor set K is a bijection of K onto 1. Let g be the function defined by g (x ) = x -tf(x); then g is a homeomorphism of 1 onto 21= [0,2], such that g(K) is a compact set of measure 1 (with respect to Lebesgue measure), although K is negligible. If h(x) = 3-'(2x), then h is a homeomorphism of I onto I such that h-'(K) is not negligible with respect to Lebesgue measure. Moreover, if A is a nonmeasurable set (with respect to Lebesgue measure) contained in g(K), then the set B = h(&A)is negligible and therefore measurable. The function F~ h is not measurable, although h is continuous and pB is measurable (cf. (16.23)). 0

18. (a) Let f be a finite continuous real-valued function on

R. Show that there exists a real-valued function g on R which is universally measurable and such thatf(g(x)) = x for all x E ~ ( R(cf. ) Section 12.7, Problem 1). (b) Assume that there exists a partition of I = [0, 11 into a family (HJnSz of nonmeasurable sets (with respect to Lebesgue measure) such that Card (H.) = Card(R) for all n E 2. Let K be the Cantor set (Section 4.2, Problem 2) and F = (K t n),

u

n s z

so that F is a negligible closed set. Show that there exists a bijection f:R .+R such thatffF n In, n 1[) = H, for all n, and such that f is linear in each of the component intervals of IF. Then f is measurable with respect to Lebesgue measure, but f - I is not.

+

19.

Let X be a locally compact space and p a positive measure on X. Let D be a denumerable dense subset of R.Show that a mappingf: X R is p-measurable if and only if, for each r E D, the set of points x E X such thatf(x) 2 r is p-measurable.

20.

Suppose p bounded and p ( X ) = I . If f

20

s

is an integrable function, show that

(1 -tf2)''2is integrable and that, if A = fdp, then

(Use Problem 14(a) of Section 13.8.) 21.

-

Let X, Y, Z be three locally compact spaces. iff: X + Y and g :Y Z are universally measurable, show that g of is universally measurable. (To show thatg o f is p-measurable where p is any positive measure on X, reduce to the case where X is compact and consider the measure f(p) on Y (Section 13.4, Problem 8).) In particular, if B c Y is universally measurable, then f -'(B) is universally measurable.

9 MEASURABLE FUNCTIONS 22.

151

Let X be a locally compact space, Y a compact space, p a positive measure on X, and (fn)">, a sequence of p-measurable mappings from X to Y . Show that there exists a mapping ( p , X)HU~(X) of N x X into N with the following properties: (1) for each integer p 2 0 and each integer n 2 0, the set upl(n) is p-measurable; (2) for each x E X, the sequence (f~,cx,(x)),soconverges in Y . (One method is as follows. Taking a distance function on Y , define universally measurable subsets A, of Y satisfying conditions (i) and (ii) of Section 4.2, Problem 3(a) and such that (in the same notation) A,, n A , - = a . Then define subsets B, of X inductively as follows: if B, has been defined in such a way that, for each x E B,, infinitely many terms of the sequence ( f , ( x ) ) belong to A,, we define B,. as the set of all points x E B, such that infinitely many terms of the sequence (fn(x)) belong to A,, , and we define B,. to be the complement of B,, in B,. Now let x E X and let (&,JP>, be the sequence of 0's and 1's such that, if s, = then x E B,, for all p . Define up(x) by induction as follows: uo(x) is the smallest integer n such that fn(x) E A,, , and up(x)is the smallest integer n > O,-~(X) such that f.(x) E A,, . Use Problem 21 to verify that this definition of u,(x) satisfies the conditions of the problem.)

23. Let p be a bounded positive measure on X and let f > 0 be a pintegrable function.

Show that there exists a lower semicontinuous function g such that g 2 l/f (with the convention that l / O = to) and such that gf is integrable (with the convention that the product of 0 and cc is 0). (Reduce to the case wherefis bounded; consider the sets A. of points x E X such that f ( x ) 5 Ijn, and apply (13.7.9) to these sets to construct 9 . )

+ +

24. Let X, Y be locally compact spaces, p a positive bounded measure on X, and 7r a p-measurable mapping from X to Y . For each f c Xw,(u>, show that f o n is p-integ-

rable and that the mapping f~

s

( f 0 n)dp is a posiriue measure v on Y (called the

image of p under T , and denoted by ~ ( p ) )Generalize . the results of Section 13.8, Problem 12, and Section 13.9, Problems 10 to 14.

25.

Let X be a compact space and let (m,)be a sequence of finite partitions of X consisting of integral sets, such that m ,is finer then w. whenever m > n (Problem 7). An elementary martingale relative to the sequence (m,)is by definition a sequence (f.) of integrable functions 2 0 such that: (i) fn is constant on each set of the partition m,; (ii) If m > n, for each set A em. we have

(in other words, on each set A E wnthe value offn is equal to the average offm over A). (a) Let a , b be real numbers such that 0 5 a < b, and let Eabbe the integrable set of points x E X such that

lim inffn(x) < a < b < lim supfn(x). n-m

n-m

Show that E,, is negligible. (Suppose if possible that p(E.J > 0. For each integer p > 0, let F, be the union of the sets A E m. (n > p ) such that A n Eubi 0; if F = F,, then p(F) 2 p(Enb)> 0. Show that for each integer p and each A E m, (n 2 p )

n P

such that A n Eabf

a,we have b . p(A n F,)

a . p(A) for all q

2 n, by using the

152

XI11 INTEGRATION

+

+

definition of a martingale; now let q+ co, then p - + co, and so obtain a contradiction.) (b) Deduce from (a) that the sequence (f.) converges almost everywhere to a n integrable functionf: (Give a , b rational values, and use Fatou’s lemma.)

J^

(c) Give an example where f dp is not equal to the (constant) value of the integrals

J^hdp. (Take X

~7

[0, 11 and take p to be Lebesgue measure; take wnto consist of the

intervals [0, 2-”[, [2-”, 2-”+’[, ..., [B, 11.1 (d) Let F(x) = supf.(x). Show that there exists a constant C > 0 such that, for all n

a > 0, if B“ is the set of points x such that F(x) > a, we have p(B.) that B, is the union of the sets Ba, = { x E X : f.(x) > a}.)

5 Cia. (Remark

26.

With the hypotheses of Problem 25, let (T be another finite partition of X into pintegrable sets. For each x E X and each integer n, let B E u and A E w. be the sets of the partitions u and m, which contain x ; put f.(x) 0 if p(A) = 0, and f&) = p(B n A)/p(A) if p(A) > 0. Show that the sequence ( f . ( x ) ) converges almost everywhere to a limit f ( x ) 5 I , and that the sequence (f.)converges in mean tof: (For each set B t u, observe that the functions ysfA form an elementary martingale for the measure yB. p, and use Problem 25.)

27.

Let X be a compact space with measure p(X) = 1 , and let a = (Ai)1E16nbe a finite partition of X into integrable subsets. For each x t X, if Ai E a is the set containing x, the number i(a;x ) =

- log

p(Ad

is called the information at the point x corresponding to the partition a . The number

(with the convention that f log t = 0 when t = 0) is called the entropy of the partition a (relative to the measure p). It is the “average” of the information corresponding to a. I f / = (BJ, Eks,,, is another finite partition of X into integrable subsets, the number H(a//)

=

- C p(AI n Bk) log(p(A, n Bk)/p(BxN2 0 i, k

(in which the terms for which p(BJ = 0 are replaced by 0) is called the entropy ofa relative ro p. If (0is the partition consisting of the single set X, then H ( a / w )= H(a). (a) If ( C L , ) ~ ~is, 0, then h(u) = 0.

10. INTEGRALS O F VECTOR-VALUED F U N C T I O N S

Consider first a mapping f of X into ajnite-dimensional real vector space

E. Let (ei)l5is,n be a basis of E, and put f(x) =

cfi(x)ei for each x m

i= 1

E X.

Then f is said to be integrable (with respect to p ) or p-integrable if each of the real-valued functions fi is integrable, and we define (1 3.10.1) It is immediately checked that this definition is independent of the basis chosen. If llzll is a norm defining the topology of E, then we have the following criterion: (13.10.2) A mapping f : X -+E is integrable ifand only iff is measurable and XH Ilf(x)JIis then integrable.

j*llf(x)ll dp(x) < + co. Thefunction

For f is measurable if and only if the f;: (1 S i 5 n) are measurable, by (13.9.6); also there exist two constants a, b such that a

Ci Ifi(x) I IIIf(x>IIS b 1 I fi(x>I i *

by (5.9.1). The result now follows from (13.9.13) and (13.9.6). In particular, a complex function f on X is integrable if and only if Bf

10 INTEGRALS OF VECTOR-VALUED FUNCTIONS

155

and 9 f are integrable, and we have

from which it follows immediately that S f d p = J Z

The set Y,!.(X, p) of p-integrable functions on X with values in C is a vector space over C , and f~ I f d p is a linear form on this vector space. Furthermore, iff is any complex-valued integrable function, then I f [ is integrable and

(13.10.3)

+

For since 1.f I = ( ( W f ) 2 ( Y f ) 2 ) ” 2it, follows that If I is measurable (13.9.6),and we have p * ( l f l ) 5 p*(lWfl) p*(lXfl). Hence the first assertion, by (13.9.13).Also there exists a complex number c of absolute value 1 such that Mf)= Mf)L and hence IP(f’)l = W ( P ( c f ) ) = P ( W ( i f ) ) 5 Alfl), which establishes (13.1 0.3).

+

If E‘ is the dual of the finite-dimensional vector space E, then to say that thef, are integrable is equivalent to saying that, for each z’ E E’, the mapping x w ( f ( x ) , z ’ )is integrable (because it is a linear combination of the A). We generalize this as follows: if I is any set, then a mapping X H ~ ,of X into the vector space K’ (where K = R or C) is scalarly p-integrable if for each c1 E I the function x ~ f , ( a )is integrable.

(1 3.1 0.4) Let E be a Frkchet space, E‘ its dual (1 2.15). Let X H f , be a scalarly integrable mapping of X into E’ such that, for each convergent sequence (a,,) in E , there exists a ,function g 2 0 defined on X such that p*(g) < + 03 and Ifx(a,)l 5 g(x)for all n, almost everywhere in X . Then there exists a continuous linear form z’ E E’ on E such that (Z , z’) = / f x ( z )dp(x)f o r all z E E.

sf,(.)

By (3.13.4)it is enough to show that if (a,,) is a sequence in E with limit z then dp(x)= lim sfx(a,) dp(x);and this follows from (13.8.4). n+m

sf,

The linear form z’ so defined is called the integral (or weak integral) of the function x-f, with respect to p , and is denoted by dp(x). Hence, for

Xlll INTEGRATION

1%

all x E E, we have (13.10.5)

If in particular E is a Hilbert space, then since there is a semilinear isometry z w j ( z ) of E onto its dual E’ (12.15), we can define the notion of a scalarly integrable mapping xl-*f(x) of X into E: this means that, for each z E E, the complex-valued function x.-,(f(x) Iz) is integrable. If the function XH(I f(x)ll is integrable, then by (13.10.4) there exists a unique element of E, denoted by jf(x) &(x) and called the integral (or weak integral) of f, with the property that (1 3.1 0.6)

for all z E E. PROBLEMS 1. If a mapping x H f , of X into R’is scalarly p-integrable, then the element

j

H fx(a) dp(x)

of R’ is called the integral of the mapping and is denoted by f,

J

f,

dp(x). If, for all x

E

X,

belongs to a weakly closed convex set A in R’ and if X is integrable and p(X) = 1,

then the integral jf,dp(x) also belongs to A. (Use Problem 13 of Section 12.15.) 2.

Let E be a separable real Frkhet space and E’ its dual (12.15). It follows from the Hahn-Banach theorem (Section 12.15, Problem 4) that the linear mapping cE which maps each z E E to the linear form Z’H (I,z’) on E’, is an injective mapping of E into RE’and is continuous with respect to the product topology on RE’.If K is any compact subset of E, the restriction CE I K is a homeomorphism of K onto the subspace ce(K) of Re’(12.3.6). (a) Suppose that K is a compact conuex subset of E. A mapping f of X into K is said to be scalarly p-integrable if, for each z’ E E’, the real-valued function XH (f(x), z’) is p-integrable. Using the remarks above and Problem 1, show that there then exists a unique vector z E K such that (z, z is denoted by

s s

2’) =

I

0 for all k . If nc > 1 and E > c(1 - c)/(n - l), show that there exist two distinct indices i, j such that p(Afn A,) cz - E . (Argue by contradiction, using the CauchySchwarz inequality.)

5.

Let (fn)be a sequence of functions belonging to -Yk(X, p) ( p = 1 or 2) which converges almost everywhere to a functionf. (a) Show that, if N,(f.) 5 a for all n, then f~ U k ( X , p) and N,(f) 5 a. (Use Fatou's lemma.) Give an example (with p = 1) where N,(f,) does not tend to N,(f). (b) Now suppose in addition that N,(f.)+N,(f) as n-, S m . Show that N,(f-f,) + 0. (Show that for each E > 0 there exists a compact subset K of X and an

If. 1

dp 5

E

for all n 2 no .)

6 . In the space -Yi(I,A), where I = 10, 11 and h is Lebesgue measure, consider the functions t' ( a real), which belong to this space provided that a > - 3. Show that a sequence of distinct exponents a, > -3 is such that the functions tanform a total sequence in Yi(1, A) if and only if the sequence (an) satisfies one of the

following three conditions: there exists a subsequence of the sequence (an) tending to a finite limit > - 4; 1 (2) lima,=- - - and n-m 2 lim a. = m a n d z a;' = + co. (3) (1)

n-r m

+

"

(Using Weierstrass' theorem (7.4.1), calculate for each integer m > 0 the minimum of Nz(t"'- f ( r ) ) asfruns through the set of all linear combinations of the first n functions tuk. For this purpose, use Problem 3(b) of Section 6.6, and the formula (Cauchy's determinant) (or - aj)(bi - 6,) det - = (at 6,) ' 0,

(c) Deduce that, for each measurable subset A of X and each E > 0, there exists an integer n > 0 with the property that for each integer m > 0, there exists an integer k such that m 5 k =< m t n - 1 and p(A n u - ~ ( A >= ) ) (p(A))’ - E (“Khintchine’s statistical recurrence theorem ”). (d) For each f E and each integer n , put

Show that the limit F A X ) = lim

m-rm

1

m-1

- hC ( ( U h.f,>(x) - (P . f ) ( x ) Y , =o

m

s

(where P .fdenotes a function in the class P . f ) exists almost everywhere and that lim F, dp = 0. (Use Birkhoff’s ergodic theorem.)

n-r m

(e) LetfeY’(X, p) be such thatf(x)z 0 almost everywhere. Show that, for almost all x E X , either f ( x ) = 0 or ( P . f ) ( x ) > 0 . (Consider the set N of points x E X at which ( P . f ) ( x )is either undefined or equal to zero.) 11. Let X be a metrizable compact space and let u : X + X be a homeomorphism. The set I of measures 2 0 on X of total mass 1 and invariant under u is then a nonempty vaguely compact subset of M(X) (Section 13.4, Problem 8). A point x E X is said to be quasi-regular (relative to u) if the sequence of measures

+

converges vaguely as n + 03 (necessarily to a measure px E I). Let Q denote the set of quasi-regular points of X. A point x E Q is said to be ergodic (relative to u ) if the measure pxis ergodic (Section 13.9, Problem 13). Let E be the set of ergodic points x E Q. A point x E Q is said to be dense if x belongs to the support of px. Let D be the set of dense points x E Q. The points belonging to R = E n D are said to be regular (relative to p). (a) Show that the complement of R (and hence also the complement of Q, E and D) is negligible with respect to any invariant measure v E I. (To show that Q has measure 1 for any measure v E I, apply Birkhoff’s ergodic theorem to the functions belonging to a dense sequence in V?(X). To show that D has measure 1, consider a denumerable basis (U,) for the topology of X, and for each pair (m.n ) such that 0, c U,, a continuous

168

Xlll INTEGRATION

mapping gmn: X + [0, 11 such that gmn(x)= 1 for all x E U, and gmn(x)= 0 for all x E X - U.; apply Problem 10(e) to each of the functions gmn. Finally, to show that E has measure 1, show (with the notation of Problem 10) that, for any measure v E I,

j((P . f ) ( Y ) - (P. f ) ( x V W Y ) = 0 for almost all x E X and allfE U ( X ) ,by applying Problem 10(d); now letfrun through a dense sequence of functions in U( X) .) (b) With respect to the vague topology on M(X) show that, for every measure v E I,

in the sense of (13.10), and that the external points of I are the measures which are ergodic with respect to u (cf. Section 13.10, Problem 9). (c) For every ergodic measure v E I, let QV (resp. R,) be the set of points x E Q (resp. x E R) such that px = v . The measure v is concentrated on R, (13.18). The sets QYand R, are called the quasi-ergodic set and the ergodic set corresponding to v . The Qv(resp. R,) form a partition of Q (resp. R), and we have u(Q,)= Q. and u(RJ = R,. Show that, for each closed set F such that u(F) = F, we have either F n R,= R, or F n Ri= 0. (d) For every nonempty closed set F such that u(F) = F, show that R n F # @ (consider the points which are regular with respect to u [ F). (e) Let A be the monoid generated by l x and u. Show that if 2 is any minimal closed orbit with respect to A (Section 12.10, Problem 6) we have u(2) = 2 and hence R n 2 # @. Deduce that, for each x E X, the closed orbit O(x) of x with respect to A intersects R, and that O ( x ) n R is a union of ergodic sets R, . (f) For each x E X,let p be a measure which is a cluster point of the sequence (pn,x) in M(X). Show that p(O(x))= 1. (g) Show that for a measure p E I to be ergodic it is necessary and sufficient that there should exist an ergodic set R, such that p(RJ = 1, and that then we have p = v . (Use (b) to prove that the condition is sufficient.) (h) If x E X is such that O(x) contains only one ergodic set, show that x is an ergodic point (use (f) and (g)). (i) Suppose that I consists of a single measure y o . Then Q = X, and R = D is the support of vo (use (h)) and is the only minimal closed orbit with respect to M. More1”-l

over, for each function f~ %‘(X),the sequence of numbers f(uk(x)) converges to n k=O f / d v , uniformZy in x

E

X. (Observe that, for each n , the mapping x ~ p . ,of X into

M(X) is continuous with respect to the vague topology, and use (7.5.6))

(s’

12. For every real number p such that 0 < p
1, put q = p / ( p - I). I f f , g are any two mappings of X into R,show that N l ( f g ) 5 N ,(f)N,( g) (“Holder’s inequality”). (Show that the set of points (rl, tz) E R2 such that t 1 2 0 and t 2 2 0 and t : l P t Y 2 1 is convex, and argue as in Section 13.8, Problem 14(c), making use of (13.5.6). Notice that the proof of (13.11.2.2) is a particular case.) (b) I f f 2 0,g 2 0 and p > 1, then N,( f ’ + g ) I N , ( f ) N,(g) (“ Minkowski’s inequality”). (Same method as (a): consider the set of points (II, t z ) such that t l L 0 and t 2 2 0 and ti’’ I:/’ 2 I . )

+

+

11 THE SPACES L’ AND

L2

169

(c) Extend the results of Section 13.11 to the case where p is any number such that 1 < p < +a. (d) Suppose that p > 1. Let f~ U i , g E -3’2 be two functions 2 0. Show that we have Nl(fg) = N,(f)N,(g) if and only if there exist two constants a > 0, p > 0 such that a(f(x))” = p(g(x))‘I almost everywhere. (e) Suppose that the measure p is bounded and that p(X) = 1. For each r > 0 and each measurable function 0 such that f‘ is integrable, show that the mapping p - N p ( f ) is an increasing function on the interval 10, r ] (use HGlder’s inequality).

fz

As p - 0 , its limit isexp(J* log

If1

d p ) , or 0 if

I’

log

If1

dp = - co. If the limit is

# 0, it follows that f ( x ) # 0 almost everywhere. If exp(l* log

the function (f) Let

If1

if1

dp)

=

/If1

dp,

is constant almost everywhere.

fz0 be a p-measurable function, and for each a > 0 let A, be the set of all

2 a. I f f e &;, then app(A,) 6/fp dp. Conversely, if the measure p is bounded and if there exist constants C > 0, E > 0 such that p(A.) C . a - p - c for all a > 0, then f~ 9: (cf. Section 13.9, Problem 3).

x E X such thatf(x)

+

f be a real valued function 2 0 defined on R*, = ] 0, to [, which is Lebesguemeasurable and such that I f l p is Lebesgue-integrable (1 < p < co). (a) The functionfis integrable on every compact subset of LO, co [,and in particular

13. Let

c +

F(x) =

f ( f ) dr

+ +

is defined for all x > 0 (use Holder’s inequality). As x tends to 0

or t o co, the quotient F ( X ) / X ( ~ tends -~)’~ to 0 (same method). (b) Show that the function F(x)/x is pth-power-integrable on 10,

+ co [ and that

(“Hardy’s inequality”). (Consider first the case where f~ X(R+,).For each compact interval [a, b] c R*, , majorize the integral using Holder’s inequality.)

Jab

(F(r)/fy df, by integrating by parts and

f be a complex p-measurable function such that ( I ) I f l p is p-integrable and ( 2 ) e r f is p-integrable for 0 < f < f o . Show that N p ( r - ’ ( e r f - 1) -f) -0 as r +O. (Let f = u iv where u and v are real; observe that if we put wI= erY- 1 - ru, we have w,/s 2 wI/f for 0 6 s 5 t. If 0 c p < 1, use the elementary inequality (a b)’ 5 u p bP when n , b are 2 0 . Finally, apply (13.8.1) and (1 3.8.4).)

14. Let p be a real number > O and let

+

+

15.

+

Let (An)”s1be a strictly increasing sequence of integers >0, and let h be a rational integer. For each integer N 2 I , put

Show that the series

170

Xlll

INTEGRATION

converges. Note that if mz 6 N < (rn

+ l)',

we have

~ to 0 for almost all x E [0, I ] (with and deduce that the sequence ( & ( x ) ) ~ >tends respect to Lebesgue measure). Hence show that, for almost all x E [0, I], the sequence (x.x - [x,x]) is equirepartitioned with respect to Lebesgue measure (Section 13.4, Problem 7).

16. Let U be a continuous endomorphism of the space LA(X, p) with norm 5 1 (in other words, such that NI(U . f ) 5 N,(f)) and such that the relationfz 0 implies U .f& 0 (13.6). l f f i s any function in the class we denote by U . f a n y function in the class V .J: (a) Let (f,) be a sequence of functions in 2 L p :which converges almost everywhere to a function f and is such that there exists a function h E Y i satisfying 1f n l 5 h for all n. Show that under these conditions the sequence ( U .f,) converges almost everywhere to U -f (Reduce to the case f = 0, and consider the sequence of functions

r:

9. = SUP

lf"+Pl.)

P

(b) For each finite sequence (rk)l

k 0, then for all (11

tl

r,,,

E

R we have

+ s,(tz, . . . , t n + d1s.(tl, ..., td.

(c) For each functionfE U:, show that

...

S"(U.f,

)

un+1.f)z U.S"(f,

. . ' ) U..f)

almost everywhere. (Use (I) and the fact that for any n functions f , , we have sup

IqkQn

u'fksu'

sup

...,f , E 14:

fk

I&k 0 and each integer n 2 0, let E , , , t ( f )denote the set of points x E X such that

t ...+ (u"-' .f)(x)>nt,

f(x)+(U.f)(x)

and let E,(f) denote the union of the En,,(f). Show that each of the sets E,,,(f) is integrable and that

(Observe that E.,,(f) is contained in the set of points x E X satisfying one or other of then relations ( U k . f ) ( x ) > t, where 0 5 k n - 1; then continue as in Problem 16, by remarking that


= n o , we have I f ( x ) -f,(x)I 5 l / m for all X E CH,. The union H of the sets H, is negligible, and f , ( x ) tends to f ( x ) uniformly in GH. (13.12.4) The normed space L,"(X, p) (resp. L,"(X, p)) is complete (Lea, a Banach space).

Let (,f,) be a sequence such that (f,)is a Cauchy sequence in L; (resp. L,"). For each integer n >= 1, there exists an integer k , such that N,(L

-f)5 1/n

12

THE SPACE L"

175

for r 2 k, and s 2 k , . For each pair (r, s) such that r 2 k , and s 2 k,, let A,,, be the negligible set of points x E X such that If,(x) -f,(x)I > I/n, and let A be the union of the sets A,,, (n 2 1, r 2 k , , s 2 k,), so that A is a negligible set. It is clear that in X - A the sequence ( f , ( x ) ) converges uniformly to a limitf(x). The functionf, which is defined almost everywhere, is measurable (13.9.10) and bounded in X - A. Hence f~ Lg (resp. f~ L,"). Clearly N , ( f - f , ) 5 I/n for all r 2 k , , and therefore 7 is the limit of the sequence (f,).

(13.12.5) r f f E S[(X, p ) (where p = 1 or 2) and g E =.YF(X,p), then PI, and N,(fg) 5 N p ( f ) N m ( 9 ) .

fgE

ym

For fg is measurable (13.9.8.1), and If(x)g(x)l everywhere; hence the result.

If(x)IN,(g)

almost

Remark

(13.12.6) Evidently we have %;(X) c S g ( X , p); but in general the canonical image of %?;(X) in L,"(X, p ) is not dense in the latter space, and L,"(X, p ) is not separable in general (Problem 1). PROBLEMS

1. If h is Lebesgue measure, show that the space Lg(R, h) is not separable. (If (A,) is an infinite sequence of nonnegligible measurable sets, no two of which intersect, consider the functions which are constant on each A, and take only the values 1.) 2.

(a) For every p-integrable subset A of X ahd every 6 > 0, let V(A, 6) denote the set of p-measurable real-valued functions f such that the set M of points x E A for which If(x)l > 6 has measure p(M) 2 6. Let Y ( X , p) denote the vector space of (finite) real-valued p-measurable functions on X. Show that the sets V(A, 6) form a fundamental system of neighborhoods of 0 for a topology on Y(X, p) compatible with its vector space structure (Section 12.14, Problem 1). This topology is called the topology of convergence in measure (with respect to p). A sequence (fn)which tends to a limitf in this topology is said to converge in measure to f. (b) Show that the intersection of all the neighborhoods V(A, 6) is the subspace .N of negligible functions, and that the quotient space S(X, p) = 9 ( X , p ) / N is metrizable. (c) If (fn) is any sequence in Y ( X , p) (12.9) such that (h)is a Cauchy sequence in S(X, p), show that there exists a subsequence (jJ of (f,)such that the sequence (fnk(x)) converges for almost all x E X. Deduce that the metrizable vector space S(X, p) is complete. (d) Every sequence (1;)of measurable real-valued functions which converges almost everywhere to a functionf, converges in measure tof. (e) For every finitep 2 1, show that the space U g ( X , p) is dense in Y ( X , p),and that the topology induced on 3g(X,p) by the topology of convergence in measure is coarser than that defined by the seminorm N, .

Xlll INTEGRATION

176

(f) Suppose that X is compact and that the measure p is diffuse (13.18). Show that, for every neighborhood V of 0 in .V(X, p) and every function f E Y(X, p), there exists an integer n such that all the functions af(where a is any real number) belong to V ... V ( n summands). Deduce that every continuous linear form on .V(X, p) is identically zero, and hence that every vector subspace of finite codimension in .V(X,p) is dense in Y ( X , p).

+ +

3. Suppose that X is compact and the measure p diffuse (13.18). (a) Let be a Hilbert basis of Lk(X, p). Show that, for each real number 6 > 0, there exists a compact subset Y of X such that p(X - Y) 5 6 and the sequence is total in L;(Y, py).(By using Problems 2(e) and 2(f), show that there exists a sequence of linear combinations of the f. ( n >= 2) which converges in measure to fl ; then use Problem 2(c) and Egoroffs theorem.) (b) Show that there exists a bounded measurable function h such that the sequence (&),,2 is total in L&(X,p). (Choose h > 0 such that f J h $ U&(X,p),and then show that no nonnegligible function can be orthogonal to hf. for all n >= 2.) 4.

Let p be a finite real number if, for each

E

>= 1. A subset H of

.PJLpg(X, p) is said to be equi-integrable

sA1fIp

> 0, there exists a compact subset K of X such that

for all f~ H, and a real number 6 > 0 such that

6.

integrable sets A of measure p(A)

sx-,lf1P

d p =( E

dp 5 E for all f~ H and all

(a) On an equi-integrable set H, show that the topology of convergence in measure is the same as that defined by the seminorm N, . Is the conclusion true if H is merely bounded in Y p ? (b) A sequence (f.) in .P&(X,p) is convergent if and only if it is equi-integrable and convergent in measure. (c) Suppose that the measure p is bounded and that p(X) = 1. Let (f,)be a sequence of functions belonging to Y & ,and suppose that lim lim Nl(I -If.[)

= 0.

"+a

s s

Show that lim

n-r m

n-r m

I Yfnldp = 0. (Use (b).)

(d) With p as in (c), let (f.) be a sequence of functions belonging to 9 ';such that

If.1

d p = lim

n-r m

s

lfnll'z

dp = 1.

Show that lirn N1(l -f,)= 0. (Reduce to the situation of (c) above by using the n-r m

Cauchy-Schwarz inequality; write and

11 - L l S 11 - ILII

+ 1If.I -f,l

11 - I L I ~= 11 - I f n I 1 / z I 5.

11

+ I.Lnt1'21.)

Let F be a real-valued function with period 1 on R which is integrable on the unit interval I = [0, 11 (with respect to Lebesgue measure). (a) Iff is any bounded measurable function on [0,1], then

12 THE SPACE L"

177

If also F is bounded on [O, 11, then this equality is valid for every integrable functionfon [0,1]. (Start with the case where F is bounded a n d f i s continuous on [0, 11, then (forfE 2")approximatefby continuous functions. When F is unbounded, reduce to the case F 2 0 and approximate F by an increasing sequence of bounded functions.) (b) Deduce from (a) that, for every function f which is integrable on an interval [a, b ] in R , we have lim j a b f ( t )sin nt dt = lim

I-m

j a b f ( t )cos nt

dt = 0.

"-4"

(c) Let q~ be the canonical mapping of R onto T = R/Z, and let p be the measure on T which is the image under q~ I I of Lebesgue measure on I; then p is invariant under translations in the compact group T (cf. (1 4.4)). Let k be an integer > 1, and let u : T + T be the continuous mapping such that u(v(t))= u ( k t ) for t E 1. The mapping u is not injective, but p is invariant with respect to u. Deduce from (a) that u is ergodic with respect to p (use Problem 13(b) of Section 13.9). Deduce that lim sup F(k"t) n-tm

is almost everywhere equal to the constant ess sup F(t), and that the same is true of

sup F(nr).

n21

6. Let

(cI

f~ 9 ; ( p ) . Show that

Isf I =!if] dp

161

d p if and only if there exists c E C with

= 1 such thatf(x) = clf(x)[almost everywhere.

S

7. Suppose that p is bounded and that p ( X ) = 1. LetfE 9;.If we have (1 for every complex number [, show that

s

+ [fl

dp >= 1

f d p = O . (For a fixed [ E C , consider

8. (a) If l s p < r < + m , then L: n L, "cLg n L: and L ; n L , " c L L n L , " . On the space L;: n L," , the function N,,, = N, + N, is a norm with respect to which the space is complete. Give an example in which the norm induced on L&n L," by n Lg we always have N,,,(f) 5 2Np,,(f).) NP., is not equivalent to N,,, . (In The space L;: n L," is a Banach algebra relative to the product 38 defined in Section 13.6; it posesses an identity element only if p is bounded (in which case it is identical with L,"). (b) The set I(X, p) (or I(p)) of idempotents in the algebra L> n L2 is independent of r , and consists of the classes @ A , where A runs through the set of integrable subsets of X. If A, B are integrable subsets of X, the relation @A = qB signifies that the set D(A, B) = A u B -A n B (Section 13.8, Problem 15) is negligible; the relation p(D(A, B)) = 0 is an equivalence relation on the set of integrable subsets of X, and the quotient set may be identified with I(X, p). We shall write A in place of qA. All the norms N,," induce the same topology on I(X, p), and this topology is therefore defined by the distance d(& 8) = NI(vA - yB)= p(D(A, B)), with respect to which I(X, p) is a complete space. The mappings (A, 8)Hsup(& 8) = (A u B)" and (A, 8)Hinf(A, 8 )= (A n B)" are continuous for this topology. (c) Let Y be another locally compact space, v a positive measure on Y. If there exists an isometry U of I(X, p) onto I(Y, v) such that U ( 4 ) = the measures p and v are said to be isometric. For each number p E [l, m [, there then exists a unique

+

4,

178

Xlll INTEGRATION linear isometry U,,of LE(X, p) onto L((Y, v) which extends U. (Show that U has a unique extension to a linear bijection of the space E(X, p) of classes of p-integrable step functions on X, onto the space E(Y, v) of_classes of v-integrable step functions on Y . Begin by proving that if inf(A, 8 ) = $, then also inf(U(A), U(8))= C$ and U(sup(A, 8))= sup(U(A), U ( B ) ) ; then deduce that U ( A ) 5 U(B)whenever A 5 8 ; finally, show that for arbitrary A,B we have U(inf(A, B)) = inf(U(A), U(@) and U(sup(A,B)) = sup(U(A), U(8)).Observe then that N,(U(f)) = N,(f) forfE E(X, p).) Show that the image of L[(X, p) n LF(X, p) under U p is L:(Y, v) n L,"(Y, v), and that the restriction of U p to Lg(X, p) n L,"(X, p) is an isomorphism of Banach algebras.

+

(d) Conversely, for a p such that 1 s p < m , let V be a linear isometry of Lg(X, p) onto Lg(Y, v) such that the restriction of V to Lg(X, p ) n L,"(X, p) is an algebra isomorphism onto L;(Y, v) n L,"(Y, v). Then the restriction of V to 1(X, p) is an isometry onto 1(Y, v) which maps I$ to

4.

9. Let X be a compact space such that p(X) = I . Let 2 be a set of integrable subsets of X, such that the relations A E 5 and B E 5 imply that X - A E 2 and A n B E 5 . Suppose also that the set {A:A E Z} is dense in the metric space I(X, p) (Problem 8). (a) Let (Cj)lsjCn be a finite partition of X into p-integrable subsets. Show that for each E > 0 there exists a partition (Aj)1Qj4,,consisting of sets in 5 such that p(D(Cj, A,)) : E for 1 5j 5 n. (Observe that if p(D(Cj, Bj)) 5 6 for 1 5 j 5 n - I , then p ( B j n B,) 5 26 for I 51j < k 5 n - 1 ; if N is the union of the Bj n B,, consider

the sets Aj =: B,

-N

(1

5 j 5 n - 1) and A,, = X

u Aj).

"- 1

-

j=1

of X into integrable (b) Deduce from (a) that for each finite partition y = (C,), subsets, and each E 1 0 , there exists a finite partition a = (Aj)lajsn consisting of sets in Z and such that (in the notation of Section 13.9, Problem 27)

H(a/y) -i-H(y/a) 5 E . (Reduce to the case where none of the C j is negligible, and use the fact that the function t ~ +t log t is zero and continuous at f = I . ) 10. Let X be a compact space such that p ( X ) = 1, and let u : X --z X be a p-measurable mapping such that u ( p ) = p . (a) Let a be a finite partition of X into integrable subsets, and let Z be the set of all finite unions of subsets of X belonging to one or other of the partitions

n-

1

V [)-'(a).

j = O

Suppose that the set of classes {A: A E 2) is dense in 1(X, p) (Problem 8). Show that we then have h(u) : h(u, a). (It is enough to show that h(u, /3) 5 h(u, a ) for every finite partition /3 of X into integrable subsets; observe that we have

and use Problem 28(d) of Section 13.9, and Problem 9 above.) is (b) Under the hypotheses of (a), show that if in addition u is bijective and p-measurable, then h(u)= 0. (Observe that the set of classes { ( u - * ( A ) ) - :A E Z} is again densc in I(X, p), and use Problem 28(c) of Section 13.9.) (c) S~tpposethat u is bijective and N - I is p-measurable. Let 5' be the set of finite

" V uj(a),and suppose J = -n

unions of subsets of X belonging to one or other of the partitions

13 MEASURES WITH BASE p

179

that the set of classes A, as A runs through 5', is dense in 1(X, p). Show that h(u) = h(u, a ) (Kolmogoroff-Sinai theorem). (Same method as in (a).) (d) Take X to be the unitcircle U : 1 z 1 = 1, let p be the image of Lebesgue measure under the mapping t H e Z Z iof t [0, I ] onto U, and let N be the mapping z H e Z n L a z . Show that h(u) = 0. (Distinguish two cases according as 8 is rational or irrational. In the latter case use (b), by taking a to be a partition into two half-open semicircles.) 11. Let X, Y be two compact spaces and let p (resp.v) be a positive measure on X (resp. Y ) such that p(X) = v(Y) = 1. A p-measurable (resp. v-measurable) mapping u : X HX (resp. u : Y t,Y) such that u ( p )= p(resp. u(v) = v) defines an endomorphism U : ~ H(f0 u) - of L&(X,p) (resp. an endomorphism V :B H (g 0 v)- of Li(Y, v)) (Section 13.1 1, Problem 10). The mappings u, u are said to be conjugate if there exists an isometry T of I(X, p) onto I(Y, v) (Problem 8) such that V o T = T o U. Show that if this is the case then h(u) = h ( v ) .

12. Suppose that the measure p is bounded, and let p E [ l , +a[. Let (Un) be a sequence of continuous linear mappings of L&(X)into the space S(X, p) (Problem 2). For any function of f e -Yk we denote by U, .f any function belonging to the class U, .f and pu$

For each a > 0 let E.,N(f) denote the set of all x E X such that (U,*. f ) ( x ) > a, and let E,(f) denote the set of all x f X such that (U* . f ) ( x )> a. (a) Show that the set of f~ Lg such that p(E.,N(f)) 5 E is closed, for each E > 0. (b) Assume that for aN f E 2 k the function U* . f i s finite almost everywhere. Show that under these conditions the number

+

tends t o 0 as a+ co (Banach's principle). (Use Bake's theorem in the complete space Lg .) (c) Under the hypotheses of (b), show that the set H of classes f such that the sequence ((U, . f ) ( x ) )converges everywhere in X is closed in Lk. (Write

RVXX) = lim .-a

remark that R(f)(x)

= R(f-

(

SUP

m,n3,

I (u,,,. f ) ( x ) - (u,. f ) ( x ) lj,

g ) ( x ) almost everywhere for @ E H, and use (b).)

13. MEASURES WITH BASE p

n.

Then the following con(13.13.1) Let g be a mapping of X into C or into ditions are equivalent: (a) For each x E X,there exists a neighborhood V of x in X such that gqv is integrable.

180

Xlll INTEGRATION

(b) The function g is measurable, and f o r each compact set K c X we have J*lc?l(P.

dP < + a *

(c) For each function h E X,(X), the function gh is integrable.

+

Since we may write g = g1 ig, , where g 1 and 9 , are 8-valued, and then g1 = g : - g ; , g 2 = g t - g ; , we reduce straightaway ((13.9.6) and (13.10)) to the case where g is a mapping 2 0 of X into To show that (a) implies (b), cover K by a finite number of open sets V j such that g q is integrable for VJ each j . Then sup(gcp,,) is integrable (13.7.4) and hence so I S

w.

j

g(PK = (PK * sup(g'PVj) i

(13.9.14). Hence (b) follows from (13.9.13). To show that (b) implies (c), observe first that g is almost everywhere finite (X being a denumerable union of compact sets), hence gh is measurable (13.9.8.1). Moreover, if L = Supp(h), we have 1gh( 5 lgpLl * Ilhll, hence it follows from (b) and (13.9.1 3) that gh is integrable. Finally, to show that (c) implies (a), consider a compact neighborhood V of x e X and a continuous mapping h : X -+ [O, 1) which is equal to 1 on V and has compact support ((3.18.2) and (4.5.2)). By hypothesis gh is integrable, hence so is g p , = (gh)pv (13.9.14).

When the equivalent conditions of (13.13.1) are satisfied, the function g is said to be locally integrable (with respect to p) or locally p-integrable. Clearly every integrable function is locally integrable. Every measurable function whose restriction to every compact subset of X is bounded almost everywhere (in particular, every function belonging to 9 2 or 2'2) is locally integrable. Every function belonging to 9;or 9 6 is locally integrable, by (13.11.7). Every function equivalent to a locally integrable function is locally integrable. We have remarked in the course of the proof of (13.13.1) that every locally integrable function is finite almost everywhere. The function on R which is equal to l/lxl when x # 0 and is 0 when x = 0 is lower semicontinuous but not locally integrable with respect to Lebesgue measure. For a complex-valued function g to be locally integrable,,it is necessary and sufficient that Seg and 9 g should be locally integrable. For a real-valued function g to be locally integrable, it is necessary and sufficient that g+ and g- should be locally integrable.

i

Let g be a locally p-integrable function. Sincef H fg dp is defined on the whole of .X,(X), it is a linear form on this complex vector space. Moreover

13 MEASURES WITH BASE p

181

this linear form is a (complex) measure on X, because if K is any compact subset of X,we have

for allfE X,(X; K), by virtue of (13.10.3). The measure so defined is called the measure with density g relative to p, and is denoted by g * p ; when g is continuous, this agrees with the definition (13.1.5). Measures of the form g p are also called measures with base p. It follows immediately from this definition that if g takes values in R (resp. 20 almost everywhere), then the measure g * p is real (resp. positive). Furthermore, g p does not change if we replace g by an equivalent function (with respect to p), and therefore we may restrict ourselves to the case where g is everywhere finite and universally measurable (1 3.9.12). If g 1 and g2 are two locally integrable functions, then g 1 + g2 and ag, are locally integrable (a being any complex scalar), and we have

-

-

For every complex-valued locally integrable function g, we have

(13.13.4) The set 9;oc, R(X,p) (resp. 9’;oc, c(X, p)) of real-valued (resp. complex-valued) locally p-integrable functions is a real (resp. complex) vector space, often denoted by 910c(X,p ) or 2’loe(p) or U,,,(X) or Y p l O c . For every compact subset K of x, the mapping p K :g w )g(pKI dp is a semi-norm on this space. We shall always suppose YlOc to be endowed with the topology defined by these seminorms. If (K,) is an increasing sequence of K, = X and K, c K,,, (3.18.3), then it is compact subsets of X, such that

i

u n

immediately seen that the topology of 9,0c is defined by the seminorms pK,. In particular, if, X is compact, then 9’;oc, R(X,p) (resp. $Plot, c(X, p)) is identical with 9 k ( X , p) (resp. 9’A(X, p)). The set of locally integrable realvalued functions g such that p&) = 0 for all compact subsets K of X is the space Jfr of p-negligible functions. We define Lioc,

p) = 9’lloc,

p)/ Jfr

and L;oc,c(X, p ) analogously. The seminorms pK(g) depend only on the = p K ( g ) ,then we obtzin seminorms defining the class of g: if we put pK(g”) topologies of these spaces, which are therefore metrizable and locally convex.

Xlll INTEGRATION

182

(13.13.5) Zfg is locallyp-integrable, then so is 191, and wehave 1g pi

=

-

19) p.

The first assertion follows immediately from (1 3.1 3.1) and (1 3.7.4).To prove the second, note first that i f f 2 0 is a function belonging to X,(X) and if u E X,(X) is such that ( u (SJ then we have

(13.10.3), and therefore (13.3.2.1)1g . pl S 19) . p. To prove the reverse inequality, let L denote the (compact) support 0f.f. Let A be the set of all x E L such that g ( x ) # 0; the set A is integrable ((13.9.9)and (13.9.2))and therefore (1 3.9.1) there exists an increasing sequence of compact subsets K, of A such that A - K, is p-negligible. Hence, by (13.8.4),for any E > 0

u n

there exists an integer n such that /A-Kn f1g1 dp S E . The same reasoning, using the measurability of g and (13.8.4),shows that there exists a compact subset KA of K, such that g I KA is continuous and JA-K,, f l g l d p 5 2 ~Now . consider on KA the continuous function x t t Ig(x)l/g(x);by applying (3.18.2) and the Tietze-Urysohn theorem (4.5.1)to the real and imaginary parts of this function, we see that there exists a function w E X,(X) such that w(x) = ( g ( x ) j / g ( x for ) all x E KA. The function u = w . inf(1, l / J w J )(with the convention 1/0 = + 03) is then continuous on X, hence belongs to X,(X), and is such that v(x) =: Ig(x)l/g(x) in KA and [u(x)l 5 I throughout X. Hence If4 S f a n d

but

and

so that finally

Since E > 0 was arbitrary, this completes the proof.

13

MEASURES WITH BASE p

183

PROBLEMS

1. Show that the locally convex metrizable space L;oc,R(X, p) (resp. Lloc,&X, p)) is complete (in other words, is a Frkhet space). 2.

Let A“ be a vector subspace of .2’:oc,Rcontaining the subspace Jy of pnegligible Jy is endowed with the structure functions. Suppose that the quotient space H = P/ of a real Hilbert space, in which the scalar product is denoted by (318) and the norm 131.Suppose moreover that, for every compact subset K ofX, there exists a constant uK 2 0 such that

for all II E 2 (cf. Section 15.11, Problem 26). (We shall also use the notation (u I u) andlulfor (ci I d) andltilwhen N and u belong to 2.)Let 2; denotethe set of bounded measurable functions on X with compact support. (a) Given any function f~ 9 2 , show that there exists a function U’ E 2 such that

s

(Uf / u) = u f d p for all u E 2 ;and that the class of U’ in H is uniquely determined by the class off. The function Uf is called the potential of f. Show that the set of classes of potentials Uf is dense in the Hilbert space H (use (6.3.2)). If J g are two elements of

s

9 2 ,then (U’ I Us) = guydp =

i

f U g dp.

(b) As f runs through the functions 2 0 belonging to YE, the set of potentials U’ is a convex cone in 2. We denote by B the closure of this cone (with respect to the topology defined by the seminorm 161 on X ) . The elements of 9’ are called pure potentials. Let P denote the image of 9’ in H. For each element ri E H let d be the projection of ti on P (Section 12.15, Problem 3(a)). Show that we have lal’ = (61li), that u(x) 2 u ( x ) almost everywhere, and that d is the only element of P satisfying these conditions. Also we have 161 5 151. Deduce that, for each L? E H , the projection of 0 on the closed convex set of points 4 E H such that u(x) 2 u ( x ) almost everywhere belongs to P. (c) Deduce from (b) that an element u E P belongs to B if and only if ( u l w ) 2 0 for all w E 2 such that w ( x ) 2 0 almost everywhere (consider the difference between d and its projection on 9). Equivalently, Id i-3 I 2 151 for all w E X such that w ( x ) 2 0 almost everywhere. (d) Suppose that for all u E 2 we have IuI E 2 and I(lu1)- I5 Iril. Show that every pure potential u is 2 0 almost everywhere (use (c)). If u, u are two pure potentials, show that inf(u, u ) is a pure potential. (Among the elements of A“ which majorize inf(u, u), consider an element w such that 1 6 1 is a minimum; then w is a pure potential, by (b), and we have ( u w 1 u - w) 5 ( u w 1 Iu - w l ) . By calculating I(inf(u, w))” 12, deduce that I(inf(u, w ) ) - I $ 131; likewise that I(inf(u, w))- 15 161;and hence that w = inf(u, w) = inf(u, w ) almost everywhere.) (e) With the same hypotheses as in (d), show that i f f € 9 2 is 20 almost everywhere, and if u E B is such that Uf(x) 5 u ( x ) almost everywhere in the set of points x E X such rhat f ( x ) > 0 , then Uf(x) 5 u ( x ) almost everywhere in X (“principle of domination”). (Observe that u = inf(Uf, u ) is a pure potential, that (Uf 1 Uf - u ) = 0 and (u I U’ - u) 2 0, and deduce that u = Uf almost everywhere.) (f) Suppose that the hypotheses in (d) are satisfied and also that for all u E 2‘ we have inf(u, 1) E 2 and I(inf(u, 1))- 15 I t i l . Show then that if u is a pure potential,

+

+

184

Xlll INTEGRATION

then so is inf ( u , 1) (same method as in (d)). Deduce that, if u and u are pure potentials, then inf(u, ti 1) is a pure potential, by remarking that u inf(u, I ) is a pure potential 2 2 is 20 almost everywhere and if u E B is and using (d). Finally show that, if such that U f ( x )5 u(x) 1 almost everywhere in the set of points x E X such that f ( x ) > 0, then U f ( x )5 u(x) 1 almost everywhere in X ("complete maximum principle": same method as in (e)).

+

+

+

+

14. I N T E G R A T I O N WITH RESPECT TO A POSITIVE MEASURE WITH BASE p

(13.14.1) Let g be a locally p-integrable function which is 2 0 on X, and let v = g p. Then iff 2 0 is any 8-valued function on X, we have (13.14.1 . I )

s'f

dv

= s"fg

4

where, on the right-hand side, the value of f g is by dejinition taken to be 0 at every point x E X at which one of f ( x ) , g ( x ) vanishes (even if the other factor is +a (13.11)).

The proof consists of several steps. (13.14.1.2) Suppose first of all that f E 3. Then (12.7.8) there exists an increasing sequence ( f , ) of functions belonging to X,(X) such that f = sup f , . n

In view of the convention about products, this implies that the sequence (f,g) is increasing and that f g is equivalent to sup(f,g), because g is almost n

everywhere finite (with respect to p ) . Moreover, the functions f . g are p integrable (13.13.1), and 1f.g dp = . j f n dv. Hence it follows from (13.5.7) that

(1 3.1 4.1.3)

Every p-negligible set N is also v-negligible.

Suppose first of all that N is relatively compact. Then ((3.18.2) and (1 3.7.9)) there exists a decreasing sequence of relatively compact open sets U, containing N, such that inf p(U,) = 0. Since g q u , is p-integrable n

by virtue of (13.1 3.1), we have Sgq", dp = v ( ~ , ) by (13.14.1.2)

and

14 INTEGRATION WITH RESPECT TO A POSITIVE MEASURE

185

(13.7.7). But, by (13.8.4), if N ' x N is the intersection of the U,, then inf lg(pun dp = /g(pN.dp = 0; and since v(N) S v(U,) for all n, we have n

v(N) = 0. Now let N be any p-negligible set, and (K,) a denumerahle covering of X by compact sets. Then the sets N n K, are v-negligible, by what has just been proved, and hence so is their union N. (13.14.1.4) Suppose now that K = Supp(f) is compact and that f l K is continuous, with values in R (and therefore bounded (3.17.10)). Then there exists a decreasing sequence (U,) of relatively compact neighborhoods of K such that K = U, (3.18.2); also, by virtue of the Tietze-Urysohn theorem

n n

(4.5.1), there exists for each n a function f, E .X,(X), with support contained in U,, which extendsf and is such that [If,[] = Ilfll. Hence we have If,dv = / f , g dp for all n. Bearing in mind (1 3.13 4 , it follows from (1 3.8.4)

that f is v-integrable and fg is p-integrable, and that Ifdv = /fg dp. (13.14.1.5) The set A of points x E X such that g(x) = 6 is v-negligible.

Since g is p-measurable (13.13.1), A is p-measurable (13.9.9), and hence is the union of a sequence (K,) of compact sets and a p-negligible set N. By virtue of (13.14.1.4) applied t o f = qK,, we have v(K,) = sg(pKn dp = 0, and v(N) = 0 by (1 3.14.1.3) ; hence v(A) = 0. (13.14.1.6) Consider now the case where Supp(f) = K is compact andfl K is lower semicontinuous on K. Then it follows from (12.7.8) that there exists an increasing sequence of finite real-valued functions u, which are continuous and >, 0 on K and such that f l K = sup u, . Let f, be the function which is n

equal to u, on K and zero on X - K, so thatf= supf,. By virtue of (13.5.7) n

and (13.14.1.4), we have

j*

f dv = s u p n

j*

f, dv = sup n

j* *! f,g d p =

fg dp

sincefg is equivalent (with respect to p) to supf,g, by virtue of the convention n

about products and the fact that g is finite almost everywhere with respect to p.

186

Xlll

INTEGRATION

End of the Proof (13.14.1.7)

For every function u E 9 such that f 5 u, we have f g 5 vg, and

J’* v dv = J’* vg dp

J’* f g dp 5 J‘* vg dp = J’* v dv.

by (13.14.1.2), so that

Hence, by definition of the upper integral, we have it remains to establish the opposite inequality

I*

f g dp S

s*

f dv. Hence

(13.14.1.8)

Let h E 9 be such that h 2 f g . Then it is enough to show that

s* s*

(13.14.1.9)

f dv

5

h dp.

The set X - A is the union of a denumerable increasing sequence of compact sets H, and a p-negligible set N, such that g I H, is continuous, finite and > O for all n. We define a mapping u of X into R as follows: u = h/g in the union of the sets H,, and u(x) = 00 in N and in A. In each of the sets H, we have ug = h, and by virtue of (13.14.1.6)

+

;:j

dv = 1:;g

dp =

f/I d p .

But v(N) = 0 by virtue of (13.14.1.3), and v(A) hence, as h 2 0 , (13.5.7) j * u dv = sp :

:1;

dv = s:p

/:/I

=0

by virtue of (13.14.1.5),

dp 5 j * h dp.

Since f 5 u, we obtain the required inequaliLy (13.14.1.9).

Q.E.D.

(13.14.2) Let g be a locally p-integrable function which is 2 0 on X, let S be the p-measurable set of points x E X such that g(x) > 0, and let v = g p . Let f be a mapping of X into E.Then the following conditions (with the conventions of (1 3.11) for products) are equivalent : (a) f is v-measurable ; (b) f q s is pi-measurable : (c) f g is p-measurable.

We may suppose g to be finite. With the conventions we have made, we have f g = ( f cps)(gcps),and since the two factors on the right-hand side never

14 INTEGRATION WITH RESPECT TO A POSITIVE MEASURE

187

take the values 0 and kc0 simulta~ieously,it follows from (13.9.8.1) tnat (b) implies (c). Conversely, it is immediately seen that the function g’ which is equal t o g-l on S and vanishes on X - S is p-measurable: for there exists a partition of S (resp. X - S) into compact sets L, (resp. M,) and a p-negligible set P (resp. Q) such that the restriction o f g to each L, is continuous, and it follows that the restrictions of g’ to L, and M, are continuous. Furthermore, we have (fg)g’ =f q s , hence by the same argument it follows that (c) implies (b). Hence it is enough to prove that (a) and (b) are equivalent. Suppose first that f q s is p-measurable. The set GS is v-negligible, by virtue of (13.14.1.5); also the hypothesis implies the existence of a partition of S into a sequence of compact sets L, and a p-negligible set N, such that f l L , is continuous for each n. Also N is v-negligible, by (13.14.1.3); hence f is v-measurable. Conversely, suppose thatfis v-measurable. Then there exists a partition of X consisting of a sequence of compact sets H, and a v-negligible set N, such thatfl H, is continuous for all n. Also there exists a partition of S consisting of a p-negligible set L and a sequence of compact sets K, such that g I K, is continuous (and >O) for all n. For all n, we have inf g(x) = a, > 0. Finally, XEK,

there exists also a partition of X - S into a p-negligible set L’ and a sequence of compact sets KI,. It is clear that the restriction of f i s to each of the sets H, n K, and KL is continuous, and it is therefore enough to prove that N n S is p-negligible. Now if p*(N n K,) > 0, then we should have g dp 2 a,p*(N n K,) > 0, which is absurd because

S”K,

0 = v(N n S)

=lNnSg dp

by virtue of (13.14.1). Hence each set N n K, is p-negligible, therefore so is N n S, and hence finally f qs is p-measurable. (13.14.3) Let g be a locally p-integrable function uhich is 2 0 on X, and let v = g . p. Then a mapping f : X -+ R is v-integrable f a n d only f ( w i t h the concentions of (13.11)) f g is p-integrable; and in that case we hare

J

P

(1 3.14.3.1)

fdv=

J

P

fgdp.

For f to be v-iniegrable it is necessary and sufficient thatf’ and f - should be (1 3.7.4) ;and since we have (fg)’ =f ‘ g , ( f g ) - =f -9, with the conventions of 13.11 about products, it is enough to prove (13.14.3) when f 2 0. The first assertion is now a consequence of (1 3.1 4.1 ), (13.1 4.2), and (1 3.9.13); and the relation (13.14.3.1) is just (13.14.1 .I).

188

Xlll INTEGRATION

(13.14.4) With the hypotheses of (1 3.14.3), the measure v is bounded ifand only i f g i s p-integrable; and v = 0 i f and only i f g is p-negligible. (13.14.5) Let g,, g 2 be two mappings of X into R, such that g1 is 2 0 and locally p-integrable. Then g2 is locally ( g l * p)-integrable Sf and only f ( w i t h the product convention of (13.11)) g 2 g 1 is locally p-integrable, and in that case we have

By considering g: and g;, we reduce immediately to the case where g 2 2 0. To say that g 2 is locally ( g l * p)-integrable signifies (13.13.1) that, for every f E XR(X), the function g 2f is ( g , p)-integrable, or equivalently (13.14.3), that g 2 g l f is p-integrable; which in turn means that g 2 g 1 is locally p-integrable. Also, if we put v = g1 . p and 1 = g 2 - ( g l * p ) , we have f dA = s f g 2 dv f g 2 g 1dp by (13.14.3); hence the relation (13.14.5.1). a

s

=s

PROBLEMS

1. Let X be a locally compact space, p a positive measure on X, and (/") a sequence of

= f. I

functions belonging to U k ( X ,p). Write p n =fn . p and pn(A)

dp for any

p-measurable set A. (a) Show that if X is not compact it can happen that the sequence (A) is unbounded in L', but that the sequence (p,) is vaguely bounded. (b) Suppose that, for every subset A of X consisting of a single point, and for every open subset A of X such that the measure induced by p on the frontier of A has finite support, the sequence &(A)) is bounded. Show that the sequence (A) is bounded in L'. (Show first that every point xo E X has an open neighborhood U such that the sequence of numbers Ip,l(U) is bounded. To do this, argue by contradiction, by showing that otherwise it would be possible to define a strictly increasing sequence of integers (nk), a decreasing sequence (uk) of open neighborhoods of xo , and a sequence (wk) of p-quadrable (Section 13.9, Problem 7) open sets, with the following properties: o k c u k - i , ~cL(ukC{Ixo})~1/k, l p n ~ l ( u k - { X o } ) ~ 1for i < k , \kkcUk--k+,, and finally

Consider the union W of the Wk, and obtain a contradiction. Then show by a similar argument that there exists a compact subset K of X such that the sequence (Ip.l(X - K)) is bounded.) (c) Deduce from (b) that if the sequence (p.(A)) is bounded for every open set A in X, then the sequence (A) is bounded in L'. (d) On the interval [0,1], take p. to be the measure defined by a mass n at the point

189

14 INTEGRATION WITH RESPECT TO A POSITIVE MEASURE

0 and a mass -n at the point I/n. Then the sequence &(A)) is bounded for every open set A which is 1p.I-quadrable for all n , but the sequence of norms Ilp.jl is unbounded. Again, if we take pnto be the measure defined by a mass n at the point l / n and a mass --n at the point l / ( n l), then the sequence (p.(A)) is bounded for every interval A c [0,1] (and therefore also for every finite subset A of [0, l]), but the sequence of norms IIpL.llis not bounded.

+

2.

For each integer n 2 1, let E. be an at most denumerable closed subset of I = 10, 1 [, and let I, be the family of component intervals of I - En. Suppose that the maximum length d. of the intervals J E 9"tends to 0 as! n + m. Let h be Lebesgue measure on I and let A be a A-measurable subset of I. Suppose that there exists a number k E 10, 1[ such that, for all n and all J E I,,we have h(A n J) < kh(J). Show that h ( A ) = 0. (By using (13.7.9), show that h(A) 5 E k ( h ( A ) E ) for all E > 0.)

+

+

+

3. For each integer n > 0, the Farey series of order n is the set F. of all rational numbers which, when expressed in their lowest terms p/q, are such that 0 $ p 5 q 5 n, and arranged in increasing order. The distance between two consecutive terms of F. is therefore 5 l/n. (a) Show that if two rational numbers r = p / q and r'=p'/q' are such that qp' - pq' = & 1, then for every pair of integers (p", q") there exist integers x, y such that p" = px p'y and q" = qx 4'y. The fraction p"/q" belongs to the closed interval with endpoints r, r' if and only if x, y are of the same sign. (b) Deduce from (a) that if r = p/q and r' = p'/q' are two rational numbers belonging to the interval [0, 11, such that q > 0, q' > 0 and qp' - pq' = f1, then r and r' are consecutive elements of the Farey series FsUpc,, ,,). Moreover the smallest of theintegers m such that the open interval with endpoints r, r' contains a point of F, is the integer q q', and this open interval contains only one point of F,,,., namely(p +p')/(q 4'). (c) Conversely, if r and r' are consecutive terms of F., show that qp' - p4' = f1 (induction on n).

+

+

4.

+

+

For each x E I = 10, I ] , put p(x) = (I/x) - [I/x], where [t] denotes the integral part of the real number t. Ifp"(x) = p(p"-'(x)) is defined and nonzero, putqn+I(x)= [l/p.(x)] (with 41(x) = [l/xl). (a) For every n such that p"(x) is defined, show that

where A.(x) and B.(x) are integers k n - 1. The fractions An-l(x)/Bn-l(x) and A,(x)/B.(x) are two consecutive terms of a Farey series (Problem 3) and x belongs to the closed interval with these as endpoints. Deduce that p"(x) is defined for all n 2 1 if and only if x is irrational. The (finite or infinite) sequence 01 numbers q.(x) is called the continued fraction expansion of x, and the fractions A.(x)/B.(x) are the convergents to x . The A.(x) and B.(x) are constant on the complement of the denumerable closed set En of points x such that P.+~(X) is not defined, and on each of the component intervals of I - E,

the function p. is monotone and varies from 0 to 1 . (b) Let h be Lebesgue measure on I and let A be a h-measurable subset of I consisting entirely of irrational numbers. Suppose that is h-invariant with respect to p (Section

190

Xlll INTEGRATION 13.9, Problems 13 and 24). Show that &A) is either 0 or 1 . (Suppose that d = &A) t 10, 1 [. Show that, for each component interval J of I - En,we have &A n

J) 5 (2d/(l + d))h(J).

For this purpose use (a) above and Problem l(c) of Section 13.21 to show that

+

for suitably chosen integers p , q, r , s with p s - qr = 1. Use Problem 2 to complete the proof.) x), is invariant with respecf (c) Show that the measure p = g A, where g(x) = l/(l fo p. Deduce from (b) and the ergodic theorem (Section 13.9, Problems 13 and 24) that, for every &integrable function f on I and almost all irrational x E I, we have

+

(Gauss-Kuzmin formula). (d) Deduce from (c) that, for each integer p k n such that qx(x)= p (x irrational), then

2 1,

if v,(x, p ) is the number cf indices

for almost all irrational x E I. (e) Deduce from ( c ) that

for almost all irrational x .

15. T H E L E B E S G U E - N I K O D Y M T H E O R E M A N D T H E ORDER R E L A T I O N O N MR(X)

(13.15.1) Let p and v he two positire measures on a focally compact space X, such that v 5 p. Then there exists a focally pintegrable function g such tl~at v=g-/&.

We distinguish two cases. (1) Suppose first that v is bounded. Since the function 1 is then v-integrable, it follows from (13.11.2.2) that for everyfE X,(X)

(13.15.1.1)

Iv(.f)12

5 V(l)V(f2)

s V(l)Pc(f*).

191

15 THE LEBESGUE-NIKODYM THEOREM

This shows first of all that iff is p-negligible then v ( f ) = 0. Passing to the quotient by the subspace Jlr of p-negligible functions, the linear form v on X,(X) therefore defines a linear form ?++v(?) on the subspace g R ( X ) of Li(X, p ) which is the canonical image of X,(X) in Li(X, p). The inequality (13.5.1.1) shows that this linear form is conrinuous (5.5.1) on the subspace g R ( X ) ;since the latter is dense in Li(X, p) (13.11.6), the linear f o r m f w v ( f ) extends by continuity to a linear form on the Hilbert space L:(X, p ) (5.5.4). Hence there exists a function g E Yi(X, p) such that v ( f ) = p(gf) for all f E X,(X) (6.3.2). Sincegislocallyp-integrable (13.13),it follows that v = g . p . (11) General case. There exists a partition of X into a sequence of compact sets K, and a p-negligible set N (13.9.2). If we put M = X - N = K,,

u n

then q M is locally p-integrable and 1 - qM is p-negligible, hence (13.14.4) q M . p = p . Put p, = qK, p and v, = qK,. v. Since qKnis the lower envelope of a decreasing sequence of functions 2 0 belonging to X,(X), the relation v 5 p implies v, p, for all n ; furthermore, v, is bounded (13.14.4). Hence, by (I) above, there exists a locally p,-integrable function g, such that v,, =g, - p , = (g,qK,) . p = )gnqK,I p ((13.13.5) and (13.14.5)), and thefunction g,qK, is locally p-integrable. Let g denote the function which is equal to Jg,qK,I on each Kn and is zero on N. Then g is the sum of the series

-

m

C

lg,(PK,(.I f f i s any function 2 0 in X,(X), and m is any integer, then

and therefore ((13.8.4) and (13.13.1)) the function g is locally p-integrable.

__

m

Also, since f = zfqKn ilmost everywhere (with respect to

sfs

,= 1

/I),

we have

(13.8.4) d p = j f q n dv. To show that v = g * p , it remains to be proved that N is also v-negligible. But by definition (13.5.5), for each E > 0 there exists a function h E 9 such that q N 5 h and p(h) 5 E . Since h is the upper envelope of an increasing sequence of functions 2 0 belonging to X,(X), the inequality v 5 p implies that v*(h) S p(h) 5 E , and the proof is complete.

The following lemma will be generalized later (1 3.1 5.8) : (13.15.2) r f ( p j ) ,j j j r is any$nite sequence of complex measures on X , there exists a positiile meamre A such that each p j is a measure with base 2.

192

Xlll INTEGRATION

By writing each p j as a linear combination of four positive measures (13.3.6), we may assume that the measures p j are all positive. Then we take r

1 = 1p j , and apply the result of (13.15.1) to each p j j= 1

A.

(13.15.3) (i) With respect to the order relation on MR(X), m y two real measures p, v have a least upper bound sup(p, v) and a greatest lower bound inf(p, v). For each real measure p put p' = sup@, 0) and p- = sup( - p , 0); then we have (13.15.3.1)

inf(p+,p - ) = 0,

lpl = p + + p - = sup(p,-p)

p = p+-p-,

and, for any two real m e a w e s p, v,

(13.15.3.2)

(ii) Let 1 be a positive measure and let g l , g2 be real-valued locally I-integrable functions on X.r f p , = g1 1 and p2 = g2 * A, then (1 3.15.3.3)

and in particular, for any real-valued locally A-integrable function g , (1 3.1 5.3.4)

W e have g1 * 1 respect to 1.

( g . A ) += g +

*A,

g 2 1 if and only

( g ' A ) - = g - -1.

if gl(x) 5 g2(x) almost everywhere with

(iii) Let (gn) be an increasing sequence of locally A-integrable real-valued functions. Then the increasing sequence of measures gn ' A is bounded above in MR(X)$and only if the function sup gn is locally A-integrable, and in that case n we have (13.15.3.5)

SUP n

( g n * 1)=

(SUP n

*

A.

To prove (i), we remark that the measures p, v can be written in the form p = p1 - 1 2 , v = v1 - v 2 , where pl, p 2 , v1 and v 2 are positive (13.3.6). Applying (13.15.2) to the four measures p l , p 2 , vl, v 2 we see (by using (13.13.2)) that p =: u , Iand v = u . A, where A is a positive measure and u, u 1

15 THE LEBESGUE-NIKODYM THEOREM

193

are locally A-integrable real-valued functions. Hence the assertions of (i) are consequences of those of (ii). To prove (13.15.3.3), we reduce to the case where pz = 0; for if sup(pl - p 2 ,0) exists and is equal to (gl - gz)+ 1, it follows immediately (13.3) that we shall have

This reduces us to proving (13.15.3.4). For this, we shall begin by proving the last assertion of (ii), or equivalently that the relation g A 2 0 implies g(x) 2 0 almost everywhere with respect to A. Let N be the set of points x E X such that g(x) < 0. If we put v = 9-1,it follows from (1 3.14.1.5) that v(X - N) = 0; so if we can show that v(N) = 0, it will follow that v = 0, and this will establish our assertion (13.14.4). Clearly it is enough to prove that v(N n K) = 0 for each compact subset K of X. If U is any relatively compact open set g - dA g + dA. For by hypothesis containing N n K, then we have /(g' - g - ) f dA 2 0 for every function f 2 0 belonging to X,(X), and it is therefore sufficient to remark that g - dA = sup g - f dA, g f dA =

1"

ju

i

ju

s sf5

lu

sup g'lf dA, the supremum being taken over all functions f E X,(X) such that 0 qrl(13.5.1). Since g'(x) = 0 for x E N n K, we have g + dA = inf U

consequently, J N (13.14.3).

g - dA = inf U

I

ju

g f dA = 0;

g - dA = 0,which shows that v(N n K) = 0

+

Now let p be a measure such that p 2 0 and p 2 g A. Putting A p = 0 , we have (13.15.1) p = u * a and A = u 8 , where u and u are locally a-integrable and u 2 0 and u 2 0 almost everywhere with respect to 8. I t follows that u 2 gu almost everywhere with respect to a, hence that u 2 (gv)' = g'v almost everywhere with respect to a ; but this implies that p =u .a

>= (g+v) - a = g + - A

by virtue of (13.14.5). Finally, we have to prove (iii). If the sequence (9,. A) is bounded above in MR(X), then for each function f 2 0 belonging to .X,(X) we have

.s

sup fgn dA
0, there exists 6 > 0 such that the relations 0 2 h 5 f and l * h dp 5 6 imply / * h dv 5 E . (c’) For every compact subset K of X and every real number E > 0, there exists 6 > 0 such that the relations A c K and p*(A) 5 6 imply v*(A) 5 E (“absolute continuity” of v with respect to p ) . (d) v = sup (inf(v, np)). n

The fact that (a) implies (b) follows immediately from (13.14.1) and

(13.6.3). Clearly (b) implies (b’). Conversely, suppose that (b’) is satisfied, and let N be a p-negligible set. Since X is a denumerable union of compact sets K,, it is enough to show that each of the sets N n K, is v-negligible, and we may therefore assume that N is relatioely conzpact. Since N then has a compact neighborhood (3.18.2),it follows from (13.7.9)and (13.8.7,(i)) that we may restrict ourselves to the case where N is a denumerable intersection of relatively compact open sets; but in this case N is v-integrable ((13.7.7)and (13.8.7,(i)), hence is the union of a sequence (H,) of compact sets and a v-negligible set P. Since by hypothesis v(H,) = 0 for all n, it follows that N is v-negligi ble. To prove that (b) implies (d) and that (d) implies (a), we remark that we may express p and v in the form p = u * A and v = v 1, where 1 is a positive measure and u, v are finite, 2 0 and locally A-integrable ((13.15.2) and (13.15.3)).If we put v’ = sup(inf(v, nu)) and o” = u - ZI’, then we have n

21’

2 0, v“ 2 0 and sup(inf(v, np)) = u‘ . A (13.15.3).Hence to show that (b) n

implies (d) it is enough to show that (b) implies that v” . A = 0. Now the set A of points x E X at which v”(x)> 0 is contained in the set of points x at which

196

XI11 INTEGRATION

u(x) = 0,and therefore is p-negligible (13.14.1), hence v-negligible by hypothesis. On the other hand, U” = u q , , hence ((13.14.1) and (13.6.3)) the relation v(A) = 0 is equivalent to u“ being hegligible, and therefore implies that u” A = 0 (13.14.4). To show that (d) implies (a), it is enough to remark that the condition (d) signifies that u” * 1 = 0, and hence (13.14.4) the set A is &negligible. Hence if we put g(x) = u(x)/u(x) at points where u(x) > 0, and g(x) = 0 elsewhere, we shall have u(x) = g(x)u(x) at all x 4 A, hence almost everywhere with respect to 1.Consequently (1 3.14.5), g is locally p-integrable, and we have v = g * p. It remains to establish the equivalence of (b), (c), and (c’). Clearly (c’) is a consequence of (c), applied to f = q K .Also (c‘) implies that v*(A) = 0 for every relatively compact p-negligible set A; since every p-negligible set is the union of relatively compact p-negligible sets, this proves that (c’) implies (b). To prove that (b) implies (c) we shall argue by contradiction. Suppose therefore that there exists a function fo 2 0 which is p-integrable and v-integrable, and a real number a > 0 such that, for each integer n > 0, there exists a function 9, for which 0 5 g, S fo, j * g , dp S 2-” and / * g n dv > a . By virtue of (13.5.5), we may replace g , by inf(fo, 9:) for a suitably chosen function g: E 9,without disturbing the above properties, hence (13.9.13) we may assume that 9, is p-integrable and v-integrable. Now let

-

h = lim sup g, = inf h , , n-tm

n

where m

Since g, sfofor all n, it follows that h, is p-integrable and v-integrable for all n (13.8.2), apd we have

(13.5.8). Hence J h d p

=0

(13.8.1), and the hypothesis (b) therefore implies

that h is v-negligible (13.6.3). But we have

s

h, dv

>= a

(13.8.1),

which gives the required contradiction. Hence (b) implies (c), and the proof is complete. (13.15.6) Let p and v be two positive measures on X. Then the,followii?g conditions are equivalent: (a) The negligible sets are the same f o r p and f o r v .

15 THE LEBESGUE-NIKODYM THEOREM

197

(b) v = g . p7 where g is locally p-integrable and g(x) > 0 almost everywhere with respect to p. If (a) is satisfied, it follows from (13.15.5) that v = g * p and p = h * v, where g (resp. h) is positive and locally integrable with respect to p (resp v), Hence (13.4.5) the function hg is locally p-integrable, and we have p = (hg) * p, which implies (13.15.3) that hg is equivalent (relative to p) to the function 1, so that g(x) > 0 and h(x) = l/g(x) almost everywhere with respect to p. Conversely, suppose that v = g . p, with g(x) 0 almost everywhere with respect to p. Since (l/g(x))g(x)is equal to 1 almost everywhere with respect to p7 the function I/g(x) is locally v-integrable, and we have p = ( l / g ) v (13.14.5), so that the condition (a) is satisfied. If p and v satisfy the equivalent conditions of (13.15.6) they are said to be equivalent positive measures on X. Clearly we have here an equivalence relation on the set of positive measures on X. The notion of a measurable function is the same for two equivalent measures (1 3.9.4). (1 3.15.7) If p is any positive measure on X , there exists a continuousfunction h such that h(x) > 0 for all x E X and such that the measure v = h * p (which is equivalent to p by virtue of (13.15.6)) is bounded.

Let (U,,) be an increasing sequence of relatively compact open sets in X with the properties of (3.18.3), and for each n letf,, be a continuous function on X with values in [0, 11, such that f,,(x) = 1 for all x E U,, andf,(x) = 0 for all x E X - Un+l(4.5.2). Let (a,,) be a sequence of real numbers > O such that

1a,, < + m

03.

n= 1

Then the series h =

m

1a,f,

n= 1

is normally convergent in X (7.1),

hence h is continuous on X (7.2.1), and h(x) > 0 for all x E X . If v = h * p, then v*(l) = l * h dp 5 Ca,,J'/;. dp ((13.14.1) and (13.5.8)). If for example we take n

a,, =

2-"

then we have n

a,,
) 0 (13.18.5). This set is clearly p-measurable, and if we write p = h . ) p ( )(13.16.3), the measure p z = ( q D h ) lpl is concentrated on D (13.14.1), and pl = p - p2 is diffuse. For we have (p21 = qD * / p ( ((13.13.4) and (13.16.3)), hence lpll= /pi- J p z ( ;for each x E D, we have I p l ( { x } ) = Jpz(({x})by construction, and for each x 4 D we have ( p l ( { x ) )= ( p z ( ( { x )=) 0 by the definition of D; hence, by (13.16.1), I p t l l ( { x ) )= 0 for all x E X. Example (13.18.7) Let A be Lebesgue measure on R. From (13.18.4) and (13.18.6) it follows that every measure p on R is uniquely of the form pi + p z + p 3 , where pl = g . A is a measure with base A, p 2 is an atomic measure and p 3 is a diffuse measure disjoint ,from 1, and therefore concentrated on a A-negligible set (necessarily nondenumerable).

Remarks (13.18.8)

(i) If p is an atomic measure, the denumerable set of points

x E X such that Ipl({x>)> 0 is the smallest set on which p is concentrated. On the other hand, for a diffuse measure v # 0, there exists no smallest set on which v is concentrated: in other words, there exists no largest v-negligible set,

because every set consisting of a single point is v-negligible. (ii) Let p be an atomic measure concentrated on a denumerable set D, and let (a,,)be the sequence consisting of the distinct points of D arranged in some order. Put Ip(({u,)) = y,, > 0. If D,, = {ul, . . . ,a,,],then 'pD = sup qD,; since ( p (= qD.Ipl, then by (13.14.1) and (13.5.7) we have

fl

for any function f 2 0 (the sum on the right is either a finite real number or +a).This implies in particular that, for any compact set K, we have C yn < -too.Furthermore, since every set consisting of a single point is aneK

p-measurable and since X

- D is p-negligible, it follows that

every mapping

of X into a topological space is p-measurable. The p-integrable functions

are therefore those for which

m n= 1

ynlf(an)l
O such that C yn < + 00 for all a,EK

compact subsets K of X. We have already seen (13.1.3) that

f

m

H

v(f) =

C rnf(an>

n= 1

is a positive measure on X. With the same notation as above, it is clear that v = sup(rpDn.v), and since q D , . v is concentrated on D, and hence on D, it n

} ) yn. For if K is a follows that v is concentrated on D. Also we have ~ ( { a , , = compact neighborhood of a,, then for each E > 0 there exists an integer m such y p 5 E . I f f : X + [0, 11 is a continuous mapping which takes that ap E K , p z m

the value 1 at a,,, and the value 0 on X - K and at points a, such that k # n and k 5 m (4.5.2), then we have v({a,}) 5 v ( f ) 5 7, + E ; on the other hand we may choose the neighborhood K such that v(K) 5 v( {a,}) E (13.7.9), and a fortiori yn 5 v ( f ) 5 v({a,}) + E . Since E was arbitrary, our assertion is proved.

+

PROBLEMS 1. Let p, u be two atomic measures on a locally compact space X, and let M, N be the smallest sets carrying lpl, 1 ~ 1 ,respectively. Show that p, u are disjoint if and only if M n N = 0 . Hence give an example of an atomic measure v on the interval I = [0, I ]

of R such that I is the support (13.19) of v + and of v-.

Let p be a positive measure on a locally compact space X. If A c X is universally measurable and not p-negligible, show that there exists a positive measure v carried by A such that v # 0 and v 5 p. (Observe that A contains a compact set K which is not p-negligible.) (b) Let M be a universally measurable subset of X. For a positive measure p to be carried by M it is necessary and sufficient that v should be disjoint from every positive measure carried by X - M (use (a)). (c) If M is closed in X, show that the vector subspace of M(X) consisting of the measures carried by M is vaguely closed in M(X). (d) Show that Lebesgue measure on I = [0, I ] is the vague limit of a sequence of atomic measures carried by a fixed denumerable subset D of I. The subspace of M(I) consisting of measures carried by D is therefore not vaguely closed.

2. (a)

Xlll INTEGRATION

210

3. (a) Let p be a positive measure on a locally compact space X, and let A be a p-integrable set such that p(A) > 0. Suppose that, for each p-integrable subset B of A, we have either p(B) = 0 or p(B) = p(A). Show that there exists a point a E A such that p({a})= p(A). (Consider the intersection of the compact sets K c A such that p(K) = p(A): show that it is not empty, has measure equal to p(A) and consists of a single point.) (b) Suppose that p is a diffuse measure. For each p-integrable set A such that p(A) > 0 and each E > 0, show that there exists a p-integrable subset B of A such that 0 < p(B) 5 E (use (a) to show that there exists an integrable subset C of A such that 0 < p ( C ) 5 fp(A)). Deduce that, as B runs through the set of p-integrable subsets of A, the set of values of p(B) is the closed interval [0, p(A)]. (For each real number b such that 0 < 6 < p(A), let c be the least upper bound of the measures of measurable subsets C of A such that p(C) 5 6. Show that c = 6 by using the preceding result, and then show that there exists an increasing sequence (C.) of measurable subsets of A such that limp(C.)=b.) n+ m

4.

(a) Let v be a positive atomic measure on a locally compact space X,and let A be a v-integrable subset of X. Show that the set of values of v(B), as B runs through the set of v-integrable subsets of A, is closed in R. (Let P be the smallest set carrying v. Assuming that A n P is infinite, and arranging the points of A A P in a sequence (xJ, consider the mapping p of the product space {0, into R defined by p(~,)= E.Y({x.)), and show that p i s continuous.) (b) Deduce from (a) and Problem 3(b) that if p is any positive measure on X and if A is a p-integrable subset of X, then the set of values of p(B), as B runs through the set of p-integrable subsets of A, is closed in R. Extend this result to the situation where p is any real measure on X. (c) Deduce from Problem 3(b) that if p is a diffuse real measure on X, then the set of values of p(A) =- p+(A) - p-(A), where A runs through the set of 1pl-integrable subsets of X, is a closed (possibly unbounded) interval of R. (d) Give an example of an atomic positive measure v on a locally compact, noncompact space X, such that the set of values of v(A), where A runs through the set of v-integrable subsets of X, is not closed in R.(Take v so that inf v ( { x } )> 0.) I E X

5. (a) Let X be a compact space and p # 0 a diffuse positive measure on X. Let (f,) be a total orthonormal sequence in Y&(X,p). Show t h a t x [f&)lz = i- a, for almost n

all x E X. (Argue by contradiction, using Problem 3(b), Bessel's inequality, and the Cauchy-Schwarz inequality for series: show that there would exist a measurable set B with measure :,O and arbitrarily small, contained in the set of points x such that C Ifn(x)lz < i-la, such that, if c, = (rpBIfn), the series c n f n ( x )converges almost

"

sh;

n

everywhere in B to a function then observe that by virtue of the hypothesis that the sequence ( f n ) is total, together with Parseval's identity, we have tiin

n-r

m

S, 1 I -

s,(x)lz ~ ( x =) 0 ,

where

y. =

2

k= 1

ckh.

Hence obtain a contradiction.) (b) Under the same hypotheses, show that there exists a sequence (b.) of scalars such that lim 6, = 0 and /b,,lzlfa(x)/z = fa, almost everywhere. (Use Egoroff's theorem "-1

m

x n

18 CANONICAL DECOMPOSITIONS OF A MEASURE

211

to obtain an increasing sequence of sets A,,, C X such that p(X - A,”) + O as m -+ and in which the partial sums s,(x)

=xlf~(x)12tend unifornily to k=l

$-co. Then

a,

remark

that the series with general term If.(x)12/s.(x) is divergent almost everywhere (Section 5.3, Problem 6). (c) Under the same hypotheses show that for almost all x o E X there exists a function g E Y&(X,p) (depending on xo) such that the series with general term (.y Ij;,)fn(xo) has sum equal to co (use Problem 23 of Section 12.16).

+

6. (a) Let 6 : R R be increasing and continuous on the right. Show that there exists a unique positive measure v on R such that v(]a,b ] ) -&b) - 6(u) for every half-open interval ]a,b ] . (This measure is called the “Stieltjes measure defined by 8”.and we --f

s

s

write f d 6 in place of f d v . ) Conversely, every positive measure on R may be obtained in this way, and two functions 6 , , Q z on R (both of them increasing and continuous on the right) define the same measure if and only if 8, 8,is constant. Under what conditions is the measure Y diffuse? What is then the image of v under 8? (b) Let K be the Cantor set (Section 4.2, Problem 2). Show that there exists a diffuse positive measure v on R with support K and total mass equal to 1 (Section 4.2, Problem 2(d)). Deduce that there exists on R a di’irse positive measure, disjoint from Lebesgue measure, with support equal to I = [0, 11. (In each component interval J of I - K choose a measure proportional to the image of v under an affine linear mapping of I onto J; then proceed by induction.) (c) Deduce from (a) that if p is a positive measure on R , a n d f a function belonging to Y:oc, c(R,, p), and (1.) a dense sequence in R such that f dp = 0 for all n , ~

!

LO,

then f i s p-negligible.

r.1

7. Let X be a compact space and p a diffuse positive measure on X with total mass equal

to 1. (a) Construct a family (U(t))oe,s of p-quadrable open sets, such that U(0) = 0, U(1) = X, u(t)C U(t‘) whenever t < r ’ , and p(U(1))= t for all 1. (First define the U(t) for r of the form k/2”, by induction on n. Use Problem 7(d) of Section 13.9 to prove that if V, Ware two quadrable open sets in X such that V c W a n d p(V) < p(W), then there exists a quadrable open set U such that c U c 0 c W and

v

:p(w - V)< p ( U - V) < $p(W - V). Also use the result of Problem 3(b).)

(b) Deduce from (a) that there exists a continuous mapping T of X onto I = [0, 11 such that the image of p under T is Lebesgue measure A on 1. (c) Show that there exists a p-negligible subset N of X, a A-negligible subset M of I and a homeomorphism r0 of I - M onto X - N such that, for every A-measurable subset A of 1 - M , the set rro(A) is p-measurable and of measure p ( r 0 ( A ) )= X(A). (Use Problem 7(d) of Section 13.9 to show that for each integer n , 0 there exists a finite partition of X consisting of a p-negligible set and quadrable open sets of diameter < l / n (with respect t o a distance defining the topology of X) and of measure s l / n . Proceeding by induction on n and passing to the limit, obtain a homeomorphism f of I - D onto a subset of X of measure I , where D is a denumerable subset of I.) 8. Let X, Y be two compact spaces, U a continuous linear mapping of the Banach space ‘dR(X) into the Banach space ‘CR(Y), and T : X + Y a continuous mapping.

212

Xlll

INTEGRATION

for all y E Y and allfE ‘C,(X). Show that under these conditions there exists a vaguely continuous mapping u: y w u Y of Y into M+(X) such that uy is carried by n - l ( y ) for all y E Y, and such that ( U . f ) ( ~ ( x )=) 0 and each integer n > 0, there exists a p-measurable subset A of X such that (i) the sets uJ(A) (0 5 j 5 n - 1 ) are pairwise --f

disjoint, and (ii) the complement of

u uJ(A) has measure 5

n- 1

j=o

E

(Rokhlin’s theorem).

(Choose a p-measurable set B such that 0 < p(B) < ~ / nand , construct the corresponding “Kakutani skyscraper” consisting of the sets uP(Em)for m 2 1 and 0 z p < rn (Section 13.9, Problem 14(d)). Show that we may take A to be the union of the sets d”(E,,) for all integers j > 0 and integers m such that m 2 ( j -t 1)n.)

19. SUPPORT OF A MEASURE. MEASURES WITH COMPACT SUPPORT

(13.1 9.1 ) If p is any measure on X, the union of all p-negligible open sets is p-negligible (and hence is the largest p-negligible open set). For if U is this union, it follows from (13.1.9) that the measure induced by p on U is zero. The complement in X of the largest p-negligible open set is called the support of p, and is denoted by Supp(p). To say that x E Supp(p) signifies that Ipl*(V) > 0 for eoery neighborhood V of x, or equivalently (13.5.1) that IpI ( Ifl) > 0 for every f~ Xx,(X)such that f ( x ) # 0; or equivalently again, that for each neighborhood V of x there exists a function , f ~ .X,(X) with support contained in V, such that p ( f ) # 0. If Supp(p) = X, the only p-negligible continuous function is the constant 0. From the definition we have Supp(p) = Supp( ]PI), and it is clear that Supp(ap) = Supp(p) for all scalars a # 0. More generally, if g is any locally Ipl-integrable function, then we have Supp(g p) c Supp(g) n Supp(p); for if we put v = g lpl, and if an open set U does not intersect Supp(g) or does not intersect Supp(p), then Ivl*(U) = 0 (13.14.1).

-

(1 3.1 9.2)

(i)

If p and v are positive measures, then SUPP(P + 4

= SUPP(P) u

SUPP(V).

19 SUPPORT OF A MEASURE. MEASURES WITH COMPACT SUPPORT

213

(ii) Zf H is any majorized family of positive measureh and v = sup H (13.15.4), then Supp(v) is the closure of the union of the supports of the measures pEH. This follows immediately from (13.16.1) and (13.15.9), applied to the characteristic 'function of a relatively compact open set (having regard to (13.1 5.4)). The complement of Supp(p) can also be defined as the largest of the interiors of p-negligible sets. Hence Supp(p) is the intersection of the closures of all sets on which the measure p is concentrated (13.18). But it should be realized that two disjoint measures can have the same support: for example, an atomic measure on R,concentrated on a denumerable dense set, and such that each point of this set has nonzero measure (13.18.8), has the same support as Lebesgue measure. (13.19.3) For a measure p on X to be such that every continuous complexualued,function on X is p-integrable, it is necessary and suflcient that Supp(p) hould be compact. The mapping fH p( f ) is ther. a continuouh linear form on the Frkchet space cS,(X) (12.14.6), and conversely every continuous linear form on %,.(X) is of this type.

If p has compact support S , then X - S is p-negligible, hencefq, is equal to f almost everywhere. Since fq, is measurable (13.9.6) and 1fqsl is bounded above by a multiple of qs (3.17.10), it follows that fqs,and hence also f , is integrable. Let us show conversely that if S = Supp(p) is not compact, then there exist continuous real-valued functions f 2 0 such that IpI*(f) = + co.By hypothesis (3.18.3) there exists an increasing sequence of relatively compact open sets U, in X such that 0, c Untl and the U, cover X. Since X is not compact, we have U, # X for all n. We now define inductively a sequence (an) of points of X and a sequence (V,) of relatively compact open sets, as follows: a, E S ; V, is a neighborhood of a, ; a,,, belongs to the intersection of S and the complement of the union of 0, and the v k with k 5 n (by hypothesis, this intersection is not empty); Vntlis a relatively compact neighborhood of a,,, such that does not intersect O,, nor any v k with k 5 n. For each n, let g, be a continuous mapping of X into [0, 13 which takes the value 1 at a, and the value 0 throughout X - V, (4.5.2). The hypothesis implies that lpI(g,) > 0. Put f, = g,/ lpl(gn), so that Ipl(f,) = 1. Then the function

v,,,

f=

m

f, is everywhere finite and continuous (because every point

n= 1

XE

X

214

Xlll INTEGRATION

belongs to some U,, which intersects only finitely many of the V k ) ,and we have Ipl*(f)

=

+ co,because Ipl*(f) 2..

n

-

k= I

Ipl(fk) for all n.

If S = Supp(p) is compact, the first part of the proof shows that E bP,(X), and therefore p is a

lpl ( f ) 5 Ipl(S) sup If(x)l for all functions f xeS

continuous function on the Frechet space U,(X). Conversely, let I be a continuous linear form on this space, so that there exists a compact set K c X and a constant c > 0 such that II(f)i 5 c * sup If(x)l for allJE bPC(X) (12.14.6). If XEK

L is any compact subset of X, we then have Il(f)l5 c * llfll for all , f ,X,(X; ~ L), and hence the restriction p of 1 to Xc(X)is a meamre on X. Moreover, if h is a continuous mapping of X into [O, 13, with compact support and such that h(x) = 1 for all x E K ((3.18.2) and (4.5.2)), then l(fh) = l ( f ) for all functionsf€ bPg,(X),becausef-fh is zero on K. From this it follows immediately that the support of p is contained in Supp(h); hence everyfe U J X ) is p-integrable, and p(f) = p ( f h ) = I ( f h ) = I ( f ) . Q.E.D. (13.19.4) If n : X -+ X’ is a homeomorphism and ,u is a measure on X, we have Supp(n(p)) = x(Supp(p)). This follows directly from the definitions. PROBLEM

Let p be a positive measure on X, and A a p-measurable set. Let i(A) be the set of points x E X such that there exists a compact neighborhood V of x in X for which p(V n (X - A)) = 0. Show that i(A) is open and that &(A) n (X - A)) = 0. (Consider S u p p ( ~ .~p).) -~

20. BOUNDED MEASURES

For every (complex) measure p on X, we define

which is a finite real number, or

+ co.

(13.20.2) We have 11p11 = lpl*(l). Hence llpll isjnite ifandonly measure lpl is bounded (13.9), and in that case 11PIl = lPI(1) = IPKW

is the total mass of X with respect to p.

if the positive

20

BOUNDED MEASURES

215

For every functionfE X,(X) we have (13.3.3) IP(f)I

s 1pK If11 s I l f

II lPI*(l) *

with the usual convention about products in [0, +a] (13.11), hence I(p(( a (13.5.1), and hence a function f~ .X,(X) such that If I 5 g 5 1 and Ip[f)l > a (13.3.2.1). This completes the proof. If 11p11 is finite, the measure p is said to be bounded. If p is a positive measure, this definition coincides with that given in (13.9). For a measure p to be bounded it is necessary and sufficient that 1p1 should be bounded, and we have II lpl II = 11p11. The set Mk(X) (also denoted by M’(X)) of bounded complex measures on X is a vector subspace of the space M,(X). We write MA(X) = M;(X) n MR[X) for the space of bounded real measures on X. The definition (13.20.1) shows that it comes to the same thing to say that the bounded measures on X are the continuous linear forms on the space .X,-(X), endowed with the norm (5.5.1), and that llpll is the usual norm (5.7.1) on the dud ML(X) of this normed space (12.15). Moreover, ML(X) is complete with respect to this norm (5.7.3). If p is any measure on X and g a locally p-integrable function, then the measure g p (13.16) is bounded if and only if g is p-integrable, and we have (1 3.20.3)

119 * pll = N,(g) =

s

191 4 P l .

- lpl (13.16.6).

This follows immediately from the relation ( g * pi = (91

It is clear that every measure p with compact support is bounded, because if S = Supp(p) then lpI*(l) = Ipl(S). If p is a bounded positive measure, then we have

because by (1 3.9.17) every bounded measurable function is integrable, and the Cauchy-Schwarz inequality (13.11.2.2) applied to g = 1, together with (13.9.13), shows that everyfe Uz(X, p) is integrable and that

N , ( f ) 5 NAf) ’ P(x>1’2. This shows also that the canonical injection Ug(X, p) + Y;(X, p) is continuous with respect to the seminorms on these two spaces; the same is true

216

Xlll INTEGRATION

for the injection 9 F ( X , p) + 9 i ( X , p) (with respect to the corresponding seminorms) because it follows from (13.12.5) that N2(f) 5 N,(f) p(X)”* for allfE 9 F ( X , p). (13.20.5) In particular (13.9.17), if p is a bounded (complex) measure, every function f~ %;(X) is p-integrable, and we have / f d p l 5 11p11 * llfll by

I

I

(13.16.5). In other words, f H f dp is a continuous linear form on the Banach space %;(X). But it should be noted that, in general, there exist continuous linear forms on this space which are not of this type. (13.20.6) The space Mk(X) is also the dual of the closure %,!(X) of X,(X) in the Banach space %$‘(X) (12.15). A function f belonging to %g(X) may be characterized by the following property: for each E > 0, there exists a compact subset K ofX such that If(x)l E for all x E X - K. For i f f € %g(X), then for each E > 0 there exists by definition a function g c X,(X) such that [ I f - 911 I E ; if K is the support ofg, then If(x)l 5 E for all x q! K. Conversely, suppose that f has the above property, and let h be a continuous mapping of X into [0, I], with compact support and equal to 1 on K ((3.18.2) and (4.5.2)); it is clear that Ilf-fhll 5 E and that fh E X J X ) , hence f E %g(X). The functions belonging to %g(X) are called (complex-valued) continuous functions which tend to 0 at infinity. (When X = R, they are indeed the continuous functions f such that lim f ( x ) = lim f ( x ) = 0.) We put X‘+W

x-+-w

%?i(X)=%g(X) n %,(X) for the corresponding space of real-valued functions. When X is compact, we have

%F(X)= %g(x)= X,(X) = %,(X). PROBLEMS

1. The space M:(X) of bounded real measures on a locally compact space X can be considered as a space of linear forms on each of the following vector spaces: (1) the space El = X d X ) of continuous functions with compact support; (2) the space E2= Wg(X) of continuous functions which tend to 0 at infinity; (3) the space E3 = Y g ( X ) of bounded continuous functions; (4) the space E, of linear combinations of (upper or lower) semicontinuous bounded functions ; ( 5 ) the space Es = @p(X)of universally measurable bounded functions.

Moreover, if v is any positive measure on X, the space M:(X,v) of bounded measures with base Y (which may be identified with the space L:(X,v) by virtue of 13.14.4)) can be considered as a space of linear forms on the vector space E4.,of bounded functions which are continuous almost everywhere with respect to v.

20

BOUNDED MEASURES

217

Let F Ldenote the weak topology on M:(X) corresponding to the vector space T4, the weak topology on M:(X, v) (or on L:(X, v)) corresponding to E4.* (Cf. (12.15)). The topology Y t on M:(X) is coarser than TIif i < j . The topology induced by F3on M:(X, v) is coarser than T4,which in turn is coarser than the topology (Consider the lower semicontinuous regularization of a function which induced by F5. is continuous almost everywhere with respect to v (Section 12.7, Problem 8).) (a) Let (p.) be a sequence of bounded real measures on a locally compact but not compact space X. Give an example in which (p.) tends to 0 vaguely (Le., for the topology Y1) but does not converge for the topology F2. A sequence (p.) which vaguely converges to 0 also converges to 0 with respect to Fzif and only if the sequence of norms (!lp.!l)is bounded (use the Banach-Steinhaus theorem). (b) Give an example of a sequence (p.) which converges to 0 for the topology T 2 but not for F 3 . A sequence (p.) of bounded real measures which converges vaguely to a measure p also converges to p for the topology T3if and only if, for each E > 0, there exists a compact subset K of X such that lp&X - K ) Z E for all n. (To show that the condition is necessary, argue by contradiction, by using the method of Section 13.14, Problem 1.) (c) Give an example of a sequence (p.) in M:(X, v) which converges to 0 for the topology F3but does not converge to 0 for the topology Y4, (take X = [0,1I). A sequence (p.) of measures belonging to M:(X, v) which converges to 0 for the topolalso converges to 0 with respect to F4,if and only if it satisfies the following ogy F3 condition: (C4.") For each v-negligible compact subset K of X and each E > 0, there exists an open neighborhood U of K such that /p,J(U)=< E for all n. (To prove that the condition is sufficient, reduce to the case where X is compact and apply (C,, ") to the set K, of points x E X at which the oscillation of an almost everywhere continuous function is Z E .To prove that the condition is necessary, argue by contradiction as in (b) above.) (d) Show that a sequence (p.) in M:(X) which is a Cauchy sequence for one of the topologies TI,F2, F3is convergent for this topology. (For F3, use (b) and argue by contradiction: form a sequence of measures which tends to 0 with respect to .T3 without satisfying the condition given in (b).) (e) A sequence (pJ of measures belonging to M:(X, v) is a Cauchy sequence with respect to the topology F4,if and only if it satisfies the condition ((24, ") (argue by contradiction as in (d)); the sequence (p,)then converges with respect to T4.? to a measure belonging to M&X, v). (f) Let (p,,) be a sequence of measures belonging to M:(X, v). For each subset A of X which is either finite or else open and v-quadrable (Section 13.9, Problem 7), suppose that the sequence (p.(A)) has a finite limit; show that (p")is then a Cauchy sequence for the topology T4. ". (Use Problem 1 of Section 13.14, and argue by contradiction.) Show that the hypothesis relative to finite sets A cannot be omitted. (g) Let (pn)be a sequence of measures belonging to M:(X, v) which converges in , ~ ~then the topology Y 3to a measure p belonging to M:(X, v). If also lim ~ ~ p=, IlpII,

EL( i # 4), and

n+ m

the sequence ( p c )converges to p in the topology F4,(use (e)). Give an example of a sequence (pn)ofpositiue measures belonging to M:(X, v), where the space X is compact, such that (pn)converges vaguely to a measure which does nof belong to Mi(X,v). 2.

The notation is the same as in Problem 1. Let (p")be a sequence of measures belonging to Mi(X). Then there exists a positive measure v on X such that p..E M:(X, v) for all n.

Xlll INTEGRATION

218

(a) Show that the following properties are equivalent: (a) The sequence (p.) is convergent for the topology r 6 . (8) For each closed subset A of X, the sequence @.(A)) has a finite limit. ( y ) The sequence (pn)converges for the topology Y3 and satisfies the following condition: (C,) For each compact subset K of X and each E > O , there exists an open neighborhood U of K such that Ip,J(U - K) 5 E for all n. (6) The sequence (pn)converges for the topology F 3and satisfies the following condition : (C,) For each E > 0 there exists 6 > 0 such that, for each universally measurable set A satisfying v(A) 5 6, we have Ipnl(A) 2 E for all n. (To show that (p) implies (y), use Problem 1 of Section 13.14 to show that the sequence l(p.11 is bounded. To establish (C5),argue by contradiction: first consider the case in which the sequence (p.) tends to 0 for the topology Ys, and then pass to the general case as in Problem l(d). Next show that ( y ) implies that the sequence (p.) is convergent for the topology F , , and in particular implies (p). To show that (6) implies (a), use the definition of measurable functions. To show that ( a ) implies (S), argue by contradiction. Finally, to show that ( y ) implies (a),consider first the case where lim lIp.ll = jlpI/,where p is the limit of (p,) for the topology Y 5 and , argue by n-m

contradiction to prove that ( y ) implies (6). To pass to the general case, reduce to the situation where p = 0, and argue by contradiction: we may assume that there exists a functionfe F-6 such that the sequence ( p n ( f ) )has a limit #O. On the other hand, we can pick a subsequence (p,,))of the sequence (p,) such that the sequence (&), and hence also (p;)), converges vaguely. Hence arrive at a contradiction, by observing that these two sequences also satisfy ( C 5 ) . ) (b) Give an example of a sequence (,urn) of measures belonging to M:(X,v) which convergesfor the topology f 4 , but not for the topology Y, (cf. Section 13.18, Problem 2(d)). on MA(X) are distinct (c) Take X = [O, 11. Show that the topologies Y5and (cf. Section 12.15, Problem 2(c), and Section 13.11, Problem 3). (d) Extend the results of Problems 1 and 2 to bounded complex measures. 3. Let X be a locally compact space, p a positive measure on X. Let (9.) be a sequence of p-integrable functions such that (1) the sequence (9")converges in measure (Section 13.12, Problem 2) to a function g; (2) the sequence of measures (gn . p) converges for the topology F6(Problem 2). Show that under these conditions the function g is p-integrable and that the sequence ( g n ) tends to g in 9 A ( X , p). 4.

Let X be a compact space, p a positive measure on X, and (fn)an orthonormal sequence in .!Z&(X, p) consisting of functions which are uniformly bounded on X. (a) Show that, for each function g E &(X, p), the sequence of numbers ( g If.) = /g(x)f(x)dp(x) tends to 0. (Reduce to the case where g is bounded and hence belongs to 9 $ ( X , p).) (b) Let xo E X . For the series C ( g I f . ) f n ( x o ) to converge for euery function n

3 g ( X , p), it is necessary and sufficient that (in the notation of Section 13.17, Problem 2) the sequence of bounded measures K.(xo , .) . p should converge for the topology 9 - 6 (Problem 2) to a bounded measure h,, . p ; hence in particular we have

gE

=I

UXO)hxO(x)fn(x)dp(x) for all n.

21 PRODUCT OF MEASURES

219

(c) Show that there exists a measurable subset A of X with measure >0, such that for each xo E A there exists a function g E -Yg(X, p) (depending on xo) for which the series with general term (g If.)f(xo) does not converge. (Take A to be the set of points x E X for which the sequence (f,(x)) does not converge to 0, and use Problem 7(a) of Section 13.11. Then argue by contradiction, using (a) and (b).) Let S be a closed subset of R. Given a sequence ( c , ) . ~ ~of real numbers, show that there exists a positive measure p on R with support contained in S and such that

j x ” +(x) = c. for all n 2 0 if and only if, for every polynomial P(X) = that P(x) >= 0 for all x E S, we have

n k=O

fkck

2 &Xk such

k=O

2 0. (As in Problem 2 of Section

13.3,

show that there exists a positive measure p with support S which extends to a linear form u defined on .f,,(R) and on the space of polynomials on R, and such that U(P) =

C &ck

k=O

for each polynomial P(X) =

is p-integrable and that

s

x” +(x)

= c,

n

C k=O

fk

Xk.Then prove that each power xn

. For this purpose, remark that for each in-

teger n >= 0 and each E > 0 there exists a number R > 0 such that If.,R(x)l 5 EX^"+^, where f,. is the function equal to 0 for 1x1 < R and to x” for [XI 2 R.) Particular cases: (1) S = R (“Hamburger’s moment problem”): the condition is that the quadratic forms

1. k = O

c,+,&[x should be positive for all n (observe that every

+

polynomial P(x) which is 20 on R is the sum of two squares (Pl(x))’ (PZ(x))’). (2) S = [0, +a[(“Stieltjes’ moment problem”): the conditiori is that the quadratic forms

5

J. k = O

c,+ktjtk

and

5 cj+k+L[]& should be positive for all n 2 o (remark

1. k = O

that every polynomial P(x) which is 2 0 on [0, +m[ can be written in the form (Pdx))’ (PZ(X))’ x((p3(x))z (P&))*).

+

+

+

Let v be a positive measure on X and (fa)a sequence of functions belonging to .Y;(X, v) such that the sequence of measures Cf. . v) converges to f. v (where f E 9;(X, v)) for the topology Y6. Show that f(x) 5 lim supfn(x) almost everywhere n- m

with respect to v. (Use (13.8.3).)

Let p, v be two bounded complex measures on X. Show that the following conditions are equivalent: (1) p and v are disjoint; (2) JJp =tvI/= IlpIl llvll; (3) IIP YII + llp - YII = 2(llpll llvll).

+

+

+

21. PRODUCT O F MEASURES

(13.21.1) Let X, Y be two locally compact spaces, 1 a measure on X and p a measure on Y. Then there exists a unique measure v on the product space X x Y such that, for each pair of functions f E Xx,(X)and g E .X,(Y), we have

220

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INTEGRATION

(1) Uniqueness. Every compact subset of X x Y is contained in one of the form L x M, where L and M are relatively compact open subsets of X and Y respectively ((3.20.17) and (3.18.2)). We shall show that, for each function 12 E Xc(X x Y) with support contained in L x M, the value of v(h) is well-determined. This will follow from (13.21 .I.2) /f L c X and M c Y are relatiueIy compact open sets, then every function h E X,(X x Y ) with support contained in L x M is a cluster point, in the Banach space Xc(X x Y ; L x M ) , of the set of functions of the form (x, y)”cfi(x)gk(y), where (A) is a finite sequence of functions in .X,(X) i, k

with support contained in L, and (gk) is afinite sequence of functions in X C ( Y ) with support contained in M .

Assuming this for the moment, by hypothesis there exists a number c > 0 such that I v(u) I 5 c . llull for all u E X,(X x Y) with support contained in L x M. On the other hand, for each E > 0 there exist two finite sequences of functions f i E X c ( X ) gk , E X , ( Y ) such that

I h(x,r) - cfi(x)gk(Y)I 5 i, k

for all (x, y ) E L x M , the support of the left-hand side being contained in L x M. Using (13.21 .I.I), it follows that i, k

and since E is arbitrary, this establishes the uniqueness of v. To prove (13.21 .I .2), let E be a positive real number. Then for each z = ( x , y ) E L x M there exists a compact neighborhood U c L (resp. V c M) of x in X (resp. of y in Y) such that the oscillation of h in U x V is S E (3.20.1). The projections S c L and T c M of Supp(h) are compact, hence for each x E S there exists a finite number of points y j ( x )( 1 5.j 5 n(x)) of T , and for e a c h j a compact neighborhood U j ( x )c L of x in X and a compact neighborhood Vj(yj(x))c M of y j (x ) in’Y, such that the oscillation of h in U j ( x ) x Vj(yj(x)) is S e , and such that the interiors of the sets V j ( y j ( x ) ) cover T. The set U’(x) = U j ( x ) is a compact neighborhood of x in X. j

Hence there exists a finite number of points x i (I 5 i 5 m ) in S such that the p the interiors of the sets U’(xi) cover S. Put A i = U ’ ( x i ) ,and let ( B k ) l s k S be family of sets obtained as follows: for each point y E T, let W ( y ) be the intersection of the (finitely many) interiors of sets V j ( y j ( x i )which ) contain y ; the sets W(y) are open and nonempty, and form a finite open covering of T which we denote by (Bk)ls k s p . Notice also that, by construction, the oscillation of h in each set A j x B, is 2 E. Now let (A),s8s,n(resp. (gk),‘ k c p ) be continuous mappings of X (resp. Y) into [0, I], such that Supp(,fi) c Ai and Supp(gk) c Bk

21 PRODUCT OF MEASURES

for each pair of indices i, k , and such that c f i ( x ) 5 1 for all x for all y E Y, and Then we have

LA(.) i

=1

i

for all x E S,

y) =

for all x E X and all y fore imply that

E Y.

5

E X,

221

Ck gk(y)S

1

1k g k ( y )= 1 for all y E T (12.6.4).

y)fi(xbk(y)

i. k

If yk is any point of B,, the hypotheses there-

xfi(x>gk(Y>

i,k

for all (x,y ) E A x B, where A

=

u i

=
c i ~ ( x is ) continuous on Y, and supp(g) c M.

MY’) - d Y ) l

=

IS

v>>W x )

( N x , Y’) - 0,

I ULE

We return to the proof of (13.21 .I). For each function h E X,(X x Y), the number v(h) = p(g) (which by abuse of notation is also written in the form p ( j h ( x , y ) dA(x))) is defined. Furthermore, with the notation used above, there exist by hypothesis two numbers aL and b, such that, for each

222

Xlll

INTEGRATION

function u E X ( X ; L) (resp. v E X ( Y ; M)), we have IA(u)l 5 aL llull (resp. Ip(u)l 5 b , Ilvll). Hence, for each function h ISX(X x Y; L x M), we have

for all y

E Y,

and by virtue of (13.21.1.3) and the definition of v(h),

Iv(h)l 5 aLb, llhll. Since every compact subset of X x Y is contained in the product of its projections on X and on Y, the proof of (13.21.1) is complete. The number v(h) = p ( / h ( x , y ) dA(x)) is also denoted by SdP(Y) Sh(x, Y ) M x ) .

s

Since we can clearly interchange the roles of X and Y, it follows that (13.21.2)

h(x, Y )

4

x

9

s s

h(x, Y ) M

Y)

= j d P ( Y )p x , Y ) W

x)

Y)=

d4x)

for all h E Xc(X x Y). By reason of this formula we write ssh dL d p or s / h ( x , y ) dA(x) d p ( y )

s

instead of h(x, y ) dv(x, y ) , and we may also interchange A and p in these notations. The measure v is called the product of 1- and p, and is denoted by A 0 p. It is clear that the mapping (2, p) H A 0 p of M,(X) x M,(Y) into Mc(X x Y) is bilinear: in other words, we have

(11 + A21 0 (PI

+ p 2 ) = A1 0 p1 + 1, 0 p 2 + A2 0 p1 + 2 2 0 p 2

and ( a 4 0 p = A 0 (up) = 4 10 p)

for any scalar a. Furthermore, if A and p are real (resp. positive) measures, then so is A@p. (13.21.3) Let A, p he two positive measures on X, Y respectively, and v = A @ p their product. For each function h E J ( X x Y), the function

21

PRODUCT OF MEASURES

223

belongs to Y(X), and we have

{*h dv = .i*(s”h(~,Y> d p W )

(1 3.21.3.1)

(with an abuse of notation analogous to those above). By (12.7.8) there exists an increasing sequence (h,) of functions belonging to X,(X x Y ) such that h = sup h, . Foieach n, it follows from (13.21 .I.3) n

that the function f n ( x )= jh,(x, y ) dp(y) belongs to .X,(X); hence f = supf, n

belongs to Y(X), and moreover we havef(x) = j*h(x, y ) d p ( y ) for all x E X, by virtue of (13.5.2). Since v(h,) = A(fn)by definition, another application of (13.5.2) completes the proof. From now on, up to and including (13.21.16), we shall assume that the measures i? E M(X) and p E M(Y) are positive, and we shall write v = i? @I p. (1 3.21.4)

If h is any mapping of X x

Y into R, then

{ * h dv 2 i?’(j*h(X,

(13.21.4.1)

Y ) dP(Y)).

Let ~ E Y ( X x Y) be such that h 5 u. Then for each j * h ( x , y ) dp(y) 5 u(x, y ) dp(y), and consequently

/*

XE

X we have

by (13.21.3). Hence the inequality (13.21.4.1) follows from the definition of v*(h) (13.15.5). We shall write /[*A di? dp or //*h(x, y ) dA(x) dp(y) instead of v*(h), and /*di?(x)/*h(x, y ) d p ( y ) in place of A*(/*h(x, y ) dp(y)). Similarly, we shall use the notations J/*h dA dp and //*h(x,y ) di?(x) dp(y) for lower integrals. Thus the inequality (13.21.4.1) takes the form (1 3.21.4.2)

j{*h(X,

Y ) W x ) d A y ) 2 j * d A ( X ) J*h(X, Y> M Y )

with equality if h E 9 ( X x Y). There is of course an analogous inequality (resp. equality) obtained by interchanging the roles of X and Y, and an

224

Xlll INTEGRATION

inequality in the reverse direction (with equality when h E 9’(X x Y)) for lower integrals :

(1 3.21.5) I f N is a v-negligible subset of X x Y, then the set of points x E X such that the section N(x) c Y of N is not p-negligible is I-negligible (in other words, we have p(N(x)) = 0 almost everywhere with respect to I).

This follows immediately from (13.21.4.2) applied to h = q N . (13.21.6) I f h is a v-measurable mapping of X x Y into a topologicul space E, then the set of points x E X for which the partial mapping y Hh(x, y ) is not p-measurable is A-negligible (in other words, the mapping y H h ( x , y ) is pmeasurable almost everywhere with respect to A).

By hypothesis, there exists a partition of X x Y consisting of a v-negligible set N and a sequence (K,),,, of compact sets, such that each of the restrictions h I K, is continuous (13.9). Let M be the I-negligible set of points x E X at which the section N(x) of N is not p-negligible (13.21.5). For each x $ M, Y admits a partition consisting of compact sets K,(x) (n 2 1) and the p-negligible set N(x), such that the restriction of y w h ( x , y ) to each of the sets K,(x) is continuous. Hence the result. It should be noted that it can happen that for each x E X the function y w f(x,y ) is p-measurable, and for each y E Y the function X H f (x, y ) is A-measurable, but that .f is not v-measurable. (13.21.7) (Lebesgue-Fubini Theorem) Let A, p be positive measures on X and Y, respectively, and v = I Q p their product. For each +-integrable mapping h of X x Y into the set of points x E X such that the partial mapping

w,

y w h(x,y ) is not p-integrable is I-negligible; the,,function

s

h(x, y ) dp(y), which is defined almost everywhere with respect to 1, is A-integrable; and X H

(12.21.7.1 )

It follows from (13.21.4.1) that the function x w J * J h ( x y)l , dp(y) is finite on the complement of a A-negligible set N, (13.6.4). On the other hand, the set N, of points x E X such that y w h(x, y ) is not p-measurable is I-negligible

21 PRODUCT OF MEASURES

225

(13.21.6). Hence, from (13.9.13), it follows that for each x $ N = N, u N, the mapping y Hh(x, y ) is p-integrable, and therefore the function X Hj

h k Y ) dP(Y)

is defined almost everywhere with respect to A. The fact that it is I-integrable, and the relation (13.21.7.1), then follow from (13.21.4.2), and (13.21.4.3). Interchanging the roles of X and Y we have also, under the hypotheses of (1 3.21.7),

But here again it needs to be said that the right-hand sides of (1 3.21.7.1) and (13.21.7.2) can be defined and equal without h being v-integrable (even if h is v-measurable) (Problem 3). (13.21.8) Let h 2 0 be a v-measurable function. Then the mapping

is A-measurable, and

Let (K,) be an increasing sequence of compact subsets of X x Y which cover X x Y (3.18.3). Then we have h = sup h , , where h, = inf(h, nqPK,), n

and h, is v-integrable ((13.9.7) and (13.9.13)); on the other hand (13.5.7), v*(h) = sup v(h,). By (1 3.6.2) and (13.21.7), there exists a A-negligible set N n

such that, for all x 4 N, all the functions y~ h,(x, y ) are p-integrable; also (13.21.7) the functions XH h,(x, y ) d p ( y ) are A-integrable and we have

s

.s

v(h,) = j d A ( x ) j h , ( x , y ) d p ( y ) ‘for all n. Hence it follows from (13.5.7) that j * h ( X , Y ) 4 0 )= SUP

h,(% Y ) d A Y )

for all x $ N. Consequently the function XH h(x, y ) dp(y) is A-measurable, s* by (13.9.11), and the relation (13.21.8.1) follows by another application of (1 3.5.7).

226

Xlll

(1 3.21.9)

INTEGRATION

A v-rneasurable,function h is v-integrable if and only

/* d W / * Ih(x, Y)l dP(Y) < + a.

if

This follows from (1 3.21.8) and (1 3.9.13). (13.21.10) Let A be a v-measurable set in X x Y . (i) The set M of all x E X such that the section A(x)is not p-measurable is I-negligible; the function x H p*(A(x)) is I-measurable; and

v*(A) = /*p*(A(x)) dI(x). In particular, i f A(x) is p-negligible except on a A-negligible set of values of x , then A is v-negligible. (ii) If A is v-integrable, then the set of all x E X such that A(x) is not p-integrable is &negligible; the ,function XI--,p ( A ( x ) ) (which is dejned almost everywhere Kith respect to A ) i3 A-integrable; and v(A) = S p ( A ( x ) )dA(x). These assertions are particular cases of (13.21.6), (13.21.8) and (13.21.7). (13.21.11). Let f (resp. g ) be a mapping of X (resp. Y) into [0, +a].With the convention of (13.11) for products, we have

By virtue of (13.21.4), we have

On the other hand, for each x E X we have

and

21 PRODUCT OF MEASURES

227

with the product convention referred to above. Hence it is enough to prove the inequality

Now this inequality is clearly valid when the right-hand side is equal to + co. So consider the case in which each of the factors on the right is finite. Then there exist two decreasing sequences (fn), (gn) such that f;,E Y(X), g n E 9 ( Y ) ,f S f , , g S g . f o r a l l n , a n d j*fd2

=

:!z j*fnd2,

[*g dp = lim j * g n dp. n-

03

By reason of our conventions and of (12.7.5), the function (x, y)wf,(x)g,(y) belongs to 9 ( X x Y), and we have f(x)g(y) 5J8(x)g,(y) for all n and all (x, y ) E X x Y. But, by virtue of (13.21.3),

L [I*f(X)S(Y) W x ) dcc(Y), from which the desired inequality follows by letting n tend to +a. Finally, to deal with the case where (for example) 1’ is A-negligible, it is enough (by virtue of our conventions) to prove that the function (x, y ) Hf(x)g(y) is v-negligible. This is a consequence of the following proposition: (13.21.12) If N is any 2-negligible subset of X, then the set N x Y is vnegligible.

For since Y is the union of a denumerable sequence (L,) of compact sets, it is enough to show that each of the sets N x L, is v-negligible. But since we have proved that (13.21.11.1) is valid whenever both of the factors on the right-hand side arejnite, we may take f = q N and g = qL, in this formula. (13.21.13) Let E, F, G be three topological spaces and u : E x F + G a continuous mapping. Let f (resp. g) be a I-measurable mapping of X into E (resp. a p-measurable mapping of Y into F). Then (x, y ) w u ( f ( x ) ,g(y)) is a v-measurable mapping of X x Y into G.

228

Xlll INTEGRATION

By (13.9.6) it is enough to show that (x, y ) ~ f ( x is) a v-measurable mapping of X x Y into E. There exists a partition of X consisting of a sequence (K,) of compact sets and a %negligible set N such that the restriction off to each K, is continuous. For each compact subset L of Y, the restriction of the mapping ( x , y )f ( x~) to each of the compact sets K, x L is continuous, and the set N x L is v-negligible (1 3.21.12), whence the result follows (13.9.4). Given two mappings f : X --f R and g :Y -+R, we denote by f Q g the mapping ( x ,y )f (x)g(y) ~ of X x Y into R (with the convention of (13.1 1) for products in R). Similarly for mappings into C. (13.21.14)

(resp. Y ) into

f (resp. g ) is a A-integrable (resp. p-integrable) mapping of X

a or C , then the function f Q g is v-integrable, and we have

By linearity we reduce to the case where f and g are mappings into R. By virtue of (13.21.12), the set of points ( x ,y ) E X x Y at which f(x) or g(y) is infinite is v-negligible, hence it follows from (1 3.21.I3) and (1 3.9.6) that f @ g is v-measurable. The fact that f Q g is v-integrable then follows from (13.21 .II) and (13.9.1 3). Finally, the formula (1 3.21.14.1) is a consequence of the Lebesgue-Fubini theorem. (13.21.15) Let A be a subset of X , and B a Jubset of Y . Then

(i) v*(A x B) = A*(A)p*(B) (with the product Convention of (13.11)). (ii) If A is I-measurable and B is p-measurable, then A x B is vmeasurable. (iii) If A is A-integrable and €3 is p-integrable, then A x B i~ v-integrable, and v(A x B) = A(A)p(B). These assertions are particular cases of (1 3.21 .II ) , (13.21.I3) and (13.21.I 4). (13.21.I 6) If ,f (resp. g ) is a locally A-integrable mapping of X (resp. locally p-integrable mapping of Y ) into or C , then f Q g is locally v-integrable, and

a

Since the set of points (x, y ) at which f (x) or g(y) is infinite is v-negligible (13.21.12), it follows from (13.21 .I3) and (13.9.8.1) thatfQ g is v-measurable.

Moreover, for each compact subset K (resp. L) of X (resp. Y) the function

21 PRODUCT OF MEASURES

229

(f@ g)'pK = (fqK)0 (g(pL)is integrable, by virtue of (13.21.14), hence f @ g i s locally v-integrable (13.13.1). Also, for each u E .X,(X) and each u E X c ( Y ) ,we have

by virtue of (1 3.21.I4) ;hence the formula (1 3.21.I6.1) follows from (1 3.21.I). (13.21.17) Let A, p be complex measures on X , Y, respectively. Then In 0 PI = I4 @ IPI.

-

We may write A =f. \A1 and p = g lpl, where If1 = / g )= 1 (13.16.3). Hence If 0 g l = 1, and the result therefore follows from (13.21.16) and (1 3.13.4). (1 3.21.18) Let A be a complex measure on X and p a complex measure on Y. (i) If A is concentrated on A c X and p is concentrated on B c Y (13.18), then A 0 p is concentrated on A x B.

( 4 Sup.p(A0 P) = S U P P ( x~ SUPP(P). (jii) With the product conuention qf (13.11) we have (13.21.18.1)

Ilf 0 PI1 = 1141- 11FIl.

In particular, $ A and p are hounded, then so is A@ p.

By virtue of (1 3.21. I 7) we may restrict ourselves to the case in which A and p are positive. Then (i) follows from the fact that X x Y - A x B is the union of the (A 0 p)-negligible sets (X - A) x Y and X x (Y - B) (13.21.12).

It follows from (i) that Supp(A 0 p) c Supp(A) x Supp(p). On the other hand, if x E Supp(A) and y E Supp(p), then for each compact neighborhood V of x in X and each compact neighborhood W of y in Y, we have A(V) > 0 and p(W) > 0, whence by (13.21.15) (A 0 p)(V x W) = A(V)pL(W)> 0. This establishes (ii), by virtue of the definition of neighborhoods in X x Y . Finally, (iii) follows from (13.21.11) applied to f = 4px and g = 'py, except when one of the factors on the right-hand side is 0 and the other i s +a,; but in this case we have A 0 p = 0 by (13.21.12), and therefore the formula (13.21.I8.1) remains valid in this case.

(13.21.19) There are analogous definitions and results for the product of any finite number of measures: if (Xi)1 i s any finite sequence of locally compact

230

Xlll

INTEGRATION

spaces, and if pi is a (complex) measure on X ifor each i, then the unique measure v on X

nXi which satisfies the relation n

=

i=1

is called the product of the measures p i (1 5 i 5 n), and is denoted by n

pl @ p2 @

*

@ p,, or @ p i , The

existence and uniqueness of v are proved

i=1

@ P , , - ~ )@ p,,, and observing that if by induction on IZ, putting v = (pl@ v' also satisfies the required conditions, then we must have

v'(h Of,) = ( ( P I O * ' . @ P n - 1Xh))pn(fn) = v(h Of,)

for all h E X

n Xi),by the inductive hypothesis. This characterization of the

n- 1

(i=1

product of measures shows immediately that the product is associative; in particular, we have also pi 0 ~2 0

*.

+

0 P n = pi 0 (PZ0

*

*

0 pn>.

We write

s

instead of f d v , and analogous notations for upper and lower integrals. We shall leave to the reader the task of formulating and proving the results corresponding to those established above for the case n = 2 ; the proofs require nothing but simple arguments by induction on n. The product of the Lebesgue measures on the n factors of R" is called Lebesgue measure on R".

PROBLEMS

1. Let X be a locally compact space, p a positive measure on X, and f a real-valued function 2 0 defined on X. In the product space X x R,let D, denote the set of points (x, t ) such that 0 5 t 5f(x). Also let denote Lebesgue measure on R.

(a) F o r f t o be p-measurable it is necessary and sufficient that D, should be measurable with respect to the product measure v = p Oh. (To show that the condition is necessary, prove that iffis p-measurable then D, is the union of a v-negligible set and a denumerable family of sets of the form A x I, where A is p-measurable and I is an interval in R + To show that the condition is sufficient, prove that if it is satisfied there exists a dense subset H of R such that f - ' ( [ a . i-031) is p-measurable for all tl E H, by using (13.21.10).)

.

21

PRODUCT

OF MEASURES

s

231

(b) Show that for f to be p-integrable it is necessary and sufficient that

D, should be v-integrable, and that we have then v(D,)

=s+a

=

fdp. Moreover,

if g is the decreasing real-valued function on R + defined by g ( r ) = p * ( f - ' [ t , then

if

dp

+a]),

g ( r ) dr.

(c) Suppose that X is an interval [0, a ] (where a > 0) and take p to be Lebesgue measure on X. Iffis a decreasing p-integrable function, show that

for every p-measurable subset A of X. (Use (b).) 2.

(a) With the notation of Problem 1, let I?, be the graph off, i.e., the set of all points ( x , f ( x ) ) in X x R.Iff is p-measurable, show that rf is v-negligible (use (13.21.13) and (13.21.10)). (b) Let 0 < a < 1 and let Q(a) be the subset of R2 which is the complement of the union of 1- I , 1 [ X I -a, a[and I-a, a [ x 1- 1 , I [in the squareQ = [- 1 , 1 ] x [- 1,1]. The set Q(a) is the union of its four connected components Ql(a)= [ - I ,

-a1 x [ - I ,

Qda)= 1-1,

-a1 x [a, 11, Q4(a)= [a, 11 x [ - I , -a].

-a],

Qda) = [ a , 11 x [ a , 1 I,

For i = 1, 2 , 3 , 4 let hi, denote the similitude mapping Q to Q1(a)for which

h1,=(-1, - 1 ) = ( - 1 , - I ) , h2..(-1, - I > = (-1, 4, h P I ( - l , --I)=(a,a), h4.d-1. - - 1 ) = ( 1 , -4,

h l . d , -])=(--I, -a), h Z . . ( l , - 1 ) = ( - a , a>. ha,.(l, - 1 ) = ( l , a), h4,d1, - l ) = ( l , -1).

Also, in the interval [0,71 in R, put I k = [ k, k i 11 for 0 6 k 5 6 and let uk be the increasing similitude which maps [0,7] to 1,. Let f. be the continuous mapping of [0,7] into Q which is affine-linear on each of the intervals I k and is such that the images of 0, 1, 2, . . . , 7 are respectively the points(--l, - 1 ) , ( - - 1 , - - 4 , ( - 1 , a ) , ( - = , a ) , ( a ,a ) , ( l , a ) , ( l , --a),(l, -1). We shall now define a sequence (gn)of continuous mappings of [0, 71 into Q as follows. Let be a decreasing sequence of numbers belonging to the interval 10, 1 [. Define go =fa, . For n 2 1, suppose that 9.- has been defined, and consider all sequences s = (il, . . ., in)in which each ix is one of the integers 0, 1, . . , 6. Put us = uI10 . . . UI.. Then it is sufficient to define gn(us(f))for 0 5 t 5 7 and for each =g.-l(us(r)). of the 7" sequences s. If at least one of the it is odd, we put gn(us(f)) If on the other hand il = 2j, for 1 S I5 n (with 0 5 j r 2 3), we putgn(us(t)) = ws(hn(r)), where w ~ = ~ J ~ o + ~ I J, ~~ +~0 I , ~h ,~n + l , a n .Show that thesequence(g,)converges uniformly to an injecfiue confinuous mapping g : [O, 71 + R2,and that the simple arc g([O, 71) (Section 4, Appendix to Chapter IX) is nonnegligible with respect to Lebesgue measure on R2, if the sequence (an)is suitably chosen.

.

0

3. (a) Give an example of two compact spaces. X and Y , positive measures h and p on X and Y, respectively, and a h 0p-measurable function f such that the two

integrals

1

dp(y) f ( x , y ) dh(x) and

unequal (cf. Section 5.2. Problem 5).

i s

dh(x) f ( x , y ) d p ( y ) are both defined and are

Xlll INTEGRATION

232

(b) For each integer n > 0 , let A;= [2-", 3 . 2-"-'[ and A = [ 3 . 2-"-', 2-"+l[. Let BA=AAxAA, B:=A:xA;, CA=ALxA:, C:=A;xAA in R2. Define f:RZ+ R as follows: f ( x , y )= 4"+' for ( x , y ) E BA LJ B:; f(x, y ) = -P+' for ( x , y ) E CA u C:, for each integer n > 0; f ( x , y ) = 0 otherwise. Show that f is measur-

ss

ss

able and that the two integrals dy f ( x , y ) dx and dx f ( x , y ) dy are defined and equal, but thatfis not integrable with respect to Lebesgue measure on R2. 4.

Let X, Y be two locally compact spaces, A a positive measure on X and p a positive measure on Y . Let f be a mapping of X x Y into a metrizable space G, such that: ( I ) for each x E X, the mappingf(x, .) is p-measurable; (2) for each y E Y, the mapping f(.,y ) is continuous. Show that under these hypotheses f is (A 0p)-measurable. (Reduce to the case where X and Y are compact; by using Egoroff's theorem and the fact that X is metrizable, show that f is almost everywhere (with respect to A 0p) the limit of a sequence of ( A @3 measurable functions.)

5. Let X, Y be two locally compact spaces, a positive measure on X and p a positive measure on Y. Letfbe a real-valued function 2 0 on X x Y which is bounded on every compact subset of X x Y and such that: ( I ) for almost all x E X the function f ( x , .)

is p-measurable; (2) for each function h E X ( Y ) ,the function

XH

s

f ( x , y)h(y) d p ( y ) ,

which is defined almost everywhere, is A-measurable. Show that under these conditions there exists a (I\ 0p)-measurable function g such that for each x E X we have f ( x , y ) = g ( x , y ) except at the points of a p-negligible set A, (depending on x ) . (Show that, for each function U E X(X x Y), the function f ( x , . ) u ( x , .) is p-integrable for almost all x

E

X, and that the function

XH

s

f ( x , y ) u ( x , y ) d p ( y ) , defined

s s

almost everywhere, is A-integrable; for this purpose approximate u by functions of the form u(x)w(y). Then remark that the linear form

I( H

d/\(x) f ( x , y)u(x, y ) d p ( y )

is a positive measure on X x Y with base 0p,and apply the Lebesgue-Nikodym theorem. Finally use the fact that Y is a denumerable union of relatively compact open sets U, and that there exists in X ( Y )a denumerable set of functions D such that every function belonging to X ( Y ) is the uniform limit of functions belonging to D and having their supports contained in some Un.) (b) Show that the conditions of (a) are satisfied if: (1) for almost all y E Y,the function f( . ,y ) is hneasurable; (2) for almost all x E X, the function f ( x , . is continuous almost everywhere with respect to p. (Use Problem 7(c) of Section 13.9.) (c) Take X = Y = [0, 1 ] and take A and p to be Lebesgue measure. Assuming the continuum hypothesis, let x < y be a well-ordering on X for which there exists no greatest element and such that, for each x E X,the set of all z E X such that z < x is denumerable. Show that the characteristic functionfof the set of pairs ( x , y ) for which x

< y satisfies the conditions of (a), but that

jf(x, y ) +(y)

=1

for all x

E

y ) dh(x) = 0 for all y

E

Y and

X.

6 . Let u, u be two increasing real-valued functions on R, each continuous on the right, and such that u(x) = u(x) = 0 for x < 0. Let w be the increasing function on R, continuous on the right, defined by w ( f )= u(t)u(t) for t 2 0 and w ( t ) = 0 for f < 0. Let A, p,v be the Stieltjes measures associated with u, u , w , respectively (Section 13.18, Problem 6).

21 PRODUCT OF MEASURES

233

To each real-valued function f o n R we associate the function fo on R2defined by fo(x, y ) = f ( x ) if y < x , f o ( x ,y ) =f(y) if y 2 x . Show that f i s v-integrable if and only

I SIf

if fo is ( h 0p)-integrable, and that in this case f d v =

dh d p . (Prove the result

first for characteristic functions of intervals.) Hence deduce the formula

If(4M x ) =I f ( X ) v ( x - 1du(x) + S f ( x ) u ( x + 14 x 1 . In particular, if u and u are contintrous on by purrs :

R,we have the formula of integration

IObu(x)du(x) = u(b)u(b)- u(a)u(u) - s b u ( x ) du(x). II

Consider the case where u and v are constant on each interval [n,n integer n 2 0 ("Abel's partial summation formula").

+ 1 [ for each

7. Let p be a finite real number 2 I , let X and Y be locally compact spaces, h a positive measure on X and p a positive measure on Y . Let 0 be a function on X x Y such that f and f p are ( h 0p)-integrable. Prove that

fz

5

( [ * ( J f d p ) ' dh)'"

J*(Ifp

dh)'" d p .

(For each X E X ,apply Holder's inequality (13.11, Problem 12(a)) to the function

y ~ f ( xy), , in the forrnf(x, y ) = g ( x , y ) conjugate t o p . ) 8.

l/Prl

f p ( x ,y ) dh(x))

, where 9 is the exponent

Let X I(1 2 i 5 n ) be locally compact spaces and pi a positive measure on X i for 1 5 i 5 n. For each i , let E, denote the product XI. Let fi be a function 2 0 which

"

is measurable with respect to p = @ p I on X i=I

n n X i , and does not depend on

J+i n = I=I

f;-' is integrable with respect to the measure @ p,, for 1 5 k function

nfiis p-integrable and that n

J+k

5 n, show

xi. If

that the

I= 1

where J x =

S. . . I

f;- dpl

'

. .d p k -

+&+

.. . d p . for 1 5 k 5 n. (By induction on

n , using the Lebesgue-Fubini theorem and Holder's inequality.) Deduce that if A is a p-measurable subset of X, and Ai its projection on E l , and if Al is integrable and of measure m,with respect to the measure @ pJ on El, then A J#i

is p-integrable and p(A) 2 ( m l. . . mn)'/("-'). Generalize to the case in which, instead of considering the products of the XI n - 1 at a time, we consider the (g) products of the X Ip a t a time, and integrateoverx a product of (;) functions 20, each of which depends on only p of the variables X I , . . . , X" .

234

INTEGRATION

Xlll

9. Let (X,),,, be an infinite sequence of compact spaces, and let pa be a positive measure on X, with total mass 1 , for each n. (a) Show that on X =

n X, there exists a unique positive measure m

"=O

of functions f;E %',,(XI),we have

each integer n and each finite sequence ( f i ) 0 6

p(f) =

n p,(fi), where

I=O

p such that, for

f ( x ) = n f , ( p r i x ) . (Observe that the continuous functions 1=0

of the form n f i ( p r i x ) , for all choices of n and off, E ~ R ( X , )form , a total set in the 1=0

Banach space 'ZR(X).)The measure p is called the product of the family (p.) and is

"

written p = @ p.. *=O

(b) For each n, let A. be a p,-measurable subset of X,. Show that the product A=

n m

"=O

A. is p-measurable, and that p(A) =

m

"=O

p.(An).

n

(c) Letf? 0 be a p-integrable function on X . For each subset L of N, put L' = N - L, and identify X with the product XL x X,, , where XL= X. and X,. = X,. If x

"€L

E

n

E

L'

X, put xL =: prLx and x L p= prL,x, and identify x with (xL,xL.). Finally, let pL,

be the product measure

J

0 , p, on XL,, and putfL(x) = ~ ( x LXL,) , dpLt (x,~).

"EL

Now let (L.) be an increasing sequence of subsets of N, and put g = supf,, h = supf,,. . For each c > 0, let A, be the set of points x

E

X at which g(x) > c,

and B, the set of points x E X at which h(x) > c. Show that c . p(A,) c

and

5

i

f dp and

. p(B,) 5 s f d p . (Remark that A, is the set of points x E X for which at least one

$IG"

of thefln(x) is >c, and express A, as a denumerable union of mutually disjoint sets G . such that c p(G.)

fdp.)

(d) Suppose that (L.) is an increasing sequence of finite subsets of N whose union is N. Show that the fLn tends to f almost everywhere and that f ~ tends " to the constant

[fdp almost everywhere. (For each E > 0, consider a continuous function g, depending on only finitely many variables, and such that to the function

If-

gl.)

s

If-gI

dp

s . ~and, apply (c)

On each factor X. ( :D) 10. Let D be the discrete space (0, 1) and X the product space DN. of X let p,, be the measure for which pn(0)= p.(l) = 1, and let p = @ pn be the n e N

product measure on X (Problem 9). (a) For each point x = (x,),,,

in X with x,

Lo

=0

or x.

=

I , put T(X) =:

C ~.2-"-~.

"=O

Show that T is a continuous mapping of X onto the unit interval I [C, I ] in R, and that the image n(p) of p is Lebesgue measure on I . For each t E I, the set n - ' ( t ) consists of a single point, unless I is of the form k . 2-" with k integral and 0 < k < 2". Hence the mapping f w / o T induces, on passing to the quotients, an isometric isomorphism of LE(I,x) onto LE(X, p) for 1 2 p 5 I a. (b) For each t E 1 and each n 2 I , put r,(t) = I - 2pr,- , ( r - ' ( t )if) f is not of the form k . 2-" with k integral and 0 k 5 2". and r.(t) =: 0 otherwise. The function rnis called the nth Radeniacherfitncfion.These functions form a nontotal orthonormal 7

21

PRODUCT OF MEASURES

235

system in 9&(1, A) (show that they are all orthogonal to the functions cos 2kn-r for k integral and 2 0 ) . Show that r,(t) = sgn(sin 2 ” d ) . Show that the measure p is (c) For each x = (xJns0in X, put u ( x ) = (x.+ invariant with respect to u and that u is ergodic with respect to p (use (a) and Problem 5(c) of Section 13.12). Deduce that lim

n-tm

I n

- ( r l ( t )i . .

+ r.(t)) = 0

almost everywhere with respect to Lebesgue measure (“ Borel-Cantelli theorem ”). (Observe that r.(l - t ) = -u,(t) for 0 5 t 5 4.) (d) For each finite strictly increasing sequence (6), s f s k of integers >= 1, show that

jolr n l ( t ) r n z ( f.). . r m k ( fdt) (e) For each finite sequence > 0, show that

bxbn

=

0.

of complex numbers, and each real number

p

(“Khintchine’s inequality”). (First consider the case where p =: 2h, with h an integer 2 1, by using (d). For 2h - 2 < p 5 2h, use problem 12(e) of Section 13.11.) (f) With the notation of (e), show that

(use (e) for p

=

4,and Holder’s inequality for p

=

3 and q = 3).

11. (a) The hypotheses are those of Section 13.17, Problem 2, and in addition the functions f. are assumed to be real. Let XH~(X)be a p-measurable mapping of X into the set I = { I , 2, . . . , n}, and let k~ w ( k ) be an increasing function on I with values >O. Then the mapping (s, u)t+KJ(.)(s, u ) / w ( j ( s ) ) of X x X into R is p 0p-

measurable. For each p-measurable subset A of X, show that

where h(s, t ) = inf(j(s),j(r)), (Express the square of the integral on the left as a double integral, so as to reduce the problem to the evaluation of a triple integral over A x A x X, and use the fact that the j,are orthogonal.) Show next that

(split up A x A into two measurable sets, defined respectively by j ( s ) O and a p-measurable subset A of X such that IH.(s)/ 5 c * w(n) for all s E A and all n 2 1, where c > 0 is a constant. Show that, for each function g E 9 i ( X , p), the sequence (s.(g)(r))/w(n) is bounded for almost all t E A. (Consider the increasing sequence of p-integrable functions u.(t) = sup (sk(g)(t))/w(k),and show that the sequence of

j(s) >j(t)

lbkBn

236

Xlll

INTEGRATION

=jA u.(t) + ( t )

integrals J.

is bounded above. For this purpose, note that we may

write u,(t) = ( s j c , , ( g ) ( t ) ) / w ( j ( r ) ) for a suitably chosen p-measurable mapping j of X into I, and majorize J,' by using the Cauchy-Schwarz inequality and (a) above.) (c) Suppose that the hypotheses of (b) are satisfied and also that lim w h ) = f a . If the a. are real and if the series

c m

n=l

n-m

a.' w2(n)converges, show that the series

c a.f.(t) m

"21

converges almost everywhere in A. (Start by using Problem 8(b) of Section 13.11. Then, in order to majorize Is.(t) - s.,(t)l for nk I n < n k + l ,determine an increasing sequence (c,) of numbers >O such that lim c.= n-m

+ a and

m

10.2 w'(n)c,' < +co

"= 1

).fI

(Section 5.3, Problem 6), so that the numbers b. = a. w(n)c. are of the form (9 for some g E Yi(X, p). Then use the fact that, by virtue of (b), the partial sums

Ik1:, I b&t)

are bounded for almost all t E A, and apply Abel's partial summation

formula.)

(d) Deduce from (c) that if IH.(s)l $ c for all s E A and all n, and if the series converges, then the series 6 of Section 5.3 again.)

m

series

I=1

c. r.(t)

a.'

"= 1

a.f.(t) converges almost everywhere in A. (Use Problem

I= 1

12. (a) Let (c.) be a sequence of real numbers such that m

m

m

;I:e,' < fa.Show that

I =1

the

(the notation is that of Problem 10) converges almost everywhere in

I with respect to Lebesgue measure h. (Express the Rademacher functions as linear combinations of the functions of the Haar orthonrmal system (Section 8.7, Problem 7), and note that Problem 1 Id) is applicable to the latter orthonormal system (cf. Section 13.17, Problem 2)) (b) Given any sequence (a") of real numbers, a real E > 0 and a measurable set A c I, show that there exists an integer no such that

whenever rn > n 2 n o . (Use the Cauchy-Schwarz inequality, the fact that the functions r l ( t ) r j ( t ) with ic j form an orthonormal system (Problem 10(d)) and Bessel's inequality applied to this system and to the function (c) Let (amn)be a double sequence of nonnegative numbers such that for each m the set of integers n 2 1 for which anla# 0 is finite, and such that lim am"= 1 for each m-m

n. Also let ( c ) be a sequence of complex numbers such that, if we put

the sequence (S,"(t)) is convergent in an integrable set A c I of positive measure. Show that

c m

n=

1

lc.I2

< + m . (By virtue of Egoroff's theorem, there exists an integer

no and a set B C A of positive measure, such that for all rn

2 no and q > p 2 n o , we

21

have

1

a m k c k r k ( t5 ) 1 for

all t

E

OF MEASURES

PRODUCT

237

B, and therefore

Minorize this integral with the help of (b).) (Rademacher-Kolmogoroff theorem.)

X be a locally compact space, p a positive measure on X, and (uJnao a sequence of p-integrable complex functions such that, as H runs through the set of finite subsets of

13. Let

N, the set of numbers m

series

I

JIZ"u.(x)

dp(x) is bounded. Show that in these conditions the

x lun(x)lz is convergent almost everywhere in X. (Observe that, for all

Klc

n=D

I

ak(x)rk(t) dp(x) is

t E I = [0, 11, the set of numbers

bounded above by a number

independent of n and t ; then use Problem 10(f) and the Lebesgue-Fubini theorem.)

X be a compact space, p a positive measure on X, and (f,) an orthonormal sequence in (t&(X,p) which is uniformly bounded in X.

14. Let

(a) Let (6.) be a sequence of real numbers such that not possible that

x Ib,(g If,)l

2 b,2 = + 03. m

"=l

Show that it is

< + 03 for all continuous functions g on X.(Argue by

m

"=I

contradiction: the functions u.(x) = b,fn(x) would satisfy the conditions of Problem 13, by virtue of the Ranach-Steinhaus theorem applied to the linear forms g

Hz b.(g If,) "EH

on V(X). Hence

zb;lfn(x)Iz < + m

*=I

co almost everywhere in X.

Using Egoroff's theorem, the fact that the functions f. are uniformly bounded and

If,(x)1' dp(x) = 1, obtain a contradiction of the hypothesis

5 b.2

"= 1

=

+

03.)

(b) Let 9 be a real number such that 1 < 9 < 2. Show that it is not possible that m

I(g

"=l

1f J q

< +a, for

all

continuous functions g on X.(If p

= 949 -

I), show that

if the assertion were false there would exist a sequence (b.) of real numbers > 0 such that b.2 = co, 6,"< a and C I b,(g Ifn)l < m for all continuous

x n

+

c

+

I

+

n

functi0ns.g.) (c) Deduce from (b) that there exists a continuous function g on X such that, for all 9 satisfying I < q < 2, we h a v e x I(g ljJq = +a.(Use the principle of condensation

"

of singularities (Section 12.16, Problem 14).) 15. On the space R",let h be Lebesgue measure and let ljxll be a norm for which the unit ball llxll 6 1 has measure 1. Let ( x k k a l be an infinite sequence of distinct points in an integrable bounded set B such that h(B) = 1. For each integer m > 1, let d,,, denote the smallest of the numbers jlx, - xjll with 1 6 i < j 5 m. Show that lim inf m d,,,5 c; where m+ m

l(1-

t)"

dt

238 XI11 INTEGRATION

(cf. Section 12.7, Problem 6). (Argue by contradiction, and assume that for some E > 0 there exists mo such that md," > h: for all m 2 m o , where h: = c;' 4E . For 1 i 2 m,let B, be the ball with center x f and radius )h. m-"",and for m 2 i 6 2"m let B, be the ball with center x i and radius )h.(2i-"" - m-""). Show that the 2"m balls B, are pairwise disjoint, and calculate the measure of their union by the EulerMacLaurin summation formula.) 16. Let X, Y be two locally compact spaces and A a universally measurable subset of x x Y. (a) For each x E X, show that the section A(x) of A at x is a universally measurable subset of Y. Also, for each measure p 2 0 on Y , the function x ~ p * ( A ( x )is) universally measurable (use the Lebesgue-Fubini theorem). (b) If Y is compact and if A is closed in X x Y, then the function x-p*(A(x>> is upper semicontinuous. (c) Let p be a positive measure on Y such that the section A-'(y) is denumerable for almost all y E Y (with respect to p ) . Show that the set N of points x E X for which p*(A(x)) > 0 contains no nondenumerable compact subset. (By using Section 3.9, Problem 4, reduce to the case where this compact set contains no isolated point, and show that such a set is the support of a diffuse measure #0, by using Section 13.18, Problem 6(b) and Section 4.2, Problem 3(b).) 17. Let p be a bounded positive measure on X, with total mass 1 . I f f € 9; is such that

/log11

+ Cfl

d p 5 0 for every complex number (, then f is p-negligible. (Use the

formula

(which is valid for all

5 E C) together

with the Lebesgue-Fubini theorem, to evaluate

the integral jlog+lRfl d p , where R > 0, and deduce that this integral must be zero.)

..., m - 1}. Let X denote the compact product space Iz, and p a positive measure on I with total mass 1 (so that p is defined by the

18, Let I denote the discrete space {O, 1,

finite sequence of masses p ,

=

p ( { j } )such that p , 2 0 and

X, of X let p" denote the measure p , and let

Y

m-1

I=o

p,

=

1). On each factor

be the product measure @ p,, on X n € Z

(Problem 9). (a) For each x = ( x ) " ~in~X , put u(x) = ( x . + ~ ) Show ~ ~ ~ that . u is a homeomorphism of X onto X and that the measure Y is invariant under u. The triple (X,Y , u) is called the Bernoulli scheme B ( p o , .., p a (b) Consider in particular the Bernoulli scheme B(),i). For each x = ( x J n G ZE X , where x. = 0 or 1 for all n E Z, letf(x) denote the canonical image in TZof the point

.

( y , z ) E R1,where y =

m

"-0

x - n 2 - " - 1 and z =

Q

n=l

xn2-".Let

fl be the normalized

Haar

measure (14.3) on the compact group T2.Show thatfis continuous and that f(v) = !?t; also that the set of points t E T2such that f - l ( r ) does not consist of a single point is fi -negligible.

21

PRODUCT OF MEASURES

239

Let v be the restriction of the canonical mapping T : R2+ T 2to the set K of points ( x , y ) such that 0 2 x < 1 and 0 y < 1, so that p is a continuous bijection of K onto TZ.For t E T2and ( x , y ) = v-I(t),put

Show that the mapping u : T2 + T2(“baker’s transformation”) is continuous almost everywhere and that u o f = f o u (where u is the mapping defined in (a)).

We shall see later that u and 2) are ergodic (relative to v and p, respectively) (Section 15.1 1, Problem 16). (c) Show that for the Bernoulli scheme B ( p , , . . . ,p b - , ) , the entropy h(u) (Section 13.9, Problem 28) is given by h(u) ==

+

IogpO

-(PO

“’

f p k - 1 logpk-1).

(Use the Kolmogoroff-Sinai theorem, by starting with the partition cc = ( A j ) o s j q k - l , where A, is the set of (x.),.~ such that xo j . Use Problem 9(a) and the definition of u, and remark that the set :

Ajon u - ’ ( A J 1 n ) . . . n U P +(AJII-1) ’

is the set of ( x n ) n s zsuch that x,, =j o , x , calculate the sum

=j , ,

.. . , x,- I

=

The problem is then to

For this purpose, observe that

Deduce from this calculation that the bijections u of the Bernoulli schemes B(1, 8) and B(i, 1, a) are not conjugate (Section 13.12, Problem 11). 19. Let X be a locally compact space, let p be a positive measure on X, and let A g be two nonnegative p-measurable functions on X. For each cc > 0, let A, be the set of x E X such that f ( x ) > u . Suppose that: (i) p(A,) < 03 for all u > 0; (ii) g E =Y;(X, p) for some p E ]I, m[;

+

(iii) for each cc > 0, p(A.) 5

U

1’

A.

+

g dp.

Show that these conditions imply that f~ =Yg(X, p) and that

(Wiener’s inequality). (Consider first the case where f is bounded and of compact support, and integrate from 0 to co the inequality

t P - ’ j V + t tdp 5

tp-’!gP)At

dp

240

Xlll INTEGRATION

(cf. Problem 1). For the general case, consider the functions ~ u p ( n , f g ) ~ ~where ), (K.) is an increasing sequence of compact sets whose union is X.) 20. Let U be a continuous endomorphism of the space L:(X, p) satisfying the hypotheses of Section 13.1 1, Problem 17, so that U extends to an endomorphism of each of the spaces Le(X, p), 1 < p 5 CD (loc. cit.). For each functionfe 9;,put

+

(Use Problem 19 above, and Section 13.11, Problem 17(c).) (b) For each functionf E =Y;, let P be the limit in L: of the sequence ((R.(f))”) (Section 12.15, Problem 12(c)). We have U P = P U = P . Show that the sequence (R.(f)) converges almost everywhere to P .f (Dunford-Schwartz ergodic theorem). (Put

S(f) = lim SUP RAf), n-m I(f)

= lim inf n- m

R.(f).

Observe that I(f) 5 S(f) 5 R*(f), and that S(R,(f) - P .f)= S(f) - P .f and I(R,(f) - P -f)= I(f) - P .ffor all rn 2 1, and use (a) with R,(f) - P * f i n place of f.) (c) Show that N I ( P .f)5 N l ( f ) for all f~ 9; n Y;, and hence that P extends to a contraction on the space LA. Deduce that the sequence (R.(f)) converges almost everywhere to f, for each function f~ 2:.(Put L(f) observe that L(f) R*(f- g) Section 13.11, Problem 7(d).)

I R A f ) - P .fl,

= lim SUP n-m

+ P . If-

g

1

for all g

E 9: n 9

g , and make use of

21. The notation is that of Section 13.17, Problem 7. For each t~ > 0 and each function g E 9P:o,,c(Y, v), let A&) denote the set of points y E Y such that ig(y)/ > a. If p E [I, +a[,the endomorphism U is said to be of weak type (p,p) if there exists a constant C > 0 such that, for each p-integrable step function f and each CL > 0, we have

21

PRODUCT OF MEASURES

+

241

Let p, q be such that 1 z p < q < w . Show that, if U is of weak type ( p , p ) and also of weak type (q, q), then U is of type (r, r) (Section 13.17, Problem 7) for all r such that p < r < q (Marcinkiewicz's interpolation theorem). (For each f~ 9A(X, p) and a > 0, let f: be the function which is equal to f ( x ) if If(x) I > a, and zero otherwise, and put f = f -f:. Show that, for each t > 0, t'-'p(Azt(U .f))5 C't'-"' Integrate from 0 to

00

1 If:lp

and use Problem 1.)

dp

+ C"t'-@-'

CHAPTER X I V

INTEGRATION IN LOCALLY COMPACT GROUPS

Haar measure and convolution on arbitrary locally compact groups have become indispensable tools for the modern analyst, as they were in classical analysis on the real line and in finite-dimensional Euclidean spaces. Together with convolution of distributions, which generalizes convolution of measures and which we shall introduce in Chapter XVII, they are the fundamental notions in harmonic analysis (Chapter XXII) and the theory of linear representations of compact groups (Chapter XXI). We have again followed, though in less detail, the exposition of N. Bourbaki [22]. I n fact, since in this treatise practically the only locally compact groups we shall consider will be Liegroups (Chapters XVI, XIX, and XXl), for which there is a much simpler proof of the existence of a Haar measure, we could have restricted ourselves entirely to Lie groups. Nevertheless, it seemed worthwhile to bring out the fact that the theory of integration on a locally compact group is entirely independent of any differential structure; and the totally discontinuous locally compact groups have ceased to be mere curiosities since the advent of p-adic and “adelic” groups in the theory of numbers [36]. Throughout this chapter, the phrase L L locally compact group separable metrizable locally compact group.”

”

will mean

1. E X I S T E N C E A N D U N I Q U E N E S S O F H A A R M E A S U R E

Let G be a (separable, metrizable) locally compact group. Iff is any mapping of G into a set E and if s is any element of G, we define the left and right translations y(s)fand 6(s)foffby s, which are mappings of G into E, by 242

1 EXISTENCE AND UNIQUENESS OF HAAR MEASURE

243

the formulas

It follows immediately from this definition that we have

for all s, t E G . If p is a (complex) measure on G, we denote by -y(s)pand S(s)p the measures on G which are the images of p under the homeomorphisms X H S X and x w x s - ' , respectively (13.1.6), so that (14.1.2)

(f,Y ( ~ ) P = ) ( ~ ( s - ' ) fp), ,

(f,WS)P)

= (W-')f, r ~ )

for all functions f E .fX,(G).From this definition it follows that

for all s, t E G. The measure p is said to be /eft (resp. right) inuuriant if Y(4P = P (resp. W

(14.1.2.2)

P = PI

for all s E G. If a measure p # 0 on G is left invariant, then Supp(p) = G, because Supp(y(s)p) = s Supp(p) for all s E G by virtue of (1 3.19.4), and Supp(p) # 0. Similarly for right-invariant measures. Let p be a left-invariant measure on G and letfbe a p-integrable mapping of G into R or C. Then for each ~ E the G function X H ~ ( S - ' X ) is also p-integrable, and we have

-

(14.1.2.3)

J W X )

44x) =

s

f(x)44x)

by (13.7.10). In particular, if A is a p-integrable set, then so is sA, and

Iffis any mapping of G into a set E, we write (14.1.3)

f(x) =f(x-')

for all x E G.

244

XIV

INTEGRATION

IN LOCALLY COMPACT GROUPS

If p is any measure on G, we denote by ji the image of p under the mapping X H X - ' of G onto G, so that we have

F> ( J P>

(14.1.4)

(f, =

for all f E Xc(G).It follows immediately from the definitions that we have (y(s) f ) - = 6(s)f, each side being the function X H f ( s - ' x - ' ) ; and therefore, for any measure p on G, we have (y(s)p)" = S(s)fi. Hence if p is a leftinvariant measure, ,ilis a right-invariant measure, and vice versa. (14.1.5) Let G be a locally compact group. Then there exists a nonzero leftinvariant positive measure p on G , and every other left-invariant measure on G is of the form ap, where a E C. (1) Existence. Let Xx*,denote the set of functions g E XR(G) which are 2 0 and not the zero function. For each f E XR(G) and each g E .X: , there exist positive real numbers cl, . . . , c, and elements sl, . . . ,s, in G such that

(14.1.5.1)

(Le., such that f ( x ) S

r

1 c,g(s;'x)

i=1

for all x

E G).

For there exists a non-

empty open subset U in G such that a = infg(x) > 0; since Supp(f) is comXOU

pact, there exist a finite number of points si E G (1 5 i 5 r ) such that the s,U cover S u p p ( f ) , and then (14.1.5.1) is satisfied by taking ci = llfll / a for all i. We shall denote by ( f :g) the greatest lower bound of the numbers i=1

c i , for all systems (cl,

. .. , c,,

sl,

. . . ,s,) satisfying (14.1.5.1).

The symbol

(f:g) has the following properties:

(0 ( W f g:) = (f:9)

for all f E XR(G), g E S ' f , S E G ; (ii) ( a f :g) = a(f:g) for all f ' XR(G), ~ g E X*, , a 2 0 ; (iii) (fl+ f 2 :9) 5 (fl :g) + ( f 2 : g ) for h,f2 S'R(G), g X*,; (iv) (f:g) 2 supf(x)/sup g(x) for all f~ XR(G), g E X : ; xsG

x s 0

.

for all S , f o , g in .X*,

1 EXISTENCE AND UNIQUENESS OF HAAR MEASURE

245

Properties (i), (ii), and (iii) are immediate consequences of the definitions. Given (14.1.5.1), there exists (3.17.10) s E G such that

and (iv) follows. For (v), observe that if we have g5

c bjy(rj)h,then f 5 1a i b j i

i, i

fr Caiy(si)g

and

y(si tj)h, and therefore

(f:h)sCaibj= i, i

(T

a,

Chi.

l ( j

1

Since we may take c a i (resp. c b j ) arbitrarily close to ( f : g) (resp. (g :A)), i

i

we have (v). Finally, (vi) follows from (v) applied to fo,f, g and to f, j b , g. By hypothesis, there exists a denumerabli fundamental system (V,) of neighborhoods of the neutral element e in G. For each n, let g, be a function belonging to X*, such that Supp(gn)cV, (4.5.2). Let fo be a function in X * ,, fixed once and for all, and put (14.1.5.2)

In(f)

= (f: gn>/(fo: gn)

for ali f~ X*,; let I,(O) = 0. From (ii) and (iii) above it follows immediately that the mappings f-I,( If/) are seminorms on XX,(G),and from (i) that I,(y(s)f) = I,(f) for all s E G. Next, there exists a sequence of relatively compact open sets Up which cover G and are such that 0, c (3.18.3). The space %(Up)is separable (7.4.4) and therefore so is X ( G ; 0,) n X r (3.10.9). Hence there exists a dense sequence (fmp),,,21 in the latter space. Since the sequence of values In(fmp) (n 2 1) lies in the compact interval with endpoints l/(fo : f m p ) and (fmp :fo)by (vi), it follows from (12.5.9) that if we replace the sequence (9,) by a suitably chosen subsequence, we may assume that the sequence (In(fmp)),,2 tends to a limit > O for all m,p . Also it is clear that iff, f ’ in X,* are such that then I,(f) 0 be a real number, and let h be a function 5 0 belonging to XR(G) such that h(x) 2 1 on the union of the (compact) supports offandf’ ((3.18.2) and (4.5.2)). Then it is enough to prove that there exists a compact neighborhood V of e in G, such that, for all g E %*, with Supp(g) c V, we have

(f:g ) + (f‘:g) 6 (f+f’: 9) + 4 h : g).

(1 4.1.5.4)

To prove this, put u = f + f ’ + + E / I , and let u (resp. u’) be the function which coincides withflu (resp. f ’ / u ) on Supp(f+f’) and is zero on the complement of this set. Since at every frontier point x of Supp(f+f’) we have f ( x ) + f ’ ( x ) = 0, and therefore f ( x ) = f ’ ( x ) = 0, it follows that u and u’ belong to XR(G) and are 2 0 , The functions u and 11’ are therefore uniformly continuous with respect to a left-invariant distance defining the topology of G ((12.9.1) and (3.16.5)), and therefore, for each q > 0, there exists a compact neighborhood V of e such that Iu(s) - u(t)l 6 q and Iu’(s) - u’(r)l 6 q for all pairs (s, t ) such that s - ’ t E V. Now let g EX: be such that Supp(g) c V. For each s E G we have u y(s)g 6 (u(s) q) y(s)g. (This is obviously true at points where y(s)g vanishes, hence at points outside sV;and if r E sV we have u ( t ) u(s) q.) Similarly, we have D‘ . y(s)g 6 (u’(s) q)y(s)g. This being so, let ci (1 5 i 511)be real numbers 2 0 , and si (1 5 i 5 n) elements of G such

-

+

1

+

that u 6

+

c ciy(si)g.Then we have n

i= 1

n

n

and a similar set of inequalities withfreplaced byf’. Consequently

c n

(f : 9) + (f’: 9) 5 i = 1C

i ( W

+

+ 2rl) 6 (1 + 277) c c i , n

“’(Si)

i= 1

because u + u’ 5 I . From the definition of u, and properties (ii), (iii), and (v), we obtain

(f:

g)

+ (f’:8) 5 (1 + 2q)(u: g) 5 (1 + 2 q ) ( ( f + f ’ : g) + +E(h :9)) 6 ( f + f ’ : g) + t 4 h : 9)+ 2 t l ( ( f + f ’ ) : h)(h : g) + El](h : g)

so that, by taking q such that

q(2((f +f ’ ) : h) + E ) 5 38, we get (14.1.5.4)

1 EXISTENCE AND UNIQUENESS OF HAAR MEASURE

247

Now extend I to the whole of X X , ( Gby ) putting I(0) = 0 and, for each -f2 with fI, f i in if:, I(f) = I(fi) - I(fi). From (14.1.5.3) it follows immediately that I(f) depends only o n f an d not on the choice of expression f = f i - f2. Clearly (14.1.5.3) remains valid for all J; f' in X R ( G ) Further. more, the original definition of I shows immediately that I(If) = I I ( f ) for all f E XT and all real numbers I > 0; and this relation obviously extends to the case where f~ XR(G) and I E R. We may therefore conclude from (13.3.1) that I is a positive measure on G ; also I is not zero and, by construction, I(y(s)f) = I(f) for all f e X R ( G ) . In other words, we have constructed a nonzero left-invariant positive measure on G.

f=fi

(2) Uniqueness. Let p (resp. v) be a nonzero left (resp. right) invariant measure on G. Then 5 is a left-invariant measure. We shall show that p and 5 are proportional, and this will complete the proof of (14.1.5). Let f~ X,(G) be such that p ( f ) # 0, and consider the function D, on G defined by

D,(s> = pc(f>- J f ( t - 1 r ) We shall show that D, is continuous on G. In fact this will be a consequence of the following more general result: (1 4.1.5.5) Ler G be a locally compact group, H a closed subgroup, a a measure on H, and f : G -+C a continuous mapping. Suppose that either Supp(f) is compact or Supp(a) is compact. Then the mappings SH

s

f(st)da(t)

and

SH

are continuous on G .

s

f(ts)da(t)

Consider, for example, the first of these integrals. Let so E G and let V, be a compact neighborhood of so. Given E > 0, we have to find a neighborhood V c V o of so such that for all s E V we have f ( s t ) - f ( s o 1 ) ) d a ( t ) )S E . If K = Supp(f) is compact and L = VG'K, then

I/(

n

, l

since f is uniformly continuous (with respect to a right-invariant distance on G) (3.16.5), there exists a neighborhood W of e in G such that the relation s E Ws, implies I f ( s t ) -f ( s o t)l &/IaI(L)for all t E G, and we may take V = V, n Ws, . If on the other hand S = Supp(a) is compact, we have

248

XIV

INTEGRATION IN LOCALLY COMPACT GROUPS

j ( f ( s t ) - f(so 0 )Wt)= (f(st) - f ( s 0 0) d 4 0 , 1s

and if r E S and s E V,, then st E V, S , which is compact. The restriction f l Vo S is uniformly continuous and therefore we may take W so that the relation s E Ws, implies If(sf)- f ( s o f ) l S E / Ial(S) for all t E S ; as before, we take V = V, n Ws,. Now let g be any function belonging to X,(G). Then the function (s, t ) H f ( s ) g ( t s ) is continuous on G x G and has compact support. By (13.21.7) and the left-invariance of p, we have

and since p ( f ) # 0 by hypothesis, it follows that vtg) = AD,

- s).

This shows first of all that D, does not depend onA for iff’ is another function in X,-(G) such that p ( f ’ ) # 0, then it follows from above that D, p = D,. p, and hence (13.15.3) that D, and D,. are equal almost everywhere with respect to p. But because p # 0 and is invariant, its support is the whole of G; also we have seen that D, and D,, are continuous on G; hence the set of s E G at which D,(s) # Df@) is open and negligible, therefore empty. In other words, D, = Df, = D, say. We have therefore, by the definition of the function D,

-

PL(f)W) = $ ( f ) for every f~ X,(G) such that p ( f ) # 0. This formula, being true in the complement of a hyperplane in Xc(G),is true in the whole of Xc(G),since both sides are linear forms in J Since v # 0 we have D(e) # 0; therefore p and t are proportional, and the proof of (14.1.5) is complete. Q.E.D. Any left (resp. right) Invariant positive nonzero measure on G is called a feft (resp. right) Haar measure on G . From (14.1.5), any two left (resp. right) Haar measures on G are proportional.

2 PARTICULAR CASES AND EXAMPLES

249

PROBLEMS

Let G be a locally compact group, A a dense subset of G, p a left Haar measure on G, and H a p-measurable subset of G with the following property: for each s E A, the sets s H n CH and H n CsH are p-negligible. Show that either H or its complement is negligible (prove that the measure P)" . p is left-invariant). Let G be a locally compact group, p a left Haar measure on G, and A, B subsets of G. (a) Suppose that one of the following two conditions is satisfied: (a)A is p-integrable; (p) p*(A) < co,and B is p-measurable. Show that in either case the function f ( s ) = p*(sA n B) is uniformly continuous on G with respect to a right-invariant distance on G. (For any two subsets M, N of G, put

+

p(M, N) = p*((M n CN) u (N n CM)).

Consider first the case where A is compact, and show that for each E > 0 there exists a neighborhood U of e in G such that p(sA n B, stA n B) 5 E for all s E G and all t E U. Then apply Problem 5 of Section 13.9. If B is p-measurable and p*(A) < co, observe that there exists a decreasing sequence (A,) of p-integrable subsets of G containing A and such that inf(p(A,)) = p(A), and show that

+

p*(sA n B) = inf(p*(sA. n B)) (Section 13.9, Problem 2(a)). Use the fact that p(A. - Am+')tends to 0 with I/a) (b) If A is p-integrable and p*(B) < co, then the function f is also uniformly continuous with respect to a left-invariant distance on G. If also A-' is p-integrable, then

+

lo

f(s) ds = p(A-')p*(B). (Reduce to the case where B is pintegrable, and

observe that in that case p(sA n B) = p(A n s-'B), and that V,A B = vSApnand VsA(t) = P E A - 1(s).) (c) Deduce from (a) that in the two cases considered there, the interiors of AB and BA are nonempty if A and B are not p-negligible. (d) In the group G = SL2(R), give an example of a compact set A and a p-measurable set B such that the functionf(s) = p(sA n B) is not uniformly continuous with respect to a left-invariant measure on G. (Observe that there exists a sequence (t.) of elements of G tending to e and a sequence (s.) of elements of G such that the sequence s; It. s. tends to the point at infinity.) Let G be a locally compact group, p a left Haar measure on G, and A an integrable subset of G such that p(A) > 0. Show that the set H(A) of elements s E G such that p(A) = p(A n sA) is a compact group. (Use Problem 2 to show that H(A) is closed in G. To show that H(A) is compact, consider a compact subset B of A such that p(B) > &(A), and prove that H(A) c BB-I.) Let G be a commutative locally compact group, written additively. Let p be a Haar measure on G and let A, B be two integrable subsets of G. (a) For each s E G let B = T,(A, B) = (A - s)nB. A' = u,(A, B) = A u (B s),

+ +

Show that p(A) + p(B') = p(A) p(B) and that A A + 4 = 4 for all subsets A of G.)

+ B'

C

A

+ B.

(Note that

250

XIV

INTEGRATION IN LOCALLY COMPACT GROUPS

(b) Suppose that 0 E A n B. A pair (A', B') of integrable subsets of G is said to be derived from (A, B) if there exists a sequence ( s ~ ) , $ x Qof~ elements of G and two Seand ( B k ) O Q k of Q nsubsets of G such that quences (Ak)oQkan Ao=A,

Bo-B,

Ax- Osk(Ak-1,%-1),

Bk-Tsk(Ak-1,Bk-l)

for 1 5 k 5 17, and sk E A x - ] for 1 5 k 5 n , and A' =~A,, B' = B,. Show that there exists a sequence (En, F.) of pairs of subsets of G such that (i) Eo = A and Fo B, (ii) ( E n + , ,F n + l )is derived from ( E n ,F,J, ( i i i ) p((E, - s) n F,) 2 p(F"+])- 2-" for En,F, = F, . Show that for each s E E, we have all n and all s E En. Let E, 7

u

n

p((E* - s) n Fa,)= p(F&). (c) Suppose that p(F,) =. 0. Show that the function

f ( s ) = p((E,.

- .P)

nF d

takes only the values 0 and p(Fn,). Let C be the set of elements s E G such that f ( s ) = p(F,). Show that C is open and closed, that p ( C ) == p(Em,)and that C is the closure of E m (use Problems 2(a) and 2(b)). Let D be the set of all s E F, such that the intersection of F, with every neighborhood of s has measure > O . Show that p(D) = p(Fm)and that E, + D c C. Deduce that D is contained in the subgroup H(C) defined in Problem 3 , and that H(C) is a compact open subgroup of G . Show also that C H(C) = C, that p(C) > p ( A ) p(B) p(H(C)) and that C C A 4 B (consider the measure of E, n (c - F,) for each c E C). (d) Let A, B be two integrable subsets of G . Deduce from ( c ) that either p+{A B)? p(A) p(B), or there exists a compact open subgroup H in G such that A -t- B contains a coset of H, and that in this case p*(A ~1 B) 2 p(A) -t p(B) - p(H). Consider the case when G is connected.

+

+~

5.

+

~

+

(a) Let A (resp. B) be the set of real numbers x - xo

+ ,xx , 2 - ' ou

,=I

where xo is an

integer, each xi ( i 2 1) is 0 or I , and x i = 0 for all even i > 0 (resp. x i = 0 for all odd i > 0). Show that each of A, B has zero Lebesgue measure but that A B = R. (b) Deduce from (a) that there exists a Hamel basis H of R (over Q) contained i n A u B and theiefore of measure zero. The set PI of numbers of the form rh, where r E Q and h E H, is also of measure 0. (c) Let P, denote the set of real numbers which have at niost 17 nonzero coordinate5 relative to the basis H . Show that if P. is negligible and P.,, Lebesgue-measurable, in then P,. I is negligible. (Let h, E H , and show that the set S of numbers x E P., which the coefficient of h , is nonzero, is negligible. Using Problem 2(c), show that if Pn+,were not negligible there would exist two points x', x'' in P.+l n I S such that (x' - - x")//iOwere rational. and hence obtain a contradiction.) (d) Deduce from (b) and (c) that there exist two negligible sets C, D in R such that C .I- D is not measurable (with respect to Lebesgue nieasure).

+

6.

Let G be a group acting (on the left) on a set X. A subset P (resp. C ) of X is said to be a G-parking (resp. a G-rouering) if for each s # e in G we have s . P n P = 0 (resp. if X - s . C). A subset P which is both a G-packing and a G-covering is called a

u

S C G

G -tessellation. (a) Suppose that X is separable, metrizable and locally compact, that G is at niost denumerable and acts continuously on X (with respect to the discrete topology on G)

PARTICULAR CASES AND EXAMPLES

2

251

and that there exists a positive nonzero G-invariant measure p on X. Let P (resp. C) be a G-packing (resp. a G-covering) such that P and C are p-integrable. Show that p(C) 2 p(P). (Remark that p(C) 2 1 p(C n s . P) = p(s-l . C n P).)

1

S C G

S E G

(b) Suppose that there exists a G-invariant distance d on X defining the topology of X. Let A(G) denote the greatest lower bound of the numbers p(C), where C runs through all integrable G-coverings of X. Let r be a real number > O such that there exists a point a E X for which p(B(a; Y)) > A(G). Show that there exists s f e in G such that d(a, s . a ) < 2r. (c) Suppose that X is a locally compact group, p a left Haar measure on X, and G a denumerable subgroup acting on X by left translations. Show that if A is an integrable subset of X such that p(A) > A(G), then there exists s E G n AA-' such that s # e. (d) With the same hypotheses as in (a), let F be a p-integrable G-tessellation and let Go be a subgroup of finite index h in G . If sl,. . . , s h are a system of representatives of sL . F is a Go-tessellation. the right cosets of Go in G, show that Fo =

u

1S14h

(e) Same hypotheses as in (a). L e t f > 0 be a p-integrable function on X. Show that there exist two points a, b in X such that p(C)

Z f ( s ' 42 Jxf(x)dp(x)

and

S S G

(Observe that if g

2 0 is an integrable

there exists c E E such that

SE

p(P) C f ( s . b) 5 I x I ( x ) W x ) . S E G

function and E an integrable subset of X, then

g(x) +(x) 5 g(c)p(E), and

c' E

E such that

IEg(x)dp(d 2 dC')p(E).)

2. PARTICULAR CASES A N D EXAMPLES

(14.2.1) On the additive group R , Lebesgue measure (13.1.4) is a Haar measure (left and right, since R is commutative). This follows from the formula for change of variable (8.7.4) applied to the function ( ~ ( 5 = ) 5 + E, which gives J -' m " f ( f + ct) df dt for allfEXX,(R) and all ct E R.

=StI,f(t)

(14.2.2) Now consider the multiplicative group RT of real numbers >O. This is a locally compact commutative group ((13.18.4), (4.1.2), and (4.1.4)) For each function f~ XX,(R*,),there exists a compact interval [a, b] with 0 < a < b, containing the support o f f . Hence for each interval [c, d] in RT containing Supp(f), the integral S c d ( f ( f ) d t ) /isf defined and its value,

which we denote by / o + m ( , f ( f ) df)/f, is independent of the choice of interval

[c, d] containing Supp(f). We assert that f ~ / ~ + ~ ( dt)/t f ( l )is a Haar measure on R,*. From the formula for change of variable (8.7.4), it follows that, for each s > 0,

252

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INTEGRATION IN LOCALLY COMPACT GROUPS

which proves our assertion. (14.2.3) Let G be a locally compact group with neutral element e, and let p be a (left or right) Haar measure on G. Then G is discrete if and only i f p( { e } )> 0 ; and G is compact ifand only ifp*(G) < + co (i.e., if and only if p is a bounded measure (13.20)).

It is clear that if G is compact then p is bounded. If G is discrete, { e } is an open neighborhood of L-,and hence p ( { e } )> 0 because the support of p is the whole of G. Conversely, let V be a compact neighborhood of e. If p ( { e } )> 0, we have p ( { s } ) = p ( { e } ) for all s E G, because p is invariant; hence the number of points of V i s j n i t e and s p ( V ) / p ( { e } ) Since . G is Hausdorff it follows that G is discrete. Suppcse now that p is bounded and (say) left-invariant. Consider the set (5 of finite subsets {sl,s 2 , . . . , s,} in G such that s i V n sjV = Q whenever j i # j . We have np(V) = p(s,V u A,V u - * * u s,V)

p(G),

hence I I 5 p(G)/p(V). Hence (5 contains a subset {sl,. . . ,s,} having the largest possible number of elements. For each s E G it then follows that sV must meet at least one of the siV, that is to say, s E siVV-'. Hence G is the union of the sets siVV-', which are compact (12.10.4) and so G is compact. (14.2.4) Let G be a locally compact group, V an open subset of G, and p a nonzero positive measure on V, having the following property: if U is an open subset of V and if s E G i s such that sU c V , then the image of the measure pu, inducedbyp on U (13.d.8) under the homeomorphism XHSX (13.1.6). isthe measure psuinduced by p on sU.Then there exists a unique left Haar measure a on G which induces p on V.

For each S E G , let ps be the image of p under the homeomorphism x Hsx of V onto sV. The restriction of ps to V n sV is the image of ps- lv A under the restriction of XHSX to s-'V n V. By hypothesis, this image is pv s v . By translation it follows that for all s, t in G the measures ,usand p t have the same restriction to sV n tV. By virtue of (1 3.1.9), there exists a positive measure r on G which induces ps on sV for all s E G. Clearly a is leftinvariant, and is therefore the unique left Haar measure on G which induces p on V.

2 PARTICULAR CASES AND EXAMPLES

253

We shall use this local definition of a Haar measure in Chapter XIX to construct a left Haar measure on a Lie group. Here we note the following consequence of (14.2.4) : (14.2.5) Let G be a locally compact group, H a discrete normal subgroup of G , and n : G -+ G/H the canonical homomorphism. Also let V be an open neighborhood of the neutral element of G such that the restriction of n to V is a homeomorphism of V onto the neighborhood n(V) of the neutral element of G / H (12.11.2). Ldt A be u left Haar measure on G. I f p is the image under n I V of the restriction Av of A to V, then p is the restriction to n(V) of a left Haar measure on G/H.

For every open set in n(V) is of the form n(U), where U c V is open, and the relation n(s)n(U)c n(V) is equivalent to sU c V ; hence it follows immediately from the definitions that p satisfies the condition of (14.2.4).

Example (14.2.6) The mapping cp : t w e Z n i ris a strict morphism (12.12.7) of R onto the compact group U of complex numbers of absolute value 1, by virtue of (9.5.2) and (9.5.7). The kernel of cp is the discrete subgroup Z consisting of the integers, and U may therefore be canonically identified with the quotient group R/Z = T (also called the 1-dimensional torus or the additive group of real numbers modulo 1). Apply (14.2.5) to the case where V = 3-4, +[; bearing in mind that a Haar measure p on U must be diffuse (14.2.3) and that the complement of cp(V) in U consists of a single point, we see that a function f' on U is p-integrable if and only if the function t H f(e2"") is Lebesgue-

integrable on

1-4,

,::I

$[, and that we then have I f dp =

f(e*"") dt.

(14.2.7) Let G,, G, be two locally compact groups, and p1 (resp. p,) a left Haar measure on G, (resp. G,). Then pl @ p 2 is a left Haar measure on G, x G 2 .

For each function.fe ,X,(G, x G2) and each (s,, s,) jJ/(S1Xl9

=p

E

G, x G , we have

S Z X J 4 4 x 1 ) &Z(XZ)

P ' ( X J j/O'Xlj

S Z X J dPZ(X.2)

254

XIV

INTEGRATION I N LOCALLY COMPACT GROUPS

by virtue of (13.21.2). Hence the result.

I n particular, Lehesgue measure on R" (13.21.19) is a Haar measure on the additive group R".

PROBLEMS 1.

Let G be a locally compact group, p a left Haar measure on G, A a subset of G , and B a relatively compact p-integrable subset of G such that p(B) > 0. Show that, if p*(AB) 1, and let A be a symmetric convex set in R" with nonempty interior. Show that, for each r > 0 satisfying h ( A ) r " 2 Pp"', there exists a point x # 0 in r A with integer coefficients, such that ui(x) = 0 (mod p ) for 1 5 i 5 nt. (Apply Minkowski's theorem to the subgroup Go of Z" consisting of all z E 2 such that u i ( z ) 0 (mod p ) for I 5 i 5 m , and use Problem 6(d) of Section 14.1.) In particular, show that if cl, c2 are any two integers, there exist integers x , , x 2 , not and c,xl -tc 2 x z = O (mod p ) (Thue's both zero, such that lxllg d i , lx21

sdi

theorem),

(c) Let a, h be two integers. Use (b) to show that there exist integers x , , x2,x3,x4, not all zero, such that fixI -1- hx2 = x 3 (mod p ) ,

bxl

- ax2 =- x4 (mod p

)

and y=x:

I x: I-x: t - x :

IdZp.

+ +

Show that if p is prime one can find two integers N , 6 such that a* b2 1 -0 (mod p ) (assuming p is odd, observe that when z takes the h(p I I ) integer values

3 THE MODULUS FUNCTION ON A GROUP

255

0 , 1, 2, . . . , - l ) , the residues mod p of the numbers zz are all distinct). Deduce that in this case y = p . Hence, using the multiplicative property of norms of quaternions, deduce that every positive integer is the sum of at most four squares (Lagrange's theorem).

(a) Let h be Lebesgue measure on R. Letfbe a real-valued function 2 0 on R which is h-integrable, bounded and of compact support. Put y= supf(t). For each w E R, tER

let U,(w) denote the set of t E R such that f ( t ) 2 w , and put v f ( w )= h*(U,(w)). Show that, for all (Y > 1,

-+ m (b) Let g be another function satisfying the same conditions asf, and put 6 = sup g(t). fER

Let h be the function on RZ defined by h(u, u) = f ( r t ) i -g(u) if f ( u ) g ( u ) # 0, and h(u, Y) = 0 otherwise; also let k ( t ) = sup h(u, u), so that k is positive, h-integrable, "+"=I

bounded and compactly supported. Show that, for all

j-

+m m

k"(t) dt

2 ( y i-6)61

(+/-+I

CL

f " ( t ) df -I-

> 1,

-/ 6a I

+m

-m

g"(t) d t ) .

+

(Observe that if 0 < w 5 1 we have Uk(yw-t 6 w ) 3 U,(yw) U,(6w), and use (a) above and Problem 4(d) of Section 14.1). (c) Let h, be Lebesgue measure on R",and let A. €3 be two A,,-integrable subsets of R".Show that ((&(A

i- B))""

2 (hn(A))l'n-1- (AJB))''"

(Brunn-Minkowski irrequalify).(Reduce to the case where A and B are compact. Then use induction on ti, Problem 4(d) of Section 14.1, the theorem of Lebesgue-Fubini, the inequality established i n (b) above, and Holder's inequality.)

Let p be a prime number. The normalized Haar measure p on the compact group Z, of p-adic integers (Section 12.9, Problem 4) is such that the measure of any closed ball of radiusp-' is equal top-i. (Show that Z, is the union ofpXclosed balls of radius p - ' , no two of which intersect.)

3. THE MODULUS FUNCTION O N A GROUP; T H E MODULUS OF A N AUTOMORPHISM

Let G be a locally compact group, p a left Haar measure on G. For all s, t E G we have Y(f)(WP) = W(Y(t)Pc) = W P directly from the definitions (14.1.2), and therefore 6(s)p is also a left-invariant positive nieasure on G. Hence there is a unique real number AG(s) > 0 (also written A(s)) such that S(s)p = Ati(s)p (14.1.5), a n d this number is clearly, by

256

XIV

INTEGRATION I N LOCALLY COMPACT GROUPS

virtue of (14.1.5) again, independent of the choice of left Haar measure p. The mapping S H AG(s) is called the modulus function on G. Iff is any p-integrable function, the function XH f ( x s ) is therefore also p-integrable, and we have r

P

(14.3.1 . I )

In particular, if A is any p-integrable subset of G, then As is also p-integrable, and AS)

(1 4.3.1.2)

(14.3.2) The mapping s-AG(s) multiplicative group R*,.

= A(s)p(A).

is a continuous homomorphism of G into the

This. follows immediately from the formula (14.3.1.1) and the lemma (1 4.1.5.5).

The group G is said to be unimodular if AG(s) = 1 for all s E G. In this case there is no distinction to be made between left and right Haar measures, and we call them simply Haar measures on G. Ifthere exists in G a compact neighborhood V of'the neutral element e which is invariant under all inner automorphisms of G, then G is unimodular. This is the case, in particular, when G is compact, or discrete, or commutative. (1 4.3.3)

For each s E G we have p(V) = p(s-'Vs) = AG(s)p(V) by (14.3.1.2) and (14.1.2.4); hence the result, since p(V) # 0 (14.1.2). (14.3.4) I f p is any left Haar measure on G , then fi = A-' * p. Iff is any p-integrable function on G, then the function X H f (x-')A(x)- is p-integrable, and we have

I

~ f ( X - ' ) A ( x ) - ' d P ( x= ) f ( x >4-0).

(14.3.4.1)

For each s E G we have ti(s)(A-'

*

p) = (ti(~)A-') . ( & ( ~ ) p=) (A(s)-'A-')

*

(A(s)p)

= A-'

p,

which shows that A - ' * p is a right Haar measure on G. Since jl is also a right Haar measure, there exists a constant a > 0 such that fi = aA-' * p. It follows that p = a(A-' . p)" = aA * fi = a2p (13.14.5), whence a2 = 1 and therefore

3

THE MODULUS FUNCTION ON A GROUP

257

a = 1 (because a > 0). This proves the first assertion. The second then follows (13.14.3). We deduce that iff is a locally p-integrable function, then

(f.P)"

(1 4.3.4.2)

=

*

P.

In particular, if G is unimodular, then for each p-integrable function the functions y(s)f, 6(s)f and f are p-integrable, and we have

(14.3.5)

(14.3.5.1)

s

f ( s x ) dp(x) =

s

I

In particular, if A is any p-integrable set in G, then (14.3.5.2)

s

f ( x s > dp(X) = f ( x - ' ) d p ( x ) = f ( x ) d p ( ~ ) .

~(sA= ) AS)

= p(A-') = p(A)

for all s E G. When G is injnite and compact (resp. injinite and discrete) and therefore unimodular (14.3.3), the normalized Haar measure on G is the unique Haar measure p on G for which p(G) = 1 (resp. p ( ( e ) )= 1). (14.3.6) Now let u be an automorphism of the topological group G . It is clear that the image u-'(p) of a left Haar measure p on G (1 3.1.6) is another left Haar measure; hence (14.1.5) there exists a number a > 0, independent of the choice of p, such that u-'(p) = up. This number a is called the modulus of the automorphism u, and is denoted by mod,(#) or mod@). For every p-integrable function f we have therefore

and in particular, for any p-integrable set A, (14.3.6.2)

L44-9= (mod W A ) .

In particular, for each s E G, let isbe the inner automorphism X H S - ~ X S . Then we have i; = G(s)y(s), and therefore i; '(p) = w

p =W p ,

which proves that (14.3.7)

mod(&) = A($).

258

XIV

INTEGRATION I N LOCALLY COMPACT GROUPS

If G is either compact or discrete, then we have mod(u) = 1 for euery automorphism u of G. For it is clear that u(G) = G and u({e}) = {e}, and we may apply (14.3.6.2) with A = G and A = {e}, respectively. We deduce also from (14.3.6.2) that if u and ZI are two automorphisms of G, then

(14.3.9)

-

mod(u u) = mod(u) rnod(z1)

(1 4.3.8)

0

If u is any automorphism of the uector space R", then mod(u)

=

ldet uI.

Let U = ( a i j ) be the matrix of the automorphism u with respect to the canonical basis of R". Let E i j denote the n x n matrix which has the element in the ( i , j ) place equal to 1 and all other elements equal to 0. If i # j and A E R, Put Bij(A)= I,, + AEij.

Then we have the following lemma: (14.3.9.1) Every invertible n x n matrix U is a product of matrices of the form Bij(A)and a matrix of the form I,, + (a - l)En,,.

Consider invertible matrices of the form 1 0 0 1

X=

"'

* * '

0 0

t1,n-h r2,n-h

* * .

52"

...................................... " '

tn-h-1,"-h"'

1. If 0 g 1g 1, then S,(1) is the set of points . . , tn-,)E R"-' such that

(cl,.

51

10,..., t,-* 2 0,

51

+ + tn-l5 1 - 1 * - *

and is therefore the image of S,(O) under the homothety with ratio 1 - 1. Applying (14.3.6.2) and (14.3.9) to this homothety in R"-', we obtain pfl-1(S,(1)) =(1

- ll)"-lu,-l.

Apply (13.21.8) to the compact set S,, and we get

Since clearly a, = 1, we have a,, = I/n!. (14.3.11) Application to the calculation of integrals: 11. Closed ball.

With the same notation as in (143.10) we shall now calculate the measure

V, = p,(B,) of the Euclidean unit ball B., i.e., the set of all (q,. . ., x,,) E R"

3 THE MODULUS FUNCTION ON A GROUP

such that

n

Ex:

i=1

261

5 1. The same method that was used for the calculation of

PASn) gives

vn =

j1

lpn- l(Bn(A)) d2-

Now B,(A) is the set of all (xl, . . . , x,,-~)E R"-' such that

n- 1

i= 1

x? 6 1 - A*,

and hence is obtained from B,,-l by the homethety with ratio (1 and therefore

-A')'/',

p,,-l(B,,(A)) = (1 - 22)(n-1)/2V II- 1 ' Making the change of variable A V, = V,-1

(14.3.11.1)

= sin

COS"0 d0

0, we get = 2V,-,

Put c, = ~ ~ ' 2 c o s "do, 0 and apply the formula of integration by parts (8.7.5): then if n 2 2 we obtain

so that nc,

Since V,

= (n

=2

(14.3.11.2)

- l ) ~ , , -Since ~ . co = ) n and c1 = 1, we have finally

and V, = n we obtain from (14.3.11.1)

J

In terms of the gamma function,* these formulas can be written in the form (14.3.11.3)

+

V, = nn/2/r(+n 1).

* See for example my book, Calcul infinit6simal (Herniann, Paris, 1968).

262

XIV

INTEGRATION I N LOCALLY COMPACT GROUPS

PROBLEMS

1. Let G be a locally compact group containing a compact open subgroup H. For each automorphism u of G, show that mod@) is a rarimialnumber. (Observe that u(H) n H is a subgroup of finite index in both H and u(H).) Show that the set of elements s E G such that A&) = 1 is an open subgroup of G, containing H. 2.

Let G be a compact group, p a Haar measure on G, and u a (continuous) endomorphism of G such that u(G) is open in G and the kernel G, = u - ' ( e ) is afinite subgroup of G. (a) Show that there exists a real number h(u) > 0 and an open neighborhood U of e in G such that, for every open set V c U, the set u(V) is open in G and p(u(V))= h(u)p(V)(use (14.2.5)). (b) Show that h(u) = Card(G/u(G))/Card(G,). (Calculate p(u(G)) in two different ways, using (a) above and (14.4.2).)

3. Let Q* be the set of nonzero rational numbers, endowed with the discrete topology. On the locally compact space G = R x Q * a law of composition is defined by the formula ( x , r ) ( x ' , r ' ) = (rx' x , r r ' ) . Show that with respect to this law of composition G is a locally compact group which is locally isomorphic to the additive group R, but not unimodular.

+

4.

(a) Let G be a locally compact group and let x be a continuous homomorphism of G into the multiplicative group C*. Show that if a complex measure v on G is such that y(s)v = x(s)v for all s E G, then v = ax . p where p is a left Haar measure on G, and a is a complex constant. (b) A complex measure v on G is said to be left quasi-invariant if y(s)v is equivalent to v (1 3.1 5.6) for each s E G. Show that v is left quasi-invariant if and only if v is a measure equivalent (1 3.1 5.6) to a left Haar measure p on G. (Reduce to the case v 2 0; use the criterion (b') of (13.15.5), consider the double integral JJf(X)%(XY)

where A is compact, f E X(G), and f

444 W Y ) ,

2 0, and use the theorem of Lebesgue-Fubini.)

Let G be a locally compact group and let X, Y be two closed subgroups of G such that X n Y = { e } and such that the set R = XY (i.e., the set of all xy, where x E X and y E Y ) contains a neighborhood of e in (i.Show that R is open in G and that the mapping ( x , y ) ~ x y - ' of X x Y onto R is a homeomorphism (cf. (12.16.12), by considering X x Y as acting on R by the rule ( x , y ) . z = x z y - I ) . (b) Let pG , px , py be left Haar measures on G, X, Y , respectively, and let p be the restriction of pc to R. Show that, up to a constant factor, p is the image of px 0 p y ) under the homeomorphism ( x , y ) w x y - ' of X x Y onto R,where denotes the restriction of AG to Y . Deduce that a real-valued functionfdefined on 0 is p-integrable if and only if the function ( x , y)t-+f(xy)AG(y)Ay(y)-' is ( p x0py)integrable, and that we then have

5. (a)

(x-'

where a is a constant independent off:

x

3 THE MODULUS FUNCTION ON A GROUP

263

(c) Suppose that Y is a normal subgroup of G. Then the measure p is (up to a constant factor) the image of ,ux0pu under the homeomorphism (x, y) H xy of X x Y onto R, and for each x E X and y E Y we have Ac(xy) = A,(x) Av(y) mod(i,), where i, is the automorphisni O F + X - ' V X of Y (cf. (14.4.6)). (d) Consider the locally compact space G = R x R*,with the law of composition (x, y)(x', y ' ) = (yx' x , yy'). Show that this locally compact group is not unimodular (use W). (e) Let G be a locally compact group. On the locally compact space E = R x G, a law of composition is defined by the formula ((, x ) ( P , x') = (( iAc(x)P, xx'). Show that, with respect to this law of composition, E is a unimodulur locally compact group (use (b)). The group G (which is not necessarily unimodular) is isomorphic to a subgroup and to a quotient group of E.

+

6.

Letp be a prime number. On the group Z, (Section 12.9, Problem 4) we define a ring structure as follows: if z = (z,,) and z'= (z:), then zz'= ( z , z:) (the Z/p"Z being quotient rings of Z). The canonical injection of Z into Z, (loc. cit.) is a ring homomorphism, which identifies 2 with a dense subring of Z , . Show that Z, is an integral domain. Its field of fractions is called the field of p-udic nunhers and is denoted by Q,. The fundamental system of neighborhoods of 0 in Z, formed by the balls with center 0 is also a fundamental system of neighborhoods of 0 in Q, for a topology compatible with the additive group structure of Q,. With respect to this topology, Q, is separable, metrizable and locally compact (Section 12.8, Problem 1). The field Q of rational numbers is dense in Q,, and the p-adic distance on Q (3.2.6) extends to a distance d on Q, defining the topology of Q,. For z E Q, we write 1 zI, = d(0, z). For each S E Q , , show that the modulus of the homothety Z-SZ of Q, is ( s / ~ . Deduce that, for each automorphism u of the vector space Q; over Q,, we have mod(u) = ldet ul,.

7. Let A. denote Lebesgue measure in the space R" endowed with the Euclidean scalar product (x I y ) = 2 f i ? l j . For every universally measurable bounded set A, let a,(A) .i= 1

be the set defined as follows. Identify R" with the product RP x R"-",and for each point x ' ~ p r , ( A )let Bn-,(x') be the closed Euclidean ball in R"-, with center 0 such that the measure An-p(Bn-,(x'))is equal to the measure An-,(A(x')) of the section of A at x'. Then a,(A) is defined to be the union of the sets {x') x Bn-,(x') as x' runs through prl(A). Hence prl(a,(A)) = pr,(A). (a) Show that if A is compact, then so is a,(A). (Use Problem 16(b) of Section 13.21.) Deduce that if A is universally measurable, then o,(A) is &measurable and that X,(a,(A)) = X.(A). (b) Show that if A and B are universally measurable bounded sets in R", then h.(o,(A) n o,(B))

= h,LA

n B)

+ A.(a,(A

n CB) n up(Bn CAN.

(c) Show that the mapping A wa,,(A) is not continuous with respect to the topology on the set of nonempty bounded closed subsets of R" defined by the distance of Section 3.16, Problem 3. (Observe that in this topology every compact set can be approximated arbitrarily closely by a finite set, which i s of measure zero.) (d) If A , B are compact subsets of R", show that

264

XIV INTEGRATION IN LOCALLY COMPACT GROUPS (cf. Section 14.2, Problem 4(d)). In particular, we have un-~(Vr(A))3 VJun-l(A)) for all r > 0, in the notation of Section (3.6). (Observe that V,(A)

8.

=A

+ B(0; r).)

Let H c R" be a hyperplane passing through the origin 0, and let T be a rotation transforming H into R"-'. For each universally measurable bounded subset A of R", put uH(A) = T - ' . I J " - ~T( A) (the "Steiner symmetrization" of A with respect to H). (a) Suppose that A is closed and contained in a closed ball B with center 0. Show that, if W is an open subset of the sphere S which is the boundary of B and if W n A = then uH(A) does not intersect W nor the image of W under the symmetry with respect to H. (b) Deduce from (a) that, under the same hypotheses, there exist finitely many hyperplanes H I , . . ., H, passing through 0, such that the set uHruHr- . . . uH~(A) is contained in B and does not meet S unless A = B (use the compactness of S). (c) Suppose that A is compact. If Bo is the closed ball with center 0 such that &(A) = X.(Bo), show that there exists a sequence (HnJ of hyperplanes passing through 0, such that the sequence of compact sets A,,, = uHmI J ~ , , , - ~ ... uHI(A) tends to Bo in the topology defined in Section 3.16, Problem 3. (Use the result of this problem, by showing first that the sequence (A,,,) can be assumed to have a limit A' such that A' C B(0; R), where R is the greatest lower bound of the radii of closed balls with center 0 containing a transform of A under the composition of a finite number of Steiner symmetrizations uH with respect to hyperplanes H passing through 0. Then argue by contradiction and use (b) above to show that B'(0; R) = Bo.)

a,

9.

(a) Let A be a nonempty compact subset of the space R" endowed with the Euclidean scalar product. For each unit vector u E R",put h(A; u) = sup+ I u), and b(A; u) =

+

YEA

h(A; u) h(A; - u ) (the width of A in the direction u). The least upper bound of the numbers b(A; u ) as u varies on the unit sphere S n - , is the diameter &A) of A (Section 6.3, problem 2). For any pair of compact sets A, B and any real number tc > 0 we have h(A B ; u ) = h(A; u ) h(B; u ) and h(aA; u ) = ah(A; u) , from which it follows that &A B) 5 &A) 6(B). (b) Let s be a finite sequence ((a,, U,))l j d , , 2 of pairs in which the aJare real numbers 2 0 such t h a t x a, = I , and the U j are rotations about 0 (i.e., elements of SO(n, R)).

+

+ +

+

I

The rotnfionol memi of A corresponding to the finite sequence s is defined to be the compact set p,(A) tlJ U,(A). We have h,(p,(A)) 2 AJA) (Section 14.1, Problem

=c J

4(d)). For each u E Snv1, show that h(pdA);

11)

=x

CLJ

h(A; U y '(u)).

(c) Let A be a nonempty compact set contained in a closed ball B with center 0, and let S be the frontier of B (a sphere). Show that if there exists a nonempty open subset W of S which does not meet A, then there exists a rotational mean p,(A) such that p,(A) c B and S n p,(A) = 0 (same method as Problem 7; use the compactness of S). (d) Let R be the greatest lower bound of radii of closed balls with center Ocontaining a transform of A under the composition of a finite number of rotational means. If Bo = B(0; R), show that there exists a sequence of sets A, = p.,p.,. . . p,,(A) tending to Bo , relative to the distance defined in Section 3.16, Problem 3. (Argue by

3 THE MODULUS FUNCTION ON A GROUP

265

contradiction, by first showing that the sequence (A,) may be assumed to have a limit A' c Bo , and then using (c) above to prove that A' contains the frontier So of Bo; finally remark that $3, &So = Bo .) (e) If A is any compact subset of R",show that

+

where the diameter &A) is relative to the Euclidean distance, and V. is the Lebesgue measure of the ball B, ("Bieberbach's inequality"). (Use (d) and the inequality &(A)) 5 10. With the notation of Problem 7, for any compact subset A of R" the upper (resp. lower) Minkowski area of A is defined to be the number

(resp.

.+(A)

= lim

sup (X.(V,(A)) - h.(A))/r

m-(A)

= lim

inf (X.(V,(A))

,-0

,-0

- h.(A))/r)

where r tends to 0 through positive values. (a) Give an example of a compact set A such that a+(A)= 1 and a-(A) = 0. (Take n = 2, and consider a union of finite sets of the form ck A*, where the sequence (ck) tends to 0,and for each k the set Ak is a product 1, x J k . where I k , Jk are finite (qbk)O 0, show that the set I?, of measures a

compact (prove that sup a ( K ) < =Err

E

I

I' such that f(s) da(s) = 1 is vaguely

+ m for every compact subset K of G). Deduce that

the mapping a + + ( a ( f ) ,a / a ( f ) )is a homeomorphism of I? onto the product space

RI, X F f .

(c) For each a E r, let H. = Supp(a): it is a closed subgroup of G. Let d be a rightinvariant distance on G such that d(x, y ) 2 1 for all x, y in G (12.9.1). Let (K,) be an increasing sequence of compact sets, covering G and such that K, is contained in the interior of K,+ I for each n (3.18.3). If h is the Hausdorff distance on G corresponding to d (Section 3.16, Problem 3), then for any two nonempty closed sets M, N in G we define h.(M, N ) = 1 if either M n K. or N n K. is empty, and h,(M, N ) = h(M n K,, N n K.) otherwise. Endow the set S(G) of nonempty closed subsets of G with the topology defined by the pseudo-distances h,. Show that the mapping a w H . is a homeomorphism of I71 (endowed with the vague topology) onto the set C of closed subgroups of G , endowed with the topology induced by that of %(G). 6. With the notation of Problem 5 , consider the subset I'O of I? consisting of measures a E I' such that H. is unimodulur. This is also the set of measures a E I' such that a ( f ) = a ( f ) for all f~ X ( G ) ,and is therefore vaguely closed in I?. For each a E Po, put Q, = G/H.; then there exists a relatively G-invariant measure p. on Q. such that

IGf(~)

= a

dp&)

1

H .

f ( x s )d a b )

for everyfE X ( G ) (where R is the coset xH.) (Problem 2). (a) If a E ro and f~ -f(G), put f.(x)

Ilf,ll

=

5

H.

f ( x s ) da(s). Show that the mapping

is vaguely continuous, and deduce that the mapping a++ ljp.jl is lower semicontinuous with respect to the vague topology. (b) Let g 2 0 be a real-valued p-integrable function, and let ro(g) be the set of a-

measures a E rosuch that a+-+IIp.ll of

I*

g(xs) da(s) 2 1 for all x

ro(g)into R is vaguely continuous.

E

G . Show that the mapping

(It is enough to show that the map-

ping is upper semicontinuous. Let h E Y + ( G )be such that

I

1g(x) - h(x)l dp(x) 5 E ,

and let K = Supp(h). If r D: G + Q a is the canonical mapping, show that pu(Qu-- r u ( K ) )5 E for all fl E I'O(g). On the other hand, consider a functionfEX+(G) such that j G f ( x s )da(s) 5 1 for all s E G and lem 2(a)). If U, is the set of measures all x

E

SG

f ( x s ) da(s) = 1 for all s E K (use Prob-

fl E I ' O ( g )

K, show that (1 - E ) ~ B ( T # ( K 5)l\pall.) )

such that

f ( x s d/?(s)> 1 - E for

5 CONVOLUTION OF MEASURES ON A LOCALLY COMPACT GROUP

271

(c) Let r: be the set of measures O L EI'O such that G/H. is compact. Show that r"(g)c for all g E Z + ( G ) . Let a E,:'I and let g E Z + ( G ) be such that / g ( x s ) da(s) = 2 for

all x

E

G (Problem 2(a)). For each compact L c G , the set W

of measures @ E r0 such that

ro

s

g(xs) d@(s)2 1

for all x

E

L is a neighborhood of a

in with respect to the vague topology. Show that if G is generated by a compact neighborhood U of e , and if we take L = U K above, then W c ro(g). Deduce that the restriction to roof the mapping a- lIpmllis vaguely continuous in this case. (d) Let rdC robe the set of measures a such that H, is discrete, and let N be the subset of rd consisting of measures a such that a ( { e } )= 1. For each relatively compact open neighborhood U of e in G, let N u be the set of all a E N such that H, n U = { e } . Show that NU is compact (observe that the relation a E N u is equivalent to a({e})2 1 and a(U) 5 1). As U runs through the set of relatively compact open neighborhoods of e in G, the interiors of the sets N u cover N. A subset M of N is relatively compact in N if and only if there exists a relatively compact open neighborhood U of e in G such that M c N u .

7. With the notation of Problems 5 and 6 , suppose that there exists a neighborhood of e in G which contains no finite subgroup of G other than { e } . Show that the mapping Q H a({e})of r d into R*,is vaguely continuous. (Show that there exists a neighborhood V of e in G and a neighborhood W of a in rd such that the relation E W implies (Vz- V) n Ho= 0 :argue by contradiction.)

8. With the notation of Problems 5 and 6 , suppose that G is cornrnirtatiue and generated by a compact neighborhood of e . Let N, denote the subset of N consisting of measures a such that Q. = G/H, is compact, so that N, = N n I':; the set N, is open in N (Problem 6(c)). A subset A of N, is relatively compact in N, if and only if it satisfies the following two conditions: (1) there exists an open neighborhood U of e in G such that H, n U = { e } for all a E A ; (2) there exists a constant k such that pL,(G/H,)5 k for all a E A. (Use Problem 6 and the fact that p a is a Haar measure on G/H. .)

9. Translate the results of Problems 6 to 8 into statements about subspaces of the space of closed subgroups of G (Problem 5(c)): in particular, the subspace of unimodular of subgroups H E cosuch that G / H is compact, closed subgroups, the subspace In particular, the subspace D of discrete subgroups, and the subspace D, = D n obtain Mahler's criterion: if for each discrete subgroup H E D, we denote by u(H) the total mass of G/H relative to the measure p a corresponding to the Haar measure a on H such that @({el)= 1 , then a subset A of D, is relatively compact in D, if and only if (1) there exists a neighborhood U of e in G such that H n U = { e } for all H E A; and (2) there exists a constant k such that u(H) 5 k for all H E A. Consider the case G = R".

c

xo

x,"

x:.

5. CONVOLUTION O F MEASURES O N A LOCALLY COMPACT GROUP

Let pl,. . . , p,,be a finite number of (complex) measures on a locally compact group G. The sequence (pl,. . . , p,,) is said to be conuolvabfe if, for each functionfE X,-(G), the function (XI,

..

' 9

X,,) H f ( X I X 2

. . . x,>

272

XIV

INTEGRATION IN LOCALLY COMPACT GROUPS

--

is integrable with respect to the product measure p1 0 p z 0 . 0 p, on G". It follows immediately from this definition and from (13.21.1 7) that the sequence (pi)l5 is convolvable if and only if the sequence ( Ipil)lsis,, of absolute values of these measures is convolvable. Moreover, it is clear ?hat

J*

fH

Sf(x1x2

* * *

xn) d 1p1I

.

* *

d I P ~(xn> I

is then a positive linear form on XR(G) and therefore a positive measure on G (13.3.1); and for all f E XR(G) we have

by virtue of (13.16.5) and (13.21.17). It follows directly (13.1.1) that

f~

j--

--

/ f ( x l x 2 x,) dpl(xl) * dp,,(x,) is also a (complex) measure on G. This measure is denoted by pl * p 2 * * * * p, and is called the convolution product or convolution of the sequence (pJljiSn. The formula (14.5.1) also shows that

-

For each functionfe X,-(G),the function ( x l , . . . , x,) w f ( x 1 x 2* . x,,) is continuous and therefore measurable with respect to every measure on G"; hence, by virtue of (13.21.lo), the sequence ( p l , . . . , p") is convolvable if and only if 4

for some permutation 0 of (1, 2, . . . , n } . An equivalent condition is that, for each compact subset K of G, the (closed) set A c G"of points (xl,x2, . . . , x,,) such that x l x 2 * * x,, E K is (pl€3 p 2 €3 * . * €3 p,,)-integrable. If a sequence (p,v ) of two measures is convolvable, we say that p and v (in this order) are convolvable, or that p is convolvable on the left with v , or that v is convolvable on the right with p. If p and v are convolvable and p', v' are two measures such that lp'l 5 lpl and lv'l 5 IvI, then it is clear from (14.5.3) that p' and v' are convolvable.

PARTICULAR CASES OF CONVOLUTION OF MEASURES

6

(14.5.4)

273

Zfthe sequence ( p i ) l 6 i 6 nis convolvable, and i f S i = Supp(pi) (13.19), S, , and the two sides are equal if all then Supp(pl * p 2 * * * p,) t S,S2 the measures p j are positive.

- -

Let x be a point not in the'closure of S1S2 *.. S , , and let V be an open neighborhood of x not meeting SlS2 * S, . Iff is any continuous function whose support is contained in V, we can write (1 3.21. I 8)

-

But if xi E Sjfor 1 5 i 5 n, we have f(xlx2 x,) = 0 by hypothesis, and the first assertion of (14.5.4) is proved. If all the p i are positive, then so is p =p,

* p2 *

*-.

* p,.

If U is a p-negligible open set, K a compact subset of U, andfE XX,(G)a function with values in [0, 11 which is equal to 1 on K and to 0 on 6U (4.5.2), then we have

J. *

*

Jf(xlx2

* * *

xn)

djLI(x1) . . * dpn(xn) = 0;

therefore the open set of points (x,, . , . , x,) such that f(xlx2 * * x,) > $ is (p, 0 p 2 0 * * 0 p,J-negligible, and consequently does not intersect the support S, x S2 x ... x S, of this measure (13.21.18), Since the mapping (xl, .. . , X , ) H X ~ X ~ .. 'x, is continuous, we conclude that K does not intersect SIS, * * S, , and therefore that S,S2 . . S, is contained in Supp(p). This completes the proof.

-

-

6. EXAMPLES A N D PARTICULAR CASES OF C O N V O L U T I O N O F MEASURES

(14.6.1) A Dirac measure E, (13.1.3) on G is convolvable (on either side) with every measure p on G, and we have

(1 4.6.1.2)

E,

* E , = E,, .

274

XIV

INTEGRATION IN LOCALLY COMPACT GROUPS

For example, let us verify the first of the formulas (14.6.1.1). We have to show that, for every functionfe Xc(G),the continuous function ( x , y )f~( x y ) is (6, 6p)-integrable. Now the function x ~ f ( x y )is &,-integrable, and

1f ( x y )dc,(x)

(sy). Since the function y~ f (sy) is continuous and has compact support, it is p-integrable. Our assertion therefore follows from (13.21 .lo), and then the theorem of Lebesgue-Fubini (13.21.7) gives f ( x y ) d.cS(x)dp(y) = f (sy) dp(y). The formula to be proved follows now from (14.1.1) and (14.1.2). =f

//

(14.6.2) Every jinite family (pl, . . , , p,,) of bounded measures on G is convolvable. The measure pl * p2 * * . . * p,, is bounded, and

To prove the first assertion, observe that the measure pl 0 * * * 0 p,, on G" is bounded (13.21.18). For every function f E X c ( G ) ,the function ( x l , . . . ,x,,)w f ( x I x 2* * x,,)is continuous and bounded on G", and therefore 0 p,)-integrable (13.20.4). To prove (14.6.2.1), it is enough to (pl 0 remark that iff E X c ( G )and IIf II 5 1, then

by virtue of (13.21.18). (14.6.3) A left Haar measure ,I on G is c o n ~ ~ o l on ~ ~the ~ ~right l e with any bounded measure p on G , and p * 1 = p( 1)i.

We may restrict ourselves to the case where p 2 0 (cf. (14.7.1.2)). For each function f E X + ( G )we have

and therefore, by virtue of (13.21.9), the function ( x ,y ) Hf ( x y ) is ( p 0 1)integrable and its integral is equal to 1(f) ( I p ( ( . The same calculation shows that I is not convolvable with itselfif G is not compact, for then the function 1 is not A-integrable (14.2.3). (14.6.4) Let ( p l , . . . , p,,) be a jinite sequence of measures on G , all of which except possibly for one have compact support. Then the sequence (pl, . . . , p,,) is convolvable.

7 ALGEBRAIC PROPERTIES OF CONVOLUTION

275

Let S i = Supp(pi), and suppose that S i is compact except possibly for one index j . Letff Xx,(G)and let K be the compact support off. It is enough to show that the set of points ( x i , . . .,x,,) E G" which belong to the support

n n

i= 1

S i of p 1 0 * *

6 pn (1 3.21.18) and are such that x l x z * * . x,

E

K, is relatively

compact in G". Now, the conditions x i E S i for all i, and x l x l imply that xi E S,7-', * * * S;'KS,' . . . ST' J+1

x,, E K,

7

and this set is compact (12.10.5). Since by hypothesis S iis compact whenever i # j , our assertion is proved (3.20.16). The result of (14.6.2) or (14.6.3) shows that two measures can be convolvable without either of them having a compact support. Later (14.10.7) we shall see examples of unbounded measures which are convolvable.

7. ALGEBRAIC PROPERTIES OF C O N V O L U T I O N

(14.7.1) Let A, p, v be three measures on G, and suppose that the pairs (A, p) and (A, v) are convolvable. Then so is the pair (A, p + v) and we have

A * (p + v) = A * p

+ A * v. For by virtue of the relation Ip + vJ S Ipl + IvI we may restrict ourselves

(14.7.1.1)

to the case where A, p and v are positive, and in this case the result follows immediately from (13.1 6.1). Similarly, if (A, v) and (p,v) are convolvable, then so is (A p, v) and we have

+

(14.7.1.2)

(A + p) * v = A * v + p * v.

Also it is clear that if the pair (A, p) is convolvable, then so is (an, bp) for all scalars a and b, and

(14.7.1.3)

(an) * ( b p ) = (ab)A* p.

(14.7.2) Let A, p, v be three measures #O on G. (i) If the sequence (A, p, v) is convolvable, then so are the sequences (A,p), ( 14 * Id, v), ( P , 4,(A 1p1 * Ivl), and we have (14.7.2.1)

A * p * v = (A * p) * v = A * ( p * v).

276

XIV

INTEGRATION IN LOCALLY COMPACT GROUPS

(ii) Zf the sequences (A,p) and (111 * lpl, v) are convoluabfe, then so is Likewise $the sequences (p, v) and (A, lpl * Ivl) are conuolvable.

(A,p, v).

We may restrict ourselves to the case in which A, p, and v are positive. Suppose that the sequence (A, p, v) is convolvable; then for every compact subset K of G the set of triples (x,y, z) such that xyz E K is (A 0 p 0 v)integrable. Let A be the set of pairs (x,y) such that xy E K. For each compact subset K‘ of G, the set A x K’ c G 3 is contained in the set of triples ( x , y, z ) such that xyz E KK’, and since KK’ is compact, it follows that A x K‘ is ((A 0 p) 0 v)-integrable. Since v # 0, this implies that A is (A 0 p)-integrable (13.21.11) and hence that (A, p) is convolvable. Consequently, for any f E X+(G),it follows from the hypothesis and the Lebesgue-Fubini theorem that

[hw p c t 44

A*

=

s’

Mz)

ss

= JJJf(XYZ)

f(xyz)

W) MY)

W) M Y )Wz)

since the function I ~ f ( t z is) in X + ( G ) for , each x E G. This shows that A * p and v are convolvable. One proves in the same way that (p,v) and (A, p * v) are convolvable. The formula (14.7.2.1) is then a consequence of the LebesgueFubini theorem. Conversely, suppose that (A, p) and (A * p, v) are convolvable, and let f be a function belonging to X + ( G ) .For each z E G, the function t H f ( t z ) belongs to X+(G),hence is (A * p)-integrable, and we have

J l ( t 4 41 * A(f)=

ss

f(xyz)

W) MY).

Hence, by Lebesgue-Fubini, it follows that

which proves that (A, p, v) is convolvable. One can give examples of measures A, p, v on G such that the pairs (A, p), ( A * p, v), (p,v) and (A, p * v) are convolvable but (A * p) * v # A* ( p * v) (Problem 1).

7 ALGEBRAIC PROPERTIES OF CONVOLUTION

277

(14.7.3) I f the sequence (pl, . . . , p,) is convohable, then so is the sequence

(fi,, . . . , PI), and we have (14.7.3.1)

(pl*pLZ*-..*pn)"=fin*ii,-l *...*fil.

For (14.1.4 and 13.7.10) we have

if and only if

and these two integrals are equal. On the other hand, if the sequence ( A , p ) is convolvable, it does not necessarily follow that ( p , A) is convolvable (Problem 2). But if G is commutative this will be the case, and we shall have A * p = p * A. In particular, it follows from the preceding results that (14.7.4) On the set MS(G) of bounded measures on G, the law of composition ( A , p ) H1 * p (together with the vector space structure) defines a C-algebra structure; the unit element is the Dirac measure E, at the neutral element e ojG. The set Mk(G) of compactly supported measures on G is a subalgebra of M,!.(G). The algebra Mh(G) is commutative ifand only ifG is comniutative.

The fact that G is commutative if Mh(G) is commutative follows from the formula (14.6.1.2). If G is discrete, the algebra M$(G) consists of all linear combinations

1U S E S , where a, = 0 for all but a finite number of points s E G (3.16.3), and

scG

the formula (14.6.1.2) shows that

This is what is called in algebra the group algebra of the group G over the field C .

278

XIV

INTEGRATION I N LOCALLY COMPACT GROUPS

PROBLEMS 1. On the additive group R, let h be Lebesgue measure, let p = pI . h, where 1 is the interval

+

and let n < b be two distinct points of R. Show that the convolution product * p) * h is defined, but that p and are not convolvable. Show that the convolutions p * ( ( F . - & b ) * h ) and ( p * (E. - E ~ ) )* h a r e both defined and are unequal. [0,

001,

((E. - Eb)

2. Let G be a locally compact group which is not unimodular. (a) Show that there exists a bounded positive measure p on G such that AG . p is not bounded (take p to be discrete). (b) Let h be a left Haar measure on G. By (14.6.3), p and h are convolvable. Show that h and p are not convolvable.

3. Let G be a locally compact group. (a) Let p, v be two positive measures on G . If p * v = E , , show that p = a & , and v = u-'E,- 1 for some x E G and a # 0 (cf. 14.5.4). (b) Give an example of a positive measure on the group 2 / 2 2 whose support is the whole group and which has an inverse (with respect to convolution product). 4.

(a) In the set M:(R), consider the two sequences of boundzd measures p n = F , and 2 c - " , both of which tend to 0 in the topology F2defined in Section 13.20, Problem I . Show that the sequence of measures pn * v. does not tend to 0 with respect to the topology 9,. (b) Let G be a locally compact group and let (p.), (v,) be two sequences of real bounded measures on G . Suppose that pn+ p in the topology Y2, and that v,, + v in the topology Y3 (notation of Section 13.20, Problem 1). Show that the sequence p n * v, tends to p * v i n the topology . T 2(Observe . that iff, g E .X(G), the function (x, y ) ~ y ( ~ , ) f ( x has . v ) compact support and can be uniformly approximated by a linear combination of functions u i@ u , , where uLand u, are i n Ty(G).) Give an example (with G = R) where p == v = 0 and pn * v. does not tend to 0 with respect t o the topology Y 3 .Show that if p n 0 with respect to Y 3and if the sequence of norms (]lv.ll) is bounded, then pn * v. + O with respect to F 2 .If pn+ p with respect to T3, and v. + v with respect to Y 3 show , that p n * v, + p * v with respect to F3(similar method). ,n + p and v,+v, (c) With the same notation, show that if, in the topology Y b p then p n * v, p * v (use Problem 2 of Section 13.20, and Egoroff's theorem). (d) Take G t o be the group RZ.Let a, 6 be the vectors of the canonical basis of G over R; let p. be the measure on I =- [O, 74 c R whose density with respect to Lebesgue measure is the function sin(2"x); let pn be the measure p. 0E~ on G, and let v n be the ~ nE~ on C . Show that pn+ 0 with respect to 9 6 and that v,,+ 0 with measure ~ b , respect to Y 3 ,but that the sequence (p. * v.) does not tend to 0 with respect to 9 6 .

"

-j

--f

5.

Let G be a compact group and p a positive measure on G such that Supp(p) = G , and p * p = p. Show that p(G) = I and then that p is a Haar measure on G. (Let

I

j'e K + ( G ) ,and put g(x) = f(yx) d p ( y ) , which is a continuous function on G. Show

J

that y(x) = g(yx) dp(y), and deduce that g is constant, by considering the set of points at which it attains its upper bound.)

8 CONVOLUTION OF A MEASURE AND A FUNCTION

279

(a) Let p be a measure on a locally compact group G. Show that p * v = v * p for all measures v such that both p * v and v * p are defined, if and only if p * E, = E, * p for all x E G . (b) Suppose that G is compact. If p is any measure on G, show that the measure pb defined by

(where

p is a Haar measure on G) satisfies pb * v = v * pb for every measure

v on G .

Let G be a compact group and p the Haar measure on G for which p(G) = 1 . Let p be a positive measure on G such that p(G) = 1 and p 2 cp, where 0 < c < 1. Show that lip*" - PI1 5 2"(1 - c)", where p*" denotes the convolution product of n measures equal to p (use (14.6.3)). Let r be a real number such that 0 < r 5 4. For each integer n > 1 , let P,,,~ denote the measure HE,. on R,and let pn.,denote p l , , * pz., * . . . * p.. ,. (a) Show that the sequence (pn.,)"? converges vaguely to a measure p on R with support contained in I = [ - I , I ] (prove that for every interval U c R the sequence (pn.W ) )converges). (b) Show that, if r < &, the measure p is disjoint from Lebesgue measure on R, but that p"lz is the measure induced on 1 by Lebesgue measure. t R. Show that (c) Let v l l J be the image of pI14under the homothety f ~ 2 on p1/4 * v , , = ~ p l I z ,although pI14and v1,4 are each disjoint from Lebesgue measure (use Problem 4(b)).

+

8. CONVOLUTION OF A MEASURE AND A FUNCTION

In the rest of this chapter we shall fix once and for all a left Haar measure P on the group G. I f f is any mapping of G into R or C, the norm N , ( f ) ( p = 1, 2 or + co)is taken with respect to the measure P. (14.8.1) Let p be a measure on G and let f be a (complex) locally P-integrable function on G (13.13.1). For the measures p and f . to be conuolrable, it is necessary and suficient that there should exist a P-negligible set N such that the function S H f ( s - ' x ) is p-integrable f o r all x $ N,and the function

(dejined almost everywhere with respect to 8) is locally P-integrable. When this condition is satisfied, the function g(x) = f (s- 'x)dp(s), defined almost ecerywhere with respect to p, is locally /I-integrable, and p We shall first prove the following lemma:

* ( f 8) is equal to g P. *

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INTEGRATION IN LOCALLY COMPACT GROUPS

(14.8.1.1) The image of the measure p @ /3 under the homeomorphism (s, x) H (s, s- ' x ) of G2 onto G2 is the measure p 0 p.

We may assume that p 2 0. Then, for every F E X + ( G 2 ) ,the function (s,x ) w F ( s , s - ' x ) belongs to X + ( G 2 ) and , we have

s s

F(s, s- 'x) dp(s) dp(x) = dp(s)

F(s, s- 'x) d&x)

by virtue of the left-invariance of fi and the Lebesgue-Fubini theorem. This proves the lemma. Now suppose that p and f * p are convolvable. Since If- PI = If I * p (13.13.4) we may limit ourselves to the case where p 2 0 and f 2 0. For each function h E X c ( G ) ,the function (s, x ) w h ( s x ) f ( x ) is ( p 0 P)-integrable by hypothesis (13.21.16 and 3.14.3); by virtue of (14.8.1.1), the same is true of the function (s,x,t+h(x)f(s-'x) (13.7.10). If A, is the set of points x E G such that h(x) # 0, it follows from the theorem of Lebesgue-Fubini that there exists a P-negligible set Nh in A, such that, for each x E A, n IN,, the function s ~ f ( s - ' x is ) p-integrable. Taking a sequence of functions h in X + ( G )such that the corresponding sets A, cover G (4.5.2), we see that the function s ~ f ( s - ' x is) p-integrable except at the points x of a a-negligible set N. Furthermore, it follows from the Lebesgue-Fubini theorem that the function x~ h(x) / f ( s - ' x ) dp(s), defined almost everywhere with respect to p, is fl-integrable, and that

s s

j J W I ( X ) d A s ) dP(x) = h ( x ) d N x ) f(s- 'x) 4 4 s ) . This proves that the condition of (14.8.1) is necessary, and that p * (f.p) = g * fi (13.13.1). Conversely, suppose that the condition is satisfied. We have to show that, for each function h E X+(G), the function (s, x ) h(sx) ~ f ( x ) is ( p 0 p)integrable. Now this function is ( p 0 @-measurable because h is continuous (13.21.13), and therefore it is enough to show that

8

CONVOLUTION OF A MEASURE AND A FUNCTION

281

But the lemma (14.8.1.1) shows that this is equivalent to the relation

ss'

h(x)f(s-'x)

dp(s) d p ( x )
0, there exists a compact subset K of G such that p([K) S E . Let V, be a compact neighborhood of x, . The function f is uniformly continuous on the compact set K-'Vo, hence there exists a neighborhood V c V, of x, in G such that the relation x E V implies

s

If(s-'x) -f(s-'xo>I

5 &/I*(K)

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INTEGRATION IN LOCALLY COMPACT GROUPS

for all s E K (3.16.5). Hence, for all x

E

V, we have

s 4 1 + 2 Ilfll),

from which (ii) follows.

(iii) Again we may suppose that p 5 0. By hypothesis, for each E > 0, there exists a compact subset H of G such that If ( x ) l 6 E for all x 4 H. Take K as in the proof of (ii) above, and suppose that x 4 KH. Then if s E K we have s-'x 4 H and therefore

(14.9.3) Every measure p on G is convolvable with every function f E Xc(G); the integral on the right-hand side of (14.8.2) is defined for all x E G, and the ,function x H f (s-'x) dp(s) is continuous on G.

s

Since the measure f p has compact support, p and f * p are convolvable (14.6.4), and it is clear that the integral f ( s - ' x ) dp(s) is defined for all x E G. The continuity of the function

XH

s

s

f ( s - ' x ) dp(s) follows from (14.1.5.5).

We shall leave to the reader the task of stating the corresponding propositions for the convolution f * p. It should be noticed in particular that (14.9.2) and its analog f o r f * p prove that if G is unimodular (14.3), then 9 S ( G )is a left and right module over the algebra MKG), and the external laws of composition of these two module structures are compatible by virtue of (14.7.2). (14.9.4) Let p , v be two measures on G, and let f E Y c ( G ) .Suppose that

fi and v are concolvable. Then the function p * f is v-integrable and (14.9.4.1)

( f , fi

* v> = ( p *"A v>.

Likewise, if p and v are bounded and f E %'g(G), the function p * f is continuous and bounded (hence v-integrable) and the formula (14.9.4.1 ) is valid. For the hypothesis implies that the function (s, x) +I f (s-lx) is (p 0 v)integrable, and the result therefore follows from the theorem of LebesgueFubini.

9 EXAMPLES OF CONVOLUTIONS OF MEASURES AND FUNCTIONS

285

PROBLEMS

1. Let G be a locally compact group and p a positive nonzero bounded measure on G , such that p * p = p. (a) Show that S = Supp(p) is compact. ( I f f € X + ( G )is not identically zero, remark that p *fmust be constant on S, and use (14.9.2(iii)).) (b) Show that S is a compact subgroup of G and that p is the Haar measure on S for which p(S) = 1. (Use fa), Problem 2 of Section 12.9 and Problem 5 of Section 14.7.) 2.

+

(a) Generalize (14.9.2(i)) to the case where 1 < p < 03, by using Holder's inequality (Sectipn 13.11, Problem 12). (b) Let p be a bounded measure on G . Show that the norm of the continuous endo~ * fp(14.9.2) is equal to IIpII. (Let (f,)be a morphism of L1(G, p) induced by f sequence satisfying the conditions of (14.11.1). If the norm in question were 11p/1- a with a > 0, we should have NI(p * f * g) 5 (Ilpll - a)N~(f.* d 5 (Ilpil - a)Nl(g). Deduce that N m ( p*f.) I/pI/- a , and obtain a contradiction by letting n tend to fa.) (c) Under the hypotheses of (b), show that the norm of the continuous endomorphism of Lm(G,p) induced by f ~ * j 'pis equal to IlpI1. (Reduce to the case where p has compact support and has a continuous density with respect to IpI.) (d) Suppose that G is compact and p is positive. Show that, for 1 < p < 03, the norm of the continuous endomorphism of Lp(G, p) induced by f ~ * fp is equal to llpll. (e) Let G be a cyclic group of order 3. Give an example of a measure p on G such that the norm of the endomorphism of LZ(G,p) induced by f ~ * fpis strictly less than llpll.

+

3. (a) Let p be a bounded measure on G . Show that we have NI(p *f)= N,(f) for all f E 9 ( G , p) if and only if p is of the form c . E~ with IcI = 1. (Using Problem 2(b), show that for each f~ X(G) we must have

1[ I

f dp = [ l f l d l p l . Deduce first that

p = clpl, where c is a constant such that IcI = 1, and then that p is a point-measure.) (b) Take G to be a cyclic group of order 3. Give an example of a measure p on G which is not a point-measure and is such that N2(p * f )= N2(f) for all real-valued functionsfon G . 4.

Let G be a,locally compact group and p a left Haar measure on G . Let f be a bounded real-valued function on G, uniformly continuous with respect to a left-invariant distance on G . If p is any bounded measure on G , show that p *f(relative to 8) is uniformly continuous with respect to a left-invariant distance on G .

5.

Let G be a locally compact group and p a left Haar measure on G . (a) With the notation of Section 13.20, Problem 1, let ( p n )be a sequence of bounded measures on G which converges to t~ with respect to the topology F3, and let (f.) be a sequence of functions in Y ( G , p) such that the sequence of bounded measures (f. . p) converges to f?.! , with respect to the topology 9 6 . Show that the sequence of bounded measures (p. * (fn . p)) converges to (p,,* (f.p)) with respect to 9 6 (cf. Problem 4(d) of Section 14.7). (Use Problems 1 and 2 of Section 13.20.) (b) Let E,/z denote the space of bounded real-valued functions on G which are uniformly continuous with respect to a left-invariant distance on G , and let

286

XIV

INTEGRATION IN LOCALLY COMPACT GROUPS

denote the weak topology on M:(G) corresponding to the vector space E5,Z; the topology F5,* is finer than Fzand coarser than F 3Give . an example of a sequence (p,) which tends to 0 with respect to F5,*, and a sequence (f,)of functions in P ( G , B) such that the sequence (f,. p) converges to 0 with respect to F6, but such that the sequence (p, * (f,. p)) does not tend to 0 with respect to F A (Take . G -= R andf,(t) to be the function which is equal to sin nt in the interval [O, n] and is zero elsewhere.) (c) Let (pn)be a sequence of bounded measures on G which converges to p with let, f E 9 ( G , 6) and let (fa)be a sequence of functions in P ( G , p) respect to F S i z such that N,(f-f,)+O. Show that NI(p.*fn-p*f)+O. (Reduce to the case where p = 0 and f. = f for all n , and then to the case where f~ X ( G ) . Show that the sequence (p, * (f.8)) tends to 0 with respect to F3, by remarking that if g is a bounded continuous function on G , the functionf* g is uniformly continuous with respect to a left-invariant distance on G.) Show that the result is no longer valid if 9-5,zis replaced by F z . (d) Let (p,) be a sequence of bounded measures on G which tends to 0 with respect 2 , (v.) a sequence of bounded measures on G which tends to Y with respect to 9 5 /and Show that the sequence (p. * v,) tends to 0 with respect t o F5/2. (By using to Fz. (14.11.1), reduce to proving that -to for any f~ X ( G ) and g uniformly continuous with respect to a left-invariant distance on G . Then use (c).) 6.

The notation is that of Problem 5. (a) Let (p.) be a sequence of bounded measures on G. Suppose that, for each function f~ P ( G , p), the sequence ( p m* (f.p)) tends to 0 with respect to F2. Show that the sequence of norms (IIpnll)is bounded. (Apply the Banach-Steinhaus theorem to the sequence of mappings f ~ ( p*f). of L1(G, 8) into L'(G, p), and use Problem 2(b).) Deduce that the sequence (p.) tends to 0 with respect to F2. (b) Suppose that, for eachfe Y ' ( G ,p), the sequence (p. * (f.p)) converges vaguely to 0. Show that the sequence (p.) tends vaguely to 0. (If K is any compact subset of G , show as in (a) that the sequence (Ip,I(K)) is bounded.) Give an example in which the sequence (IIpnll)is not bounded (take G = Z). (c) Suppose that, for each f~ 9 ( G ,p), the sequence (p. * (f.p)) converges to 0 with respect to 9 5 / 2 . Show that pn-0 with respect to 9 5 / 2 (use (a)). Give an example where N,(pn*f)+ O for each f e 9 ( G ,p) but the sequence (p.) does not tend to 0 with respect to F 3 .

10. CONVOLUTION OF TWO FUNCTIONS

(14.10.1) Let .f and g be two (conzplex) locally 8-integrable functions on G. Then the measures f 1 and g . B are convolvable if and only i f there exists a P-negligible set N such that the function s H g ( s - ' x )f ( s ) is P-integrable f o r all x $ N and the function

10 CONVOLUTION OF TWO FUNCTIONS

(defined almost everywhere, relative to condition is satisfied, the function

s

p) is locally p-integrable. When this

h(x) = s ( s - ' x ) f ( s >M defined almost everywhere relative to

s),

p, is locally /I-integrable, and

( f PI * ( 9 rc3) *

287

*

=h

*

8.

This is a particular case of (14.8.1). When the conditions of (14.10.1) are satisfied, the functions f and g are said to be convolvable (with respect to p). Any function which is equal almost everywhere (with respect to p) to the function h above is called a convolution o f f a n d g (with respect to p) and is writteiIf* g (orf * pg, where it is necessary to bring j3 into the notation). Thus for almost all x we have (14.10.2)

and likewise, using (14.8.4), (14.10.3)

s s

(f* s ) ( x ) = d s - ' x > f ( s >4 T s )

(f * g ) ( x ) =

f ( x s - ')s(s)&(s-

'1 a

s )

almost everywhere with respect to p. When one of the functions equal to h almost everywhere is continuous on G, we adopt the same convention as in (14.8) and call this function the convolution off and g. In particular, when f * g is continuous, we have n

(14.10.4)

It should be remarked that the property off and g of being convolvable does not depend on the choice of left Haar measure p, but their convolution f* pg does. If p is replaced by ap, where a > 0, then we have f * 'pg = a . f * pg. When G is discrete, to say that f and g are convolvable signifies that the family (g(s-'x)f(s)),,G is absolutely summable (5.3.3) for all x E G, and we have (f * s)(x) = s ( s - ' x ) f ( s )

1

S E G

if the Haar measure

p is such that /?({e})= 1.

288

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INTEGRATION IN LOCALLY COMPACT GROUPS

From the results of (14.9) we have, in particular: (14.10.5)

Suppose that f andg are locally P-integrable. Ifone of the functions

f,g is continuous, and if one has compact support, then f and g are convoluable, the right-hand sides of (14.10.2) and (14.10.3) are dejned for all x E G , and the function f * g is continuous. If both f and g belong to X,-(G), then so does f*s. This follows from (14.9.1) and (14.9.3). (14.10.6) Let f be a P-integrable function. (i) For p = 1, 2 or co,the function f is convolvable with every function g E 49g(G,P); the function f * g belongs to Yg(G,P); and

+

(14.1 0.6.1)

(ii) Zfp

=

+

00,

and g E Yg(G, P), the integral

s

J g ( s - ' x ) f ( s >d P 6 ) = f ( x s - l)g(s)A(s-l)dP(s)

is dejned for all x E G, and the function XH lf(xs-')g(s)A(s-') dP(s) is uniformly continuous with respect to every right-invariant distance on G. (iii) If p = 1, we have

(iv) Zfg

E

Wg(G), then also f

* g E @(G).

Parts (i) and (iv) follow from (14.9.2) and the relation 11 f * PI1 = N , ( f ) (13.20.3). T o prove (14.10.6.2), we remark that by virtue of (14.8.1.1) and the fact that the function (s, x ) ~ g ( x ) f ( sis) (P 0 P)-integrable (13.21.14),

the function (s, x ) w g ( s - ' x ) f ( s ) is also (P 0 /?)-integrable. The formula (14.10.6.2) then comes immediately from the Lebesgue-Fubini theorem and the left-invariance of P. As to (ii), for each x E G the function S w g ( s - l x ) belongs to 49Z(G),and therefore the integral on the right-hand side of (14.10.2) is defined for all x E G. If we put v = A-' P, then v is a right Haar measure (14.3.4), and we may write (14.10.3) i n the form

10 CONVOLUTION OF TWO FUNCTIONS

Consequently,

s s

tcf * g>(x>- ( f * g)(x')i 5 N d g ) If (xs- '1 - f (x's-')l = N,(g)

289

dv(s)

If(s-') - f ( X ' x - ' S - ' ) l

~v(s).

The result will therefore be a consequence of the following more general lemma: (14.10.6.3) For p = 1 or p = 2, every right Haar measure v on G and every function h E YE(G,vj, the mapping s ++S(s)h of G into Yg(G,v) is continuous and satisfies N,(S(s)h) = N,(h).

The second assertion is an immediate consequence of the right-invariance of v . To prove the first, suppose first of all that h E X c ( G ) ;then the continuity of S H 6(s)h follows from (14.1.5.5). In the general case, if (h,) is a sequence of functions in Xc(G )which converges to h in Y $ ( G ,p) (13.11.6), the relation N,(S(s)h - 6($)h,) = N,(h - h,) shows that the sequence of functions S H ~ ( S ) ~ ,converges uniJormIy on G to the function s ~ 6 ( s ) h Hence . the result (7.2.1). (14.10.6.4) In the same way, one shows that if h E 2 g ( G ,p) wherep = 1 or 2, the mapping s H y(s)h of G into Yg(G,p) is continuous, and that N,(y(s)h) = N,(h). (14.10.7) Let f e B,?(G, p) and let g E YZ(G,8). Then the integral

f g ( s - ' x ) ( f (s) dp(s) is dejined for all x E G, and the function f %g(G)and satisfies the inequality

*g

belongs to

For each x E G, the function st+g(s-'x) belongs to YZ(G,j?), and therefore the first assertion follows from (1 3.11.7). Moreover, again from (1 3.11.7),

290

XIV

INTEGRATION I N LOCALLY COMPACT GROUPS

which is (14.10.7.1). If both f and g are in Xc(G),then so is f * g ; since the formula (14.10.7.1) shows that the bilinear mapping (f,g ) H f * g of 9 i ( G ,P) x Y:(G, into 9?,(G) is continuous (5.5.1), and since X c ( G )is (13.11.6), the values of the mapping dense in Y;(G, P) and in Y;(G, (f,g ) t + f * g belong to the closure of X c ( G )in Bc(G) (3.11.4): that is, they belong to @(G) (1 3.20.5).

B)

B)

Proposition (14.10.7) implies the following corollary: (14.10.8) Let A and B be two 8-integrable sets in G. Then the function xt+fl(A n xB) is continuous on G and tends to 0 at injnity (13.20.6). If B-' is also P-integrable, the function x H P(A n xB) is P-integrable, and we have

"A

n xB)

= P(A)P(B-').

If moreover neither A nor B is @-negligible, the set AB-' has a nonempty interior. Finally, for every subset A which is P-measurable and not P-negligible, the set AA- is a neighborhood of the neutral element e of G.

'

We have qAE 9 ' ( G , P) and q B E 9'(G, P), hence we may apply (14.10.7) with f = qA and g = 3jB. Since 3jB(s-'x) = qB(x-'s) = (pXB(s), we have

I

( q *~~ B ) ( x = ) (PA x&) dP(s) = P(A n xB). This proves the first assertion. If B-' is also /?-integrable, then both qAand ( p B - l belong to 9 ' ( G , P), and

we can apply (14.10.6.2), which gives us the formula P(A n xB) dP(x) = P(A)P(B-'). If the right-hand side of this formula is nonzero, it follows that P(A n xB) is not identically zero, and hence there exists a nonempty open set U in which the continuous function P(A n xB) is >O. This implies that U c AB-'. Finally, to prove the last assertion, we observe that there exists a compact set K c A which is not P-negligible, so that we may assume that A is compact; but then /?(A)= (qA* 3jA)(e)is >O, and it follows as above that there exists a neighborhood of e contained in AA-'. (14.10.9) Let A g be two locally /I-integrable functions, p n measure on G. The two following conditions are equivalent: (a) .f andg' are convolvable and the function If I * Ig'I is P-integrable: (b) ,Z u n d f are conuoluable and the function g(l,ilI * I j l ) is /I-intqgrable. If these conditions are satisjied, we have (14.10.9.1 )

( g ?ii

* ( f . P)>

=

( f * g'? P ) .

From (13.21.9) and (14.8.1), it follows that condition (a) means that the

10 CONVOLUTION OF TWO FUNCTIONS

291

function (s, x) + + g ( x - ' s ) f ( x ) = g(s-'.x)f(x) is (p 0 /?)-integrable, and the right-hand side of (14.10.9.1) is then equal to (14.10.9.2)

Similarly, condition (b) means that the function (s, x ) H g ( x ) f ( s x ) is (p @ /?)integrable, and the left-hand side of (14.10.9.1) is then equal to nn

(14.1 0.9.3) But from the theorem of Lebesgue-Fubini and the left-invariance of /?,it follows that one of the integrals (14.10.9.2), (14.10.9.3) exists if and only if the other one exists, and then both are equal.

PROBLEMS

1. (a) Let g be a /%measurable function of compact support on G. Show that f a n d g are convolvable and that f * g is rontinltort.7 on G in the following two cases: (i) g is essentially bounded a n d f i s locally /%integrable; (ii) g E -4Pf(G,/If) i s,/&measurable a n d f 2 is locally ,%integrable. (b) Assume that G is unimodular. Show that if f e -Yg(G,8) and g E Yg(G,P), 1 1 where p 2 1 , 9 2 1 and - - 2 1 , then f and g are convolvable, f * g E -Y;(G, P) P 9 1 1 1 where - = - - - 1 , and N,(f* g) 5 N,(f)N,(g) (W. Young's inequality). (Consider r P 9

+

+

1

+1

first the case where - - = 1, then use (14.10.6.1) and the Riesz-Thorin theorem P 9 (Section 13.17, Problem 7).) 2.

Let G be a compact group, p the Haar measure on G for which P(G) = 1 , and let A, B be two p-integrable sets. Show that for each 8 > 0 there exists x E G such that P(A n xB) 2 (1 s)P(A)p(B). Deduce that, for each integer n such that B(A) 2 l/n, there exist n points x,,. . . , x, in G such that @(xlAu x2 A u ... u x , A ) 2 6. (Consider the sets x, . CA.)

+

3.

Let G be a compact commutative group, P the Haar measure on G for which P(G) = 1, and g a measurable function on G. Let (r,) be the orthonormal system of Rademacher functions (Section 13.21, Problem 10). Let CL > 0 and let A be the set of x E G such that Ig(x)l > a . Let n be an integer 2 1 such that n . P(A) 2 I . Show that there exists a P-measurable set B such that @(B) 2 i,and n points s,, . . . , s, in G such that, if we put F(x, t ) =

2

k=l

rk(f)g(.skx),the following is true: for each x E

B, there e x i m a finite union l(x) of

intervals in [O, 11 such that (i) x(l(x)) 2 4 (A being Lebesgue measure); (ii) for all

292

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INTEGRATION IN LOCALLY COMPACT GROUPS

t E I(x), F(x, t ) 'i. a . (Use Problem 2, and observe that, by virtue of the relation r.(l - t ) : - r n ( f ) , for a given integer h E [ I , n] the set of t E [0, 11 such that the x ) the same sign, has measure 2 4.) numbers rh(f)g(shx)and ~ r k ( ? ) g ( s khave k#h

4, Let G be a compact commutativegroup, @theHaar measure on G for which p(G) = 1 , Let (UJ be a sequence of continuous endomorphisnis of L:(G, p), each of which commutes with all translationsf~(y(s)f)" for all s E G . For each f~ Y:(G, p) we denote by U. .fany function belonging to the class U , .f, and we put

u* .f = sup U" .f: n For each a > 0, let E,(f) be the set of points x E G such that ( U * .f ) ( x ) > a. (a) Let f~ Y i ( G ,6) be such that N2(f) 5 1, and let n be an integer such that n.p(E,(f))> ] . L e t s l , ..., s,bepointsofGsuchthat theunionBofthesetss;'E,(f) has measure p(B)

2 4 (Problem 2), and let F(x, 1 ) =

"

k=l

r k ( t ) f ( s k x )Show . that for each

a finite union of intervals I(x) c [0, 11 such that )\(I(x)) 2 1 and such that x E E,(F( . , t ) ) for all t E I(x). (Observe that if x E B there exists an integer m and an integer j E [ I , n] such that ( U , .f ) ( s j x )= 0, and apply Problem 3 to g =

x E B there exists

u,. .f:)

(b) Let S c [0, 1 1 be a A-integrable set such that h(S) 2 i. Show that there exists t E S such that P(E.(F(. , t ) ) ) > &. (If H is the set of (x, t ) E G x [0, 11 such that (U*. F( . ,t ) ) ( x ) > a , remark that for each x E B we have A(H(x)) 2 2, and deduce that (/3 OAHH) > t.1 (c) For each M > 0, let S, be the set of t E [0, I] such that N,(F(. , t ) ) 5 M . Show that h(S,) 2 1 - n/M2. Deduce from (b) that if M2 4n there exists r E [0, 11 such that both N,(F(. , t ) )5 M and p(E.(F(. t ) ) )> &. (d) Suppose that for eachfE gi(C, p) the function U* .fis finite almost everywhere. Show that there exists a constant C > 0 such that, for each f E -LP:(G, p) and each a > 0, we have (P(E,(f)))"* 5 Ca-'N,( f ) (E. Stein's theorem). (Deduce from Section 13.12, Problem 12 that there exists a constant c > 0 such that for each function h E Y i ( G ,p) satisfying N2(h) 5 M, we have /3(EcM(h))< &. Then make use of (c) above, taking M = LY/C, h = F( . , t ) and n = [iM'].) 5. Let A be Lebesgue measure on R and let f be a compactly supported h-integrable function. Put

for each F

> 0, and 1

rh

these are lower semicontinuous functions on R (Problem 1) with compact support. @( f ) is the Hardy-Littlewood maximal function relative to f . (a) For each LY > 0, let Ee,=(f) be the set of x E R such that @ , ( f ) ( x )> a . Every compact set K c E,,.(f) is contained in the union of a finite number of compact intervals

Ik

(1 5 k

5 n) such that

a

'

5

j Ik

If(t)l dt.

11

293

REGULARIZATION

(b) Show that there exists a sequence of indices ( k , ) l s , s , such that the intervals I, are mutually disjoint and such that

(We may assume that no 11,is contained in the union of the I, such that h # k . Show br] and arrange the Ix so that a, 5 a k + l , then necessarily that if we put Ix = [ak, ax< a k + l , b2,-, < a z r + l and b 2 k < a Z X + 2by , considering three intervals with consecutive indices. Deduce that the intervals are mutually disjoint, the Izr are mutually disjoint and that the k, may therefore be taken to be either the even indices or the odd indices.) (c) Deduce from (a) and (b) that if E.(f) is the set of x E R such that 8 ( f ) ( x )> a, then

s

(d) Let g be a nonnegative function in -YA(x) such that (i) g ( - t ) decreasing on [0,

+ a,[;

(iii)

g ( t ) dt = 1 . Prove that

=g(t);

(ii) g is

I (g * f ) ( x )I 5 8 ( f ) ( x ) for

all

R . (For each CL > 0, let ]-h(a), h(cr)[be the largest open interval in which&) > a . Show that x

E

=lo+s””’

(9 * f ) ( x )

and observe that

5

da

f ( x - t ) dl

-h(a)

+m

0

h(a)da = 4 .)

(e) State and prove analogous results for functions which are integrable with respect to Haar measure on the torus T.

11. RECU LARl ZATlON

(14.11.1) Let (f,) be a sequence of P-integrable functions which satisfy the following conditions:

(a) the sequence of integrals If,(x)I dp(x) is bounded;

s

1 V

(b) the sequence of integrals f,(x) dp(x) tends to 1; (c) for each neighborhood V of e, the sequence of If,(x)l dp(x) tends to 0.

integrals

Then (i) For each bounded continuous function g on G , the sequence (f,* g ) converges uniformly to g on every compact subset of G. g g is uniformly continuous with respect to a right-invariant distance on G , then the sequence (f.* g ) converges uniformly to g in G.

294

XIV

INTEGRATION IN LOCALLY COMPACT GROUPS

(ii) Zfp = 1 or 2 and g E Z P P ( Gthe ) , sequence of norms N,((f, * g ) - g) tends to 0 as n tends to +a.For each function g E L,"(C), the sequence ( ( f ,. g ) " ) converges weakly to y" in L,"(G), considered as tlze dual of Lk(G) (12.15 and 13.17). (iii) Suppose in addition that the supports of the f , are contained in a fixed compact subset of G . Then for each measure p on G , the sequence of measures

cc * (f,*PI = (cc *f,). P converges vaguely to p (1 3.4).

(i) For each x from the definitions

E

g(x) - ( f n

G and each compact neighborhood V of e we have

* g)(x)

= g(x)

+

(1 -

I

I

fn(s)(g(x)

- lci/.(S)g(s-

1

f n ( s > dP(s)

- g(s-

'x)) dP(s)

1x1 d ~ ( s ) .

Let Vo be a compact neighborhood of e and let L be a compact subset of G . Since V i ' L is compact (12.10.5), the restriction o f g to V;'L is uniformly continuous with respect to a right-invariant distance on G (3.16.5). Hence, for each E > 0, there exists a compact neighborhood V c V, of e such that Ig(x) -g(s-'x)( =< E for all X E L and all ~ E V Now . choose no so that If,(s)I dP(s) 5 E and ( 1 - Sf. dP(s)I 0 5 E for all n 2 n o . We have then

scv

and therefore, for all x E L,

I 1 REGULARIZATION

295

Ifg is uniformly continuous with respect t o a right-invariant distance on G, the same argument applies, taking L = G. (ii) There exists a function h E X,(G) such that N,(g - h) 5 E (13.1 1A). By (14.10.6.1), it follows that N,((f, * g ) - ( f , * h)) 5 aN,(g - h) a&.Thus we are reduced to proving (ii) when g E X c ( G ) .Let S be the support of g . If V is a compact neighborhood of e, we have seen in (i) that f, * g - g converges uniformly to 0 in the compact set K = S u VS. Next, if x # K, we have

(f" * g ) ( x ) - g ( x ) =

f,(s)g(s-

XI

dP@)

Since, on the other hand, the integral I ( f , * g ) ( x ) - g ( x ) I @ ( x ) tends t o 0 SK with l/n, we see that

and as N,(f,

*g

-g)

5 (a + l ) ~ ~the g ~same ~ , is true of

Suppose now that g E 2'g(G), and let 11 E 2'A(G). If V is any compact neighborhood of e in G, we have

296

XIV

INTEGRATION IN LOCALLY COMPACT GROUPS

But we have n

and therefore,

For each E > 0, it follows from (14.10.6.4) that there exists a compact neighborhood V of e in G such that N,(h - y(s-')h) 5 E for all s E V, and hence,

Examples (14.11.2) Let (V,) be a fundamental system of neighborhoods of the neutral element e in G . Since the support of p is the whole of G, there exists for each n a function f n e X + ( G ) with support contained in V, and such that /fn(x) @(x) # 0 (13.19.1). Multiplying each f , by a suitable constant, we

s

may assume that fn(x)@ ( x ) = 1, and the sequence (f,) now satisfies the conditions of (14.11.1). We see therefore that every measure p on G may be approximated (in the sense of the vague topology) by a sequence of "regularizations " which are measures having a continuous density with respect to /I(14.9.3).

I1

REGULARIZATION

297

(14.11.3) Take G = R, and /3 to be Lebesgue measure. Put (14.11.3.1)

gn(x) =

(I:-

if X E [-I, if 1x1 > 1.

x2)"

11,

=I1

Let a, g,(x) dx, and f, = ai'g,. The sequence (f,)satisfies the conditions of (14.11.1). For 1 - x2 2 1 - 1x1 for 1 S x 5 1, hence a,

2 2 Jol(l- x)"

2 n+1'

dx = -

+

and therefore f,(x) 5 ( n 1)(1 - x')" for all x E [ - 1, 11, which proves that f,(x) + 0 uniformly on every compact interval not containing 0. Let p be a measure on R with support contained in [-f, f]. Then we have

and if x E [ -$,+I

this gives (p * jn)(x) = a,

5

112

1

- 112

(1 - (x - Y)')" ~ A Y ) ,

showing that the function p * f, is equal to a polynomial function on [ -3, f]. In particular, if p = h p, where h is a continuous function with support contained in [-$, $], we obtain from (14.11.l(i)) the theorem of Weierstrass on uniform approximation of continuous functions by polynomials on a compact interval (7.4.1).

-

PROBLEMS 1.

2.

If a locally compact group G is such that the algebra S ( G ) is commutative with respect to convolution, show that G is commutative. (Show by regularization that the algebra of measures with compact support is commutative.) Let G be a locally compact group, p a left Haar measure on G. Show that the algebra L'(G, p) has a unit element if and only if G is discrete. (Suppose that G is not discrete, and let fo E -LP'(G,p); then there exists a compact neighborhood V of e such that

< 1. Show that, if U is a compact symmetric neighborhood of e such that Uz t V, then I(rpu * f o ) ( x ) I < I for almost all x E U, and hence that f0 cannot be Jv Ifo(x)l # ( x )

the unit element of L'(G,

6))

298

3.

XIV

INTEGRATION IN LOCALLY COMPACT GROUPS

Let G be a locally compact group, /? a left Haar measure on G. If G I { e } , the algebra L'(G, /?) has zero-divisors # 0. To construct two nonnegligible functions f,g in Y ' ( G ,p) such that f * c/ is negligible, we may proceed as follows: ( I ) The case where G has a compact subgroup H #- { e } . Take for f a characteristic function cpa and for y a function of the form cpSR - cpR, where A, B, and s are suitably chosen, and remark that A,(x) = I for all x E H . (2) The case G = Z. Show that we may take

for all 17 t Z, and y =f. (3) The general case. Prove first of all that there exists (I # e in G such that A(a) = 1. The closure H in G of the subgroup generated by a is then either compact or isomorphic to Z (Section 12.9, Problem 10). In the former case, use the result of case ( I ) ; in the latter, take

where the set U and the sequences case (2).

(4 (p.) ,are

suitably chosen with the help of

4.

Let G be a locally compact group, /3 a left Haar measure on G. In order that a subset H of P ( G , 8) (I s p < 0 0 ) should have a relatively compact image R in LP(G,p), it is necessary and sufficient that the following conditions should be satisfied: (1) R is bounded in Lp(G,/?); (2) for each E > 0, there exists a compact subset K of G such - ~E )for all f~ H; (3) for each E > 0, there exists a neighborhood V that N , ( ~ $ J ~ ;g of e in G such that N,((y(s)f) -f) 5 E for aHff H and all s E V. (To prove that these conditions are sufficient, observe that if g E X ( G ) and if L is a compact subset of G, then the image, under the mappingfwg *f,of the set of restrictions to L of functions belonging to H is an equicontinuous subset of X(G).)

5.

Let G be a locally compact group, /? a left Haar measure on G , and p a positive measure on G. Show that if A is a p-integrable set and B is a universally measurable set in G, the function u : s ~ p ( n A sB) is /3-nieasurable on G. If moreover B-I is /?-integrable, then so is u and we have

+

j p ( A n sB) d/?(s) = p(A)p(B-'). (Use the Lebesgue-Fubini theorem and Problem 21 of Section 13.9.) Give an example of a measure p such that the function s w p ( A n sB) is not continuous. If p is a measure with basis /? and if A is p-integrable and B is universally measurable, then the function s ~ p ( n A sU) is continuous on G. 6.

Let G be a locally compact group, G' a topological group, f a homomorphism of G into G' which is /?-measurable, where /? is a left Haar measure on G. Show thatfis continuous. (Observe that if we put y(x) = f ( x - ' ) , there exists a compact non-pnegligible subset K of G such that the restrictions o f f and y to K are continuous. Deduce that the restriction o f f t o K K - ' is continuous, by using (12.3.8); then apply (14.10.8).)

11

REGULARIZATION

299

7. Let G be a locally compact group. For each t E R*,, let y, .f 0 be a positive bounded

measure on G. Suppose that the mapping t H p L ,is continuous with respect to the topology .F2(in the notation of Section 13.20, Problem I), and that p-,,, = ps * pt for all s, t in Rf . (a) Show that there exists a real number e such that llp,ll = ecr. (Observe that the mapping I H IIpJ is lower semicontinuous, and apply Problem 6.) (b) Show that, as t + O , pr converges (with respect to Y2) to a Haar measure of total mass 1 on a compaci subgroup of G. (Use (12.15.9) to show that there exists a sequence (s.) tending to 0 such that the sequence (ySn)tends to a limit ~1 in the topology Y2, and that y, * y = yr = y * yr for all t > 0. Deduce that ps+y as s + 0, and that y * u , = y. Complete the proof by using Problem 5 of Section 14.7.)

8.

Let h denote Lebesgue measure on R“.Let A be a bounded convex open set in R” (Section 8.5, Problem 8), and let D(A) = A - A be the set of all x - y where x , y E A. The set D(A) is convex, open, symmetric and bounded. For each x f 0 in D(A), let p(x) be the unique real number, belonging to 10, I[, such that p ( x ) - * x lies in the frontier of D(A). If we put p(0) = 0, then p is a continuous function on D(A) (Section 12.14, Problem 12). (a) For each x e D(A), show that h(A n (A -t x ) ) >. ( I - p(x))”h(A).

(Observe that if x # 0 and p ( x ) - I x (1 - p(x))A

=

b - a where a, b

t p(x)b = (1

- p(x))A

E

A, then

+ p(x)a + x ) .

(b) Show that

(c) Deduce from (a) and (b) that

Prove that this integral is equal to A(D(A)) J 1 n t n - l ( l 0

-

t)” dt

= -h(D(A)). (nYZ

(2n)!

(Split up the interval [0, 11 into rn parts by means of an increasing sequence (tk)o CL has measure zero, for all a > 0.) (b) Let (gn)be a sequence of nonnegative functions in X ( R ) , satisfying conditions (a), (b), and (c) of (14.11.1); assume also that g n ( - t ) == gn(f)and that gn is decreasing in 10, a[.Show that as n + co,(9. *f)(x) +f(x) almost everywhere in R (Lebesgue's theorem). (Same method, using Section 14.10, Problem 5(d).) (c) State and prove analogous results for Haar measure on the torus T.

+

CHAPTER X V

NORMED ALGEBRAS A N D SPECTRAL THEORY

The spectral theory of operators, which we have already encountered in an elementary aspect in Chapter XI, is one of the masterpieces of modern analysis. Its main object is to obtain, for linear operators on a Hilbert or prehilbert space satisfying suitable continuity conditions, an analog of the classical theorem of algebra which assigns a canonical form (by means of “Jordan matrices”) to an n x n matrix over C (or, equivalently, to an endomorphism of aJnite-dimensional complex vector space). In Chapter XI we have seen how this result, suitably modified, can be extended to compact operators. But there is another much less obvious generalization, due to Hilbert and his successors, which applies to a wider class of operators: in particular, to continuous self-adjoint operators (11.5), and more generally to normal operators (15.11). Just as in the classical case a self-adjoint (or normal) matrix over C has a canonical form which is a diagonal matrix, the continuous normal operators can all be described in terms of a single model: namely multiplication M,(u) : . f ~ ( u f )in- a space Li(p) by the class of an essentially boundedfunction u (15.10). The Lebesgue theory intervenes here in an essential way (even if we begin with a self-adjoint operator which comes from a differential equation as regular as we please (Chapter XXIII)), and it is no exaggeration to say that the occurrence of Lebesgue theory in spectral theory and related fields such as harmonic analysis and the theory of representations of locally compact groups is the principal reason of its importance in Analysis. A modern account of spectral theory does not follow the path mapped out by Hilbert, but uses a much more elegant and powerful method based on the theory of normed algebras inaugurated by Gelfand and his school. In this chapter we shall concentrate mainly on normed algebras with involution (15.4), because these are the ones which arise in spectral theory. But the general theory of normed algebras, and especially the fundamental concepts of 304

1 NORMED ALGEBRAS

305

spectrum and Gelfand transformation (15.3), have found many other applications in modern analysis, notably in the theory of analytic functions. Some of these applications are touched on in the problems, and the reader interested in this aspect is referred to [35] and [29]. The central part of this chapter is the study of representations of algebras with involution, which enables such an algebra, given “ abstractly,” to be “ realized ” as an algebra of operators on a Hilbert space. The essential notion in the modern development of this theory is that of a Hilbertform, which is closely related to that of a Hilbert algebra (15.7). We shall study in detail only two particular aspects of the theory of Hilbert algebras: the first (1 5.8) prepares the way for the theory of representations of compact groups (Chapter XXI), and the second (15.9) for spectral theory and harmonic analysis (Chapter XXII). The reader who wishes to go further (notably in view of the deep and difficult theory of representations of locally compact groups) is warmly recommended to read the two beautiful volumes by J. Dixmier ([24] and [25]) which dominate the subject. The Hilbert spectral theory (15.10 and 15.11) appears in our treatment as an immediate particular case of the general theorem of Bochner-Godement (15.9). It can be reached more directly and rapidly (Section 15.10, Problem 2) from the Gelfand-Neumark theorem (15.4), but it seemed to us to be more instructive to deduce it from a much more powerful theorem which is the cornerstone of harmonic analysis, even if this requires a small additional effort. The applications of spectral theory are not limited to those referred to above. Among the most celebrated we should mention at least the following: (1) one of the most elegant theories in Analysis, namely the “moment problem ” inaugurated by Stieltjes, with its many ramifications (analytic functions, orthogonal polynomials, Jacobi matrices, continued fractions, etc), which fits admirably into the theory of unbounded Hermitian operators; (2) the interesting relations between ergodic theory and spectral theory; (3) perturbation theory. Some of the important results of these theories are mentioned in the problems, and the reader is referred for more ample information to the works [20], [281, [301, and [321 in the References. I. N O R M E D ALGEBRAS

When we speak of algebras in this chapter, we shall always mean algebras over the field of complex numbers C . A normed algebra is defined to be an algebra A endowed with a norm X H llxll (5.1) satisfying the inequality (15.1.1) for all x, y in A.

llwll 5 llxll * llyll

306

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

If also A has a unit element e and is not the zero algebra (in other words, if e # 0) we shall always assume that the norm satisfies the additional condition (15.1.2)

llell = 1.

The inequality (15.1.1) shows that the bilinear mapping (x, y ) ~ x of y A x A into A is continuous (5.5.1).

A complete normed algebra (i.e., a normed algebra in which the underlying normed vector space is a Banach space) is called a Banach algebra. It is clear that every subalgebra B of a normed algebra A (if A has a unit element e, we require that B contains e), endowed with the restriction to B of the norm (IxII, is a normed algebra. If in is a closed two-sided ideal of A, then the quotient algebra A/in, endowed with the norm (12.14.10.1) induced by IIxII, is again a normed algebra. For if i, j are two elements of A/m, then for each E > 0 there exist x E i and y E j such that

+

+

from which it follows that llxyll 5 ( 1 1 1 li e)(IIj(l E ) ; since xy E i j and since E was arbitrary, this shows that Ilijll 5 llfll . 11j11. If moreover A has a unit element e, and if in # A, then i is the unit element of A/in and is ZO,hence /(C((S (le((= I by definition, and on the other hand I(t(( = (lt2(I S j(tj(2, which implies that \[ti] >= 1. Hence llell = 1, and so A/in is a normed algebra. (15.1.3) Let A be a normed algebra. Then the closure in A of a subalgebra of A (resp. of a commutative subalgebra, a left ideal, a right ideal) is a subalgebra (resp. a commutative subalgebra, a left ideal, a right ideal).

By virtue of (5.4.1) and the principle of extension of identities, this follows directly from the continuity of multiplication in A, by the same proof as in (5.4.1). If A, B are two normed algebras, an algebra isomorphism u : A + B is said to be a topologicalisomorphism if it is bicontinuous, that is if (5.5.1) there exist two real numbers a > 0 and b > 0 such that a llxll iIIu(x)II 5 b IIxI/ for all x E A (5.5.1). The isomorphism u is said to be isometric if in addition we have IIu(x)ll = (Jx(1 for all x E A.

1 NORMED ALGEBRAS

307

Examples of’ Normed Algebras (15.1.4) For each nonempty set X, the set ,%,(X) of bounded complex-valued functions on X is a commutative Banach algebra with respect to the norm (7.1 .I) and ordinary muItiplication of functions: the inequality (15.1 .I) is clearly satisfied, the constant function 1 is the unit element of &?,-(X) and satisfies (15.1.2). If X is a topological space the subspace %‘F(X)of bounded continuous functions on X is a closed subalgebra of W,(X). If X is metrizable, separable and locally compact, the space @(X) (13.20.5) of continuous functions which tend to 0 at infinity, and the space .X,(X) of continuous functions with compact support, are ideals in the algebra g;(X), the former being the closure of the latter. (15.1.5) If X is the closed disk IzI 5 1 in C, then the set &(X) c U,(X) of continuous functions on X which are analytic in the interior IzI < 1 of X is a closed subalgebra of g,(X), by virtue of the theorem of convergence of analytic functions (9.12.1 ). (15.1.6) We have already seen (11.1) that the algebra 9 ( E ) of continuous endomorphisms of a complex normed space E is a normed algebra (in general noncommutative) with the identity mapping 1, of E as unit element (the inequality (15.1.1) in this case is just (5.7.5), and the relation (15.1.2) is obvious). Furthermore, if E is a Banach space, then 2 ( E ) is a Banach algebra (5.7.3). The set of compact operators in 2 ( E ) is a closed two-sided ideal by virtue of (11.2.6) and (11.2.10). (15.1.7) Let G be a separable metrizable locally compact group. The set Mh(G) of bounded measures on G, endowed with the convolution product (A,p ) A *~ p and the norm (13.20.1) is a Banach algebra. For in this case the inequality (15.1.1) is just (14.6.2.1) with n = 2; the unit element E, (the Dirac measure at the neutral element e of G) is such that lleell = 1 ;and M;(G), being the dual of the normed space Wg(G) (13.20.6), is complete (5.7.3). The subspace of MA(G) consisting of measures with base a left Haar measure on G can be identified (together with its norm) with the Banach space LA(G, p), because N,(f) = llf. fill (13.20.3), and it follows from (14.9.2) that if G is unimodular, LA(G, 8) is a closed two-sided ideal in Mr(G). We shall always identify LA(G, p) with this ideal (the identification of course depends on the choice of p). Under this identification, X,(G) is identified with a subalgebra (14.10.5) of MA(G), which is not, in general, an ideal (unless G is compact (14.9.1)) and, in general, is not closed, because its closure is LA(G, p) (13.11.6).

a

308

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

Remark (15.1.8) An algebra A over C , endowed with a topology, is said to be normable if its topology can be defined by a norm with respect to which A becomes a normed algebra in the sense defined above. For this it is necessary and sufficient that the topology of A should be definable by a norm, and should be compatible with the vector space structure of A, and that the mapping ( x , y ) x y ~of A x A into A should be continuous for this topology. Clearly these conditions are necessary. To see they are sufficient, observe that if llxll is a norm defining the topology of A, then there exists a constant a > 0 such that llxyll 5 a * [I.v[l . (lyl( for all x , y in A (5.5.1). If we replace the norm IIx(I by the equivalent norm (5.6.1) allxll = IIxII1, then clearly llxylll 5 llxlll * llylll. This establishes our assertion in the case where A has no unit element. If A has a unit element e # 0, we consider for each x E A the left translation L, : YH xy; this is a continuous endomorphism of the normed space A, and we have L,+,, = L, + L,. and L,, = I L , for all I E C , and L,,, = L, L,,. In other words, the mapping X H L, is a homomorphism of the algebra A into the algebra Y(A) of continuous endomorphisms of the normed space A. Moreover, this homomorphism is injective, because the relation L , = 0 implies x = xe = L,(e) = 0. Now we have 11L,11 g a [Ixll (5.7) and on the other hand, llxll = llxell 5 llLxll * Jlel(.It follows that if we put IIxI12 = llL,ll, then llxl12 is a norm on A which is equivalent to the given norm llxll (5.6.1), and for which A is a normed algebra. 0

PROBLEMS 1. Let A be the C-algebra of complex-valued functions defined and k times continuously differentiable on [0, 11. Show that the function

on A is a norm for which A is a Banach algebra. 2. Let (a,,) be a sequence of real numbers > O such that a,, = 1, a,+. lim a:'"

n-. m

= 0.

Let A denote the set of all formal power series x

complex coefficients & such that

2 m

"=O

m

=

"=O

5 a.,a.

and

&Tn with the

< + a.Show that A is a subalgebra of the

algebra C[[T]] of formal power series in T, and that llxll = for which A is a Banach algebra with unit element.

m "PO

~~15.1is a norm o n A

1 NORMED ALGEBRAS

309

3. Let I ( X ) be the set of continuous functions on the closed unit disk X : IzI 5 1 in C which are analytic in the interior Irl < 1 of X. For each pair of elements x , y E I ( X ) we define (x * Y ) ( 5 ) = 5

I'4 5

- t51Y(t5) dt

for all 5 E X. Show that this product makes d ( X ) a commutative algebra without a unit element, and that with respect to the norm induced by the norm on VC(X) the algebra d ( X ) is a Banach algebra. 4. (a) Let !2 be the set of finite real-valued functions w on R belonging to L'(R,h) (where h is Lebesgue measure) such that w ( - t ) = w ( r ) , w ( t ) 2 0 for all t E R, and such that w is decreasing on [0, a)[. Every function w E !2 is the uniform limit of an increasing sequence (w,) of step functions belonging to 52. Put

+

V(w) = w(0)

+1

+m

-m

w ( t ) dt.

If wl, w z E !2, show that w I * w 2 E and that V(wl * w z ) V ( w l ) . V(w2) (consider first the case where one of wl, w 2 is a step function). (b) Let d denote the set of &measurable complex-valued functionsf on R such that, for at least one function w E R, the function If12w-' is A-integrable (with the convention that 0 . (+ 00) = 0). We have R c d.For each functionf6 d,put

Show that also

Deduce that d is a complex vector space and that N, is a seminorm on A. (Observe that if a, b, a , p a r e real and >0, then

( a + b)2 c(+p

az

bZ

I-+-.) - a j3

Show that I c YA(R, h) n 9 $ ( R , A) and that Nl(f) 2 NB(f) and N2(f) 5 N B ( ~ ) . (c) L e t f s Y 1n Y 2be the function defined as follows:f(x) = l/n2 if 2"- 1 5 x 2" (n an integer L l ) , f ( x ) = 0 otherwise. Show t h a t f e d. (d) For each function f e d,show that there exists wf E 0 such that

s

(Consider a sequence (w.) of functions belonging to R such that V(w.) such that the sequence of integrals

= N,(f)

and

Iflzw;' dh tends to NB(f). Then consider the

function lim sup w , , and use Fatou's lemma.) n-r m

(e) The subspace -4'" of hegligible functions is contained in d.By passing to the quotient, N, induces a norm on the space A = d / N . Show that, with respect to this norm (denoted by N,(f) or I l f l l ) , the space A is a Banach space. (If (f.) is a sequence in

310

XV NORMED ALGEBRAS AND SPECTRAL THEORY

I such that (A) is a Cauchy sequence in A, show that the sequence (wrn)converges in 9, and that the sequence of classes of the functions If.12wi,,’converges in L’.) (f) Let I. be the interval [-n, n ] in R and d 1 I. the set of functions belonging to d whose support is contained in I,. Show that, for eachfE I ,the sequence of functions f~,,, tends to f i n d .The topologies defined on d I I. by Ne and by the seminorrn N, are equivalent. Deduce that the space S ( R ) is dense in d ,and that the Banach space A is separable. (g) Iffand g are two functions belonging to d ,show that

- 0, then y" = 1 x , and we write

+

+

y = (e

+

+

x)lIrn,

12. With the same hypotheses as in Problem 1 1 , let (Kj)l*j 0 such that llyxll 5 kllxyll for all x , y in A, then A is commutative. (Apply the inequality with e-{' in place of x and ecxy in place of y, and use (b).) (e) Let a E A be such that

+

+

ll(a 5 . 1)xIl 5 llx(a 5 . 1)Il for all 5 E C and all x E A. Prove that a belongs to the center of A. (Prove that, for each 5 E C and all integers n 2 no (where no depends on C),

for a l l y E A ; then let n tends to

+

00

and use (b).)

15. Let A be a Banach algebra with unit element e, and let C be a commutative Banach subalgebra containing e. Show that there exists a commutative Banach subalgebra B of A containing C such that SpR(x) -.Sp,(x) for each x t C.

3. CHARACTERS A N D SPECTRUM O F A COMMUTATIVE B A N A C H ALGEBRA. T H E GELFAND TRANSFORMATION

Let A be a commutative algebra (over C). A character of A is defined to be any homomorphism x of the algebra A into C which is not identically zero. Because x(1x) = Ax(x) for all scalars 1, this condition is equivalent to x(A) = C. If A has a unit element e # 0, then x(e) # 0 (otherwise ~ ( x=x(ex) ) = x(e)x(x)= 0 for all x E A); it follows that x(e) = 1, because x(e)2 =x(e2) = x(e). An ideal in # A in A is said to be maximal if there exists no ideal n such that in # n, it # A and in c n. If A has a unit element, in is maximal if and only if A/in is a (commutative)jeld. (15.3.1) (i) (ii) (iii) onto the

Let A be a commutative Banach algebra with unit element e # 0. For each x E A and each character x of A, we have ~ ( x E) SpA(x). Every character x of A is a continuous linear form with norm 1. The mapping x++x-'(O) is a bijection of the set of characters of A set of maximal ideals of A (which are therefore closed).

Assertion (i) is a particular case of (15.2.8(i)). It follows (15.2.4(iii)) that [x(x)l 5 ((XI(, which shows (5.5.1) that x is a continuous linear form with norm 5 1 ; and since x(e) = 1 and //el/= 1, we have lIxil = 1. Finally, since

3 SPECTRUM OF A COMMUTATIVE BANACH ALGEBRA

319

x(A) = C, the quotient algebra A/m is a field isomorphic to C, hence in is maximal. Conversely, let it be a maximal ideal of A. The fact that n is closed is a consequence of the following lemma: (15.3.1 . I )

If a # A is an ideal in A, then its closure ti is an ideal # A .

For the complement C a contains the set G of invertible elements of A , which is open in A (15.2.4(i)), and hence G is contained in the complement of ii. Applied to the maximal ideal n, this lemma shows that ii= it, because n and ii# A. The quotient normed algebra (15.1) A/n is therefore a Banach algebra (12.14.9) which is a field because n is maximal, hence bv the Gelfand-Mazur theorem (15.2.5) is isomorphic to C. In other words, there exists a unique homomorphism x : A -+ C such that x(e) = 1 and x - ' ( O ) = 11, and the proof of (iii) is complete. -

11

=)

(15.3.2) Let A be a Banacli algebra with unit element e # 0. The set X(A) of characters o j A i s a subset of the unit ball J/x'// 5 1 in the dual A' of the Banach space A, and is closed in A' with respect to the weak topology (12.1 5 ) . A is separable then X(A) is metrizable and compact for the weak topology.

In view of (12.15.9) we need only prove that X(A) is a closed subset of A', or equivalently that if u lies in the closure of X(A) in A' for the weak topology, then ( x y , u ) = (x, u ) ( y , u ) for all x,y E A, and ( e , u ) = 1. But this follows from the continuity of the mapping U H (x, u ) of A' into C (for each x E A) and the principle of extension of identities. The set X(A), endowed with the topology induced by the weak topology on A', is called the spectrum of A. For each x E A, the mapping x H ~ ( x of ) X(A) into C is denoted by 9 x or 9* x, and is called the Gelfand transform of x; the mapping X H ~ Xof A into CX(A)is called the Gelfand transformation. Hence we have by definition, for all x E A and all x E X(A), (15.3.3)

(Bx)(x) =

xw

(15.3.4) Let A be a separable commufatii3e Banach algebra with unit element e # 0. (i) The Gelfand transformation XH 9 x is a continuous homomorphism of the Banach algebra A into the Banach algebra g,-(X(A)) such that ii9xlJ= p(x) 6 llxi/and such that %e is the constantfunction I . (ii) The set of values of the continuous function 9 x on X(A) is equal to SpA(x).In particular, for x to be incertible it is necessary and s u ~ c i e n tthat 9 x should not vanish on X(A).

320

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

For each x E A, the mapping xw (.Yx)(x) = ~ ( x is) continuous on X(A), by the definition of the weak topology. Moreover, we have

( % v J ) ) ( x )= x(xA = x(x)x(y>= ((Yx)(x))((Yy)(x)) and therefore 9(xy) = (Yx)(Yy).The relation Y e = 1 is obvious. The equality )19x))= p ( x ) will follow from (ii) and the definition of p(x) (15.2.7). Consider therefore the first assertion in (ii). We know already that ~ ( xE) Sp(x) for all x E A and x E X(A) (15.3.1).Let us show conversely that, for each A E Sp(x), there exists a character x such that ~ ( x= ) 1, or equivalently that x(x - Ae) = 0. Since x - 1e is not invertible, the ideal A(x - 1e) generated by this element is distinct from A. In view of (15.3.1(iii)) we are therefore reduced to proving the following lemma:

(15.3.4.1) In a separable commutative Banach algebra A with unit element e # 0, every ideal a # A is contained in a maximal ideal. Let (x,) be a dense sequence in A. We define inductively an increasing sequence (a,,) of closed ideals # A (n 2 0) as follows. Take a, = ii (which is distinct from A , by (15.3.1.1)). In the quotient Banach algebra A / a , - , consider the image 2, of x, and an element A, of its spectrum (which by (15.2.4) is not empty), so that X, - A, C? (where 2 is the canonical image of e in A/ a,is not invertible in A / a , - , . It follows that the ideal a: in A generated by a, - and x, - A, e is distinct from A, and therefore a,, = i: is distinct from A. The union in' of the ideals a,, is therefore also an ideal #A, and hence the closure in of m' is also an ideal #A. We shall show that m is maximal, which will complete the proof. If e" is the unit element of A / m , and x; the image of x, in A/m, then by construction we have x: = A,e"; in other words, all the x i belong to the closed subalgebra (5.9.2) Ce" of A" = A/in. Since they form a dense subset of A" (3.11.4), it follows that A" = Ce", and therefore in is maximal.

Remarks (15.3.5) It should be noted that the Gelfand transformation x w Y x is not necessarily injective. For example, if x is nilpotent but not zero, say xk = 0, then the image of Sp(x) under cw is (0) ((15.2.3.1) and (15.2.1)) and therefore Sp(x) = { O } , hence ~ ( x = ) 0 for all characters x, although x # 0. This example also shows that IIYxll need not be equal to ((x((. But even if 9 is isometric (in which case the image %(A)is closedin %',-(X(A)) (3.14.4)), %(A)may not be equal to %,-(X(A)) (15.3.8).

c

3 SPECTRUM OF A COMMUTATIVE BANACH ALGEBRA

321

(15.3.6) Suppose that there exists an element xo E A which, together with e, generates a dense subalgebra of A (in other words, as P runs through the set C[X] of polynomials in X with complex coefficients, the subalgebra consisting of the P(x,) is dense in A). Then the mapping x H x(xo)is a homeomorphism of X(A) onto Sp,(x0).

Notice first that the elements P(xo) for which the coefficients of the polynomial P are of the form LY + pi with LY and p rational, form a dense subset of A, and therefore A is separable. The mapping X H X ( X ~ )is continuous and surjective (15.3.4), and X(A) is compact and metrizable (15.3.2) ; hence it is enough to show that this mapping is also injective (3.17.12). But if xl(xo)= x2(x0) for two characters xi, x 2 , then also xl(P(xo)) = x2(P(xo)) for all polynomials P; since x1 and x2 are continuous on A, we deduce that x1 = x2 (3.15.2), and the proof is complete.

Examples of Spectra and Gelfand Transformations (15.3.7) Let X be a metrizable compact space and A = qC(X) the Banach algebra of continuous complex-valued functions on X. Then the characters of A are the Dirac measures E, (x E X) (13.1.3), and the mapping XH E, is a homeomorphism of X onto the spectrum X(A). For in this situation we have A' = M,(X), since every measure on X is bounded (13.20), and the weak topology on A' is by definition the vague topology (13.4). A character is therefore a measure p # 0 (because p(1) = 1). We shall show that Supp(p) consists of a single point. If Supp(p) contained two points a # b, then a and b would have disjoint neighborhoods U and V, respectively, hence we should have functionsf and g belonging to A, with supports contained in U and V, respectively, and such that p ( f ) # 0 and p(g) # 0. But then fg = 0 so that p ( f ) p ( g )= p(fg) = 0, giving a contradiction. If Supp(p) = {x}, then the relation ~ , ( f )= 0 implies p ( f ) = 0, and therefore (A.4.15) there exists a scalar c1 such that p = a&,; since p(1) = 1, it follows that a = 1. The continuity of the mapping X H E , is immediately seen: for each f e A, we have (f, E, - E , ~ ) =f ( x ) -f ( x o ) , and since f is continuous, for each S > 0 there exists a neighborhood V of xo in X such that If(x) -f(xo)l 5 S for all x E V. In order to prove that XHE, is a homeomorphism, it is enough (12.3.6) to show that this mapping is injective; and this is clear, because E,, and E , ~have distinct supports if x1 # x 2 . We have ( % f ) ( ~ , = ) ~ , ( f )=f(x) for all x E X and all , f ~A, and therefore when we identify X and X(A) by means of the mapping X H E, , the Gelfand transformation becomes the identity mapping. (15.3.8) Let D be the closed unit disk 5 1 in C, and let A = d ( D ) be the closed subalgebra of gC(D)consisting of the continuous functions on D which

322

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

are analytic in the interior of D (1 5.1 S).We shall show that the characters of A are again in this case the restrictions to A of the Dirac measures E,(X E D). From this it will follow (15.3.7) that X W E , is again a homeomorphism of D onto the spectrum X(A). Letf,, be the canonical injection of D in C;then it is enough to show thatf, and the unit element I of A generate a dense subalgebra of A (for it is clear that SpA(f,,)= D, and the result will then follow by (1 5.3.6)). Now, let A, denote the subalgebra of A consisting of the functions which are restrictions to D of functions analytic on some neighborhood of D in C.This subalgebra is dense in A, because for each integer n > 0 and each function f E A, the functi0n.h which is the restriction to D of the mapping

r ++f((n/(n+ l)>O

of ((n + I)/n)D into C clearly belongs to A,,, and .f,-+ f uniformly on D as n + + co (3.16.5). On the other hand, every function f~ A, is the uniform limit (in D) of a sequence of polynomials P, , namely the partial sums of the Taylor series for f at the point 0 (9.9.1 and 9.9.2). Hence the set of polynomials in [ is dense in A, and our assertion is proved. We remark that here the Gelfand transformation is isometric, but %(A) is distinct from gC(X(A)). (15.3.9) Let B be the Banach algebra consisting of the restrictions to the unit circle U : = 1 of the functions belonging to the algebra A of (15.3.8). Then the restriction mappingfwfl U is an isometric isomorphism of A onto B: this is an immediate consequence of the maximum modulus principle (9.5.9), which implies that llfll = l l f l UII. Here the mapping XH E, of U into X(B) is again continuous and injective; but it is no longer a homeomorphism, because all the characters of A can be identified with characters of B, and therefore X(B) I= X(A) = D.

PROBLEMS 1.

Let X be a metrizable compact space and B a subalgebra of U,(X) containing the unit element. Assume that B is equipped with a norm llxlie which makes it a Banach algebra. (a) Show that llxlls2 /IxIl (the norm on U,(X)), by remarking that V ( t ): x HX(?) is a character of B for all t E X. Deduce that 0 is the only quasi-nilpotent element of B. (b) Show that 9) is a continuous mapping of X into X(B). If B separates the points of X (7.3) then is a homeomorphism of X onto a closed subset of X(B). (c) Suppose that, for each function x E B, the conjugate 2 belongs to B, and that if x E B is such that x ( t ) # 0 for all t E X, then the inverse x - I of x in VC(X) belongs to B. Show that in these conditions the mapping 9) is surjective. (Let tn be a maximal ideal of B, and let Z be the set of all E X such that x ( t ) = 0 for all x E m.Show that

3 SPECTRUM OF A COMMUTATIVE BANACH ALGEBRA

323

Z is not empty: if Z were empty there would exist a finite open covering (V,) of X , and for each j a function xl E rn such that xi(?) # 0 for all t E V l ; now consider the function x, Z 1 , which belongs to m.) I

(d) Deduce from (b) and (c) that the spectrum X(A) of the algebra of Section 15.1, Problem 1 can be canonically identified with the interval [0,1]. 2. Show that the spectrum of the algebra A of Section 15.1, Problem 2 consists of a single point, and that the unique maximal ideal of A is the radical of A (Section 15.2, Problem 7).

3. In the algebra d ( X ) of Section 15.1, Problem 3, show that llx"II 5 Ilxll"/(n- l)!, and deduce that d ( X ) is equal to its radical. Deduce that d ( X ) has no characters (use Problem 5 of Section 15.1). 4.

Let B be the Banach subalgebra of Vc(D) (notation of (15.3.8)) generated by d ( D ) and the function Ifo[ : [ H 151. Show that X(B) is homeomorphic to the set of points (x,, xz, xJ)E R3 such that x: x i 5 x: and 0 5 x3 1. (Remark that B contains all functions of the form [+ig(l[I), where g is a continuous function on (0, 11, and that B also contains the subalgebra A. of d ( D ) introduced in (15.3.8). To show that Ix(fo)l 5 x(/fol)for all characters of B, consider the function [([ - (If1 which belongs to B for all E > 0.

+

+

x

5. The space 1; (6.5) becomes a Banach algebra without unit element under the multiplication (fn)(vn)= (&vn). Let A be the Banach algebra obtained by adjoining a unit element to 1: (Section 15.1, Problem 5). Show that X(A) may be identified with the compact subset of R consisting of co and the integers 21 (use the fact that 16 can be canonically identified with its dual). The Gelfand transformation 9, then becomes the identity mapping, and 9 J A ) is a nonclosed dense subalgebra of V,(X(A)).

+

Show that the dual of the underlying Banach space of the Beurling algebra A (Section 15.1, Problem 4) can be identified with the space of classes of A-measurable complex-valued functions g such that

6. (a)

r+m

I

where the supremum is taken over the set of functions w E i2 such that V ( w )> 0. Then IlQll is the norm on the dual A' of A . Show that

The canonical bilinear form 0. (f) Let M be a proper closed B-submodule of =.Y'(p).Show that M is of the form q W ( p ) , where 141 = 1 . (Let N = M n Z 2 ( p ) ,which is a closed B-submodule of P2(p).I f f € M is not in the closure of BoM in .ILp'(p),deduce from (e) that f = g h , with g E N and h E X 2 ( p ) ;moreover g is not in the closure of BoN in .ILp2(p).Use (c) to show that N _=q3Ep2(p)with 191 = 1, and deduce that f~ qX'(p). Applying this , thatf, E 9-@'(p) result tof+f,, wheref' is in the closure of BoM in Y 1 ( p ) deduce and hence that M c 9 2 ' ( p ) . ) 16.

We shall apply the results of Problems 10 to 15 to the Dirichlet algebra B of (15.3.9), where p is the normalized Haar measure dp(6) = ( 2 ~ ) - 'd6 on the unit circle U, and is the characterft+f(O), so that p is the unique representative measure for (Problem 9(d)).

x

x

(a) B is the closure in gc(U) of the algebra of trigonometric polynomials

"

k=O

ckek"

3

SPECTRUM OF A COMMUTATIVE BANACH ALGEBRA

with exponents k 2 0. To each function function in the disk 1 z 1 < 1 defined by

f t Xp(p)

331

there corresponds the analytic

For 1 < p = < co (resp. p = i-co) f is the limit (resp. weak limit) in W ( p ) of the functions B+-tF(re") as r - I , so that HP(p) may be identified with the space of analytic functions F corresponding to the functions f E -P(p). (Use the fact that x p ( p )is the closure (resp. weak closure) of B in P ( p ) . ) In particular, H2(p) is identified with the space of functions m

f(z) == E C" z" "=a

which are analytic for IzI < 1 and such that

m "=O

'/,cI

< 4-m (cf. the problem in

Section 9.13). (b) Let u be a measure disjoint from p which is quasi-representative for Show that e - ' @. u is also a quasi-representative measure for and hence by induction that e - n i e . v is a quasi-representative measure for for all integers n > 0. Hence show that u = 0 and consequently (Problem 14) that every quasi-representative measure for has base p. (c) Let f~ , P 1 ( p )be nonnegligible (with respect to p). Then the function loglfl is p-integrable, or equivalently, J,(lfl) > 0, and consequentlyfis the product of an exterior function and an interior function. (This follows from Problem ll(b) if

x,

s / d p # 0. If

x.

x,

x

.c

f dp = 0 then f,identified with an analytic function on the unit disk,

is of the form f(z): z"g(z) where ir > 0 and g(0) f 0. We have y E Y l ( p ) and log If1 = log 191 on U.)In particular, a function belonging to Z ' ( p ) which vanishes on a non-p-negligible subset of U is p-negligible. Every function in 2 ' ( p ) is the product of two functions in . F 2 ( p )and conversely (Probleni 15(e)). A function f> 0 belonging to .4p1(p)is equivalent to a function of the form 1.4 I ', wherey t Xi"(p), if and only if J,(f) > 0 (or equivalently if and only if log If1 is p-integrable). (See Section 22.19, Problem 19.) 17. Let B be the Banach algebra considered in Problem 16, and BI the Banach subalgebra of B consisting of functionsfcontinuous on Iz/ S 1 and analytic on 1.7 < 1, and such thatf'(0) = 0. For each complex number c such that j1 , let p c denote the measure F, . p, where F,(z) = 1 - W(Ez). Show that the measures pc are representative measures for the character :f ~ f ( 0 ) the ; representative measures which are extremal points of are the pCsuch that / c / = 1 ; the nieasure po = p is the unique Jensen measure for and hence there exists no hyperextremal representative measure for

IcI

%(x) x;

18.

x

x.

Let A be a commutative Banach algebra with unit element. (a) Let u E A be such that p ( u ) < 1. Then for -each character of A the element 1 - x(u)u is invertible. If v = ( ~ ( u-) u)(l - x(u)u)-', then /x(u)I < 1. (b) If xI, x 2 are two characters of A, let o(x,,x.) denote the least upper bound of the number Ix2(u)l as u runs through the set of elements of A such that ~ ( u5) 1 and xl(u) = 0. Show that if p(u) 5 1 we have

x

IXI(4 -

X Z W 5 O(y.1,

Xdll

-

xI(u)xz(u)l

332

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

(use (a)). Deduce that

dx1, XZ) = d x 2 ,xI) and that

Deduce from this inequality that, if

xl, xz, x3 are three characters of A, we have

and hence that the relation o(x', x") < 1 between characters x', x'' of A is an equiuulence relurion on X(A). The equivalence classes for this relation are called Gleuson parts of X(A). (c) Let u E A be such that p(u) 5 1, and let xl, x2 be two characters of A, such that xl(u)= 0. Show that there exists a complex number such that Ihl < 1 and such that, if we put v = (A - u)(1 - xu)-' E A, then xl(u) = A and x2(v)= -A. Deduce that

(d) I f

xI, xz are two characters of A, put

Show that

(use (c)). The equivalence relation defined in (b) is therefore equivalent to T(x', x") < 2. (e) Let u E A be such that Wx(u) 2 0 for all x E X(A). Show that, for any two characters xl, x 2 ,we have

(Apply (*) to the element e-'", where f > 0, and let f tend to 0.) 19. With the hypotheses of Problem 8, let xl, x2 be two characters of B. (a) Let pI,p 2 be representative measures for xl, x2 respectively (Problem 9). If a(xI, x2)= 1 (Problem 18), show that p1 and p2 are disjoint (13.18). (Write p2 = h . p1 Y where v is disjoint from pl.and by majorizing Ix2(u)- x~(u)Ifor u E B

+

1

2 I , show that the hypothesis implies that (h + 1) dpl $ j l h - 11 dp1.) (b) If o(x1,x2) < 1, show that there exist representative measures pI for x1 and and IIull

p 2 for

x2 such that

+ u(xl, ,y2))-'. Use Problem

(Put c = (1 - cr(xI,x2)(1

18(e) and the Hahn-Banach

3 SPECTRUM OF A COMMUTATIVE BANACH ALGEBRA

333

theorem to show that there exist two positive measures a, /3 on X such that

xJu> - c x I ( u )= / u dol and xl(u)- cx&)

=/ u

dp.) Deduce that if

x

x is a character

of B, then the Gleason part (Problem 18(b)) containing is the set of characters for which a representative measure has base p, for at least one representative measure p for x. (c) Suppose x is a character for which the representative measure is unique. If belongs to the Gleason part containing x , show that the representative measure p' for is also unique, and that

x'

x'

x

(d) Suppose that there exists a hyperextremal representative measure p for (Problem 11). Show that if a character x'admits a representative measure p' with base p, then p' is the only representative measure for x' with base p. (If p ' = f . j and = g . p are two representative measures for show that f- g is zero almost everywhere with respect to p, by using Problem 13(j).)

x',

PI'

20. Let A and B be two commutative Banach algebras having unit elements and h : A

+B

a homomorphism which sends unit element to unit element. show that the (a) Let I' be the graph of h in A x B. For each character x of mapping (x,y ) Hx(h(x)) - x(y) is continuous on A x B. Deduce that if (a. b) is in the closure of I?, then ,-&(a)) = ~ ( b ) . (b) Deduce from (a) that if B is without radical (Section 15.2, Problem 7) then h is necessarily continuous (use the closed-graph theorem). In particular, for a commutative C-algebra having a h i t element, in which the intersection of the maximal ideals

n,

is {O},two norms which define Banach algebra structures are necessarily equivalent. 21.

Let 9 be the algebra of indefinitely differentiable complex-valued functions on [0, I]. (a) Let A be a subalgebra of 9, containing the unit element and endowed with a norm which makes A a Banach algebra. Show that there exists a sequence (mn)n30 of finite real numbers 20 such that, for all functions x E A, we have sup lx(")(t)l = O(mA

O O on X(A); use (15.3.4). If a character x of A is not hermitian, then there exists a self-adjoint y E A such that x(y) = i, whence x(1 yy*) = 0, and 1 + y y * is not invertible.) Hence give an example of a commutative Banach algebra with involution, having a unit element, for which there exist nonhermitian characters.

+

+

+

4.

(a) If A is a star algebra with unit element, then llx112 = p ( x * x ) for all x E A (prove this first when x is self-adjoint). Deduce that A is without radical. (b) Let A be the algebra of Problem 2 of Section 15.1. If x m

=z "=O

&T", define x* t o be

&T".This defines an involution on A for which A is a commutative Banach algebra

"20

with involution, having a unit element, and such that the only character of A (Section 15.3, Problem 2) is hermitian.

5.

-

With respect to the involution f*(r) = f ( - r ) on the Eeurling algebra (Section 15.1, Problem 4 and Section 15.2, Problem 6), show that all the characters are hermitian.

6. On the algebra A of Problem 7 of Section 15.3, consider the involution induced by

f ~ f and , extend this involution to A by putting e * with involution. Find its hermitian characters.

= e.

Then A is a Banach algebra

4 BANACH ALGEBRAS WITH INVOLUTION. STAR ALGEBRAS 7.

341

(a) In a star algebra, every hermitian quasi-nilpotent element (Section 15.2, Problem

5) is zero.

(b) Let A be a star algebra with unit element, and let x be an element of A which is not left-invertible. Then x*x is not invertible in the star subalgebra B of A generated by 1 and x*x, and therefore there exists a sequence (y.) of elements of B such that lIynll = 1 and lim y.x*x = 0 (use the Gelfand-Neumark theorem). Deduce that a n+m

noninvertible element of A is necessarily a (left or right) topological zero-divisor.

8. Let A be a star algebra with unit element, x a normal element of A, and S = SpA(x). Then there exists a unique homomorphism of the star algebra V,(S) into A, transforming unit element into unit element, and such that ~ ( 1 , ) = x. This homomorphism of V,(S) onto the star subalgebra of A genis an isometry, which sends f to erated by 1, x and x*; all the elements of this subalgebra are therefore normal. Put ~ ( f=f(x), ) and show that if f i s analytic in a neighborhood of S, then the element f(x) is equal to the element so denoted in Section 15.2, Problem 11.

v(f)*,

9.

Let A be a star algebra with unit element. Let P denote the set of self-adjoint elements of A such that SpA(x) C [0, cc [. (a) Show that if x E A is self-adjoint and jle - X I / 5 1, then x E P. If x E P and llxll 5 1, then lle -xIl 5 1. (Consider the subalgebra generated by e and x . ) Show that a self-adjoint element x belongs to P if and only if IIx - IIxlleIl 5 IIxII. (b) Deduce from (a) that P is a closed convex cone in A, such that P n (-P) = (0). (c) Show that the relation x*x E - P implies that x = 0. (Observe first that also xx* E -P, and by writing x = u iu, where u and u are self-adjoint, deduce that x*x E P, whence x*x = 0.) (d) Deduce from (c) that x*x E P for all x E A. (Write x*x = u - v, where u, u are hermitian and belong to P, and uu = uu = 0 (use Problem 8). If z = xu, show that z*z E -P, and deduce that u = 0.) Deduce that e x*x is invertible in A (use Problem 8).

+

+

+

Let A be a commutative Banach algebra with unit element,and let XHX* be any involution on A. For each character x of A, put x*(x) = x(x*). Show that XI--+X* is an involutory homeomorphism of the space X(A). (b) Let X be a metrizable compact space, and v an involutory homeomorphism of X onto X. For eachfE FC(X),putf*(x) =f(v(x)). Show that this defines an involution on Vc(X), and that every involution on V,(X) may be so obtained. (c) Let X be the compact subspace of RZ which is the union of the segment y = 0, - 1 5 x 5 2, the segment x = 0, 1 5 y 5 2, the circle x 2 y z = 1 and the open halfdisk D : y > 0 , xz y z < 1. Assume that the only involutory homeomorphism of X onto X is the identity (Chapter XXIV). Let A be the Banach subalgebra of %‘,(X) consisting of functions which are analytic in D. Show that the spectrum of A can be canonically identified with X, and that the algebra A has no involution other than the identity.

10. (a)

+

+

11. Let A be a noncommutative star algebra with unit element.

(a) Show that there exists a hermitian element y E P (notation of Problem 9) such that y is invertible and y’ is not in the center Z of A. (Using Problem 9, show that if the result were false, the intersection of Z with the real vector subspace H of A consisting of the hermitian elements would contain a neighborhood of e in H, which would imply that A was commutative.)

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NORMED ALGEBRAS AND SPECTRAL THEORY

(b) Deduce from (a) that, if we put x'=y-'x*y for all X E A, then continuous involution on A such that (x')* # (x*)' for some x E A. 12.

is a

Let E be a Hilbert space. Show that a self-adjoint compact operator U is a HilbertSchmidt operator if and only if, (h.) being the sequence of eigenvalues of U each counted according to its multiplicity, the s u m z Ih.1* is finite. The sum is then equal to

13.

XHX'

I1

w:.

n

(a) The norm llullz is defined as in (15.4.8) when E is a finite-dimensional Hilbert space. In a Hilbert space of dimension n, give an example of an operator u such that

(b) Let E be a Hilbert space of infinite dimension. Give an example of a sequence (u,) of Hilbert-Schmidt operators on E such that the sequence of numbers ~ ~ u ~ u ~ tends ~ ~ z to/ 0 ~ as ~ nu tends , J ~ to ~ m. 14. Let X be a (metrizable, separable) locally compact space, and let p be a positive measure on X. Let K(x, y) be a function on X x X, belonging to 2&(X x X, p 0p).

For each function f~ 2 6 ( X , p), the function

XH

s

K(x, y ) f ( y ) dp(y) is defined al-

most everywhere, and its class belongs to L&(X,p) (use the Lebesgue-Fubini theorem and the Cauchy-Schwarz inequality). If this class is denoted by U . A show that U is a Hilbert-Schmidt operator; U is said to be associated to the kernel K. Show that IILIIIz = N,(K) and that I/* is associated to the kernel ( x , y ) ~ K ( x). y , If U1, Uzare associated to kernels K I , K2 respectively, belonging to Y & ( X x X, p @ p), then U,Uzis associated to the kernel

~ ( xy ,) = s ~ xt)K2(r, , y ) dp(t). 15.

Let E be an infinite-dimensional Hilbert space and U a Hilbert-Schmidt operator on E. If h is a regular value for U,show that the operator (U- h . lE)-l is of the form V - h-' . l E , where V is a Hilbert-Schmidt operator. Deduce that the spectrum of U in the algebra Y(E) is the same as in the algebra obtained by adjoining a unit element to Y,(E). For every function f which is analytic in an open set containing Sp(U), the operatorf(U) (Section 15.2, Problem 11) is a Hilbert-Schmidt operator.

16.

Let HI,...,H. be self-adjoint operators, each pair of which commute, on a separable Hilbert space E, and suppose that liHJll,5 1 and that IIHJHkll5 dJ-lrl for all j , k, where E is a real number such that 0 5 E < 1. Then lIH1

+ . . . + H"ll5 (1 + +)A1

-E)

(" Cotlar's lemma").

(Using the Gelfand-Neumark theorem, reduce to proving the same property for real numbers ul,. . . , u.. For each t 2 0, let v(r) be the number of indices j such that Ju,I 2 t . Remark that

2 lul

J= I

+

=Jol v ( t ) dt

and prove that v ( t ) 5 1 [2 log tllog E ] , by observing that if lull 2 t and then dJ-*l 2 IuJukl 2 t 2 . )

IUkl

2 I,

4

BANACH ALGEBRAS WITH INVOLUTION. STAR ALGEBRAS

343

17. Let A be a Banach algebra with unit element e, and let x-x*

be an involution on A which is not necessarily continuous. (a) Show that the radical ‘31 of A (Section 15.2, Problem 7) is a self-adjoint subset of A . Passing to the quotient, it follows that the involution on A defines an involution (also written zt+z*) on A/%. (b) Let a = a* be a self-adjoint element of A, such that SpA(a)is contained in the open half-plane 9[ > 0. Then there exists an element b E A, which is the limit of a sequence (u,) of self-adjoint elements which are polynomials in a, and which satisfies b Z = a. (Write a = o r ( e - ( e - .-‘a)) for a suitable a > O , and use Problem Il(f) and Section 15.2, Problem 15.) Show that 6 and b* commute, and that we may therefore write b = u iu, where u and v are self-adjoint and commute with each other and with a. Deduce that uu = 0 and u2 - v 2 = a. (c) Let C be a commutative Banach subalgebra of A which contains e, a, b, b* and is such that Sp,(b) = SpA(b) (Section 15.2, Problem 15). Let C be a commutative Banach subalgebra of A containing the image of C under the homomorphism : x-x* of C into A. If W’ is the radical of C and T : C C / W ’ the canonical homomorphism, then the homomorphism T 0 9 : C + C’/W’ is continuous (Section 15.3, Problem 20).

+

--f

1

Deduce that u = - (6* - b) E and thence that SpA(a)= SpA(uz). (Observe that 2i Sp,(a) = S p c , , a ( . ( ~ ( u-zu z ) ) = SpC./st.(7i(u2)).)Consequently u is invertible, u = u - ’ u u = 0, and finally b = u is self-adjoint. Furthermore, if Sp(a) is contained in 10, a[,then the same is true of Sp(b). %I,

+

18. With the hypotheses of Problem 17, put pA(x)= (p(x*x))’l2 (also denoted by p(x)). We havep(x*) = p(x); also if W is the radical of A and if T :A + A p t is the canonical homomorphism, then P ~ / ~ ( T ( X=)pA(x). ) The involution x-x* on A is said to be hermitian, and the involutive algebra A is said to be hermitian, if Sp(a) c R for all self-adjoint elements a E A. If A is commutative, an equivalent condition is that every character of A is hermitian (Problem 2). Assume for the rest of this Problem that A is hermitian. (a) Let x E A be such that p(x) < 1 . Show that (e i-x*)(e - x) is invertible. (Remark that e - x*x = w2, where w is self-adjoint and invertible, by virtue of Problem 17; hence show that the spectrum of w-l(x* - x)w-’ is contained in iR.) Deduce that for all x E A (Ptak’s inequality). Consequently, if x is normal, we have p(x) = p ( x ) . (Remark that if x, y are two ) by (15.3.4).) elements of a Banach algebra which commute, then ~ ( x y 0 such that Iln(x*)Il 5 clla(x)ll, and that pA(x) = ~ A / w ( T ( x ) ) . ) (h) Show that, for each x E A, Sp(x*x) is contained in [0, m[. (Argue by contradiction: suppose that there exists x E A such that - 1 E Sp(x*x). Write z = x * x , and for each positive integer n let w. be a self-adjoint element of A which commutes with z,

+

+

+

and is such that w:

= z’

+ -n1 e and Sp(w,) c 10, +

03

[ (Problem 17). Let b, = w, - z.

By considering a commutative Banach subalgebra B of A containing w, and z, and such that the spectra of z, w., and b. are the same in B and in A, show that

+

Show also that p(b,) 5 1 2p(z) = a, independent of n (use (c)). Put y. = xb,, so that y : y n : - biw. - 1 ~( l / n ) b n ;deduce that Sp(y,fy,) is contained in the interval 1- a),a/n],and then that Sp(y,y,*) is contained in the interval [- a/n, m[ (use (c)); hence that p(y,*y.) -< a/n (Section 15.2, Problem 2(b)), and consequently that 1x(y,*y.)) 5 a/n for all characters x of B. Finally, obtain a contradiction by noting that y:yn = biz, that ~(w,,) 2 0 and that there exists a character x such that x ( z ) = - 1 .)

+

19.

Let A be a Banach algebra with unit element, and let x - x * be a (not necessarily continuous) involution on A. Show that the following properties of A are equivalent: ( a ) A is hermitian. (fl) p(x) ( p ( x ) for all normal x E A. ( y ) p(x) = p ( x ) for all normal x E A. (6) p(x* x ) 5 2p(x) for all x E A. ( E ) P(X v) 5 p ( x ) P(P) for all x , y E A. (5) The set of numbers p(x), where x runs through the unitary elements of A, is bounded. (7) For each x E A the element e t-x*x is invertible in A. (To show that (7) implies (a), argue by contradiction by showing that the spectrum of a hermitian element cannot contain the number i. To show that ( y ) implies (a), argue as in (15.4.12). To show that (6) implies (fl), remark that (8) implies that p(x)” = p(x”) 5 2p(x)” for x normal and n 2 I . Finally, to show that ([) implies (a), notice first that (5) implies that Sp(x) c U for all unitary elements x , by considering the powers x” ( n E Z). Next observe that if a is self-adjoint and p(a) < 1, there exists a self-adjoint element h, commuting with a, such that b2 = e - a a 2 (Problem 17), hence a ib is unitary. Then consider a commutative Banach subalgebra containing a and band such that the spectra of a, b and a ib are the same in A and B).

+ +

+

+

+

5 REPRESENTATIONS OF ALGEBRAS WITH INVOLUTION

345

20. Let A be a Banach algebra with unit element e, and let x - x * be a (not necessarily continuous) involution on A. Show that if IIx/l= < p ( x ) ,then A is a star algebra.

5. REPRESENTATIONS O F ALGEBRAS WITH I N V O L U T I O N

Let A be an algebra with involution (not necessarily endowed with a norm, and not necessarily having a unit element), and let H be a Hilbert space. A representation? of A in H is a homomorphism SH U(s) of A into the algebra 9 ( H ) of continuous endomorphisms of H, such that (15.5.1)

U(s*) = (U(s))*

This implies in particular that if s is self-adjoint, then so is U(s). If A has a unit element e, we require in addition that (15.5.2)

U(e) = 1.,

The representation U is said to be faithfulif the homomorphism SH U(s)is injective, that is if the relation U(s) * x = 0 for all x E H implies that s = 0. Let H, H’ be two Hilbert spaces. A representation SH U(s)of A in H and a representation SHU’(S) of A in H’ are said to be equivalent if there exists a Hilbert space isomorphism T : H --+ H’ such that U‘(s)= TU(s)T-’ for all s E A. When H = H’, the Hilbert space automorphisms T of H are precisely the unitary elements of the star algebra 9 ( H ) . For T must be invertible and satisfy(T.xlT.y)= (xIy)forallx,yinH,so that(T*T.xly)=(xly)and therefore T*T = l,, hence T* = T-’; the converse is immediate. Let H,, H, be two Hilbert spaces, S H Ul(s) a representation of A in H,, and SH U,(s) a representation of A in H,. Let H be the Hilbert sum of H, and H, , so that H, and H, are identified with supplementary subspaces of H. If x = x, + x2 and y = y , + y , are two elements of H (where x i , y i in Hi for i = 1, 2), then (6.4) (x I Y>= (x1 I Y1)

+ (x2 I Y 2 ) -

If we put U(s) . (xl + x,) = U , ( s ) . x , + U,(s) . x, , it is immediately verified that U(s)E 9 ( H ) for each s E A, and that SH U(s) is a representation of A, called the Hilbert sum of the given representations.

t Strictly speaking, a unitary representation. Since we shall not consider other types of representations, we shall suppress the word unitary,” by abuse of language. ‘I

346

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

Let SHU(S) be a representation of A in a Hilbert space H. A vector subspace E of H is said to be stable with respect to this representation if U(s)E c E for aN S E A . If E is stable with respect to U , then so is the closure E of E (3.11.4). If E is closed and if E’ is the orthogonal supplement of E in H (6.3), then E’ is also stable with respect to U. For if x E E and x’ E E‘, we have (x 1 U(s) . x’) = ((U(s))*xI x’) = (U(s*)x I x ’ ) = 0 by hypothesis, hence U ( s ) .x’ is orthogonal to all x E E and therefore belongs to E‘. If U,(s)and U,(s) are the restrictions of U(s) to E and E‘, respectively, then the representation U is the Hilbert sum of U , and U , .

(15.5.3) For a closed subspace E of H to be stable with respect to U , it is necessary and sujicient that PEU(s) = U(s)PEf o r all s E A, where PE is the orthogonal projection on E (6.3). The condition is necessary, for if X E E we have P,. x = x and P,. (U(s).x) = U ( s ) . x, because E is stable with respect to U ; and if x E E’ we have P , x = 0 and P, * (U(s) * x) = 0, because E’ is also stable with respect to U. Conversely, if the given condition is satisfied, then U ( s ) x = P, U(s)* x E E for all x E E and all s E A.

On a Hilbert space H, an orthogonal projector is by definition any continuous operator on H which is an orthogonal projection onto a closed subspace of H. The importance of such projectors is due to (1 5.5.3) and to their characterization in terms of the structure of algebra with involution of Y(H) : (15.5.3.1) A continuous operator P on a Hilbert space H is an orthogonal projector if and only if it is idempotent and hermitian. The necessity of these conditions has already been proved (1 1.5). Conversely, if P z = P = P*, then ( P . X I y - P . y ) = (XI Pa y - P 2 y ) = 0 for all x , y E H. Since P(H) is also the kernel of 1, - P , it is closed, and H is the Hilbert sum of P(H) and P -‘(O). Hence the result.

-

Suppose that H is the Hilbert sum of an infinite sequence (H,) of subspaces which are stable with respect to the representation U. Let U,(s)denote the restriction of U(s) to H,, so that for each n the mapping SH U,(s)is a representation of A in H, By abuse of language, the representation U is said to be the Hilbert sum of the representations U, . For each s E A and each x = x, E H, where x, E H, for each n, we have U(s) * x = U,(s) x, , and

.

1 n

n

1 IIUn(s).xnII2 n

=

IIu(s).xIIZ.

5

REPRESENTATIONS OF ALGEBRAS WITH INVOLUTION

347

A representation SH U ( s )of A in H is said to be topologically irreducible if there exists no closed vector subspace E of H, other than (0) and H, which is stable with respect to U. From (15.5.3) we obtain the following irreducibility criterion: (15.5.4)

In order that U should be topologically irreducible, it is necessary and suficient that the only orthogonal projections P such that PU(s) = U(s)P for all s E A should be 0 and 1,. For this condition expresses that ( 0 ) and H are the only closed subspaces of H which are stable under U .

(15.5.5) Let SH U(s) be a representation of A in H , let E be the closure in H of the vector subspace generated by the elements U(s) . x for s E A and x E H , and let E‘ be the set of all x E H such that U(s) . x = 0 .for all s E A. Then E and E’ are stable with respect to U and are orthogonal supplements of each other in H . Since U(st) = U(s)U(t),it is clear that E and E’ are stable subspaces of H. Let E” be the orthogonal supplement of E in H. We have seen earlier that E” is stable with respect to U. But if x E E”, we have U(s) x E E by definition, hence U(s). x E E n E” = (0) for all s E A, so that E” c E’. Conversely, if x E E’, s E A and y E H, we have (x I U ( s ) .y ) = (U(s*) . x 1 y ) = 0 by definition, so that x is orthogonal to E, and therefore E’ c E”. The subspace E is called the essential subspace for U . If E’ = { 0 } , the representation U is said to be nondegenerate. An equivalent condition is that the elements U(s) . x should form a total subspace of H ; by (15.5.2), this is always the case if A has a unit element.

A vector xo E H is a totalizer or totalizing vector for a representation U of A in H if the vector subspace of H generated by the transforms U ( s ) .xo of xo , as s runs through A, is dense in H . (In any case, this subspace is stable with respect to U . ) A representation which admits a totalizer is said to be topologically cyclic. If U is topologically irreducible, every nonzero vector xo E H is a totalizer, and conversely.

(15.5.6) Suppose that A has a unit element. Let SH U(s) be a nondegenerate representation of A in a separable Hilbert space H . Then H is the Hilbert sum of a sequence (H,) ( f n i t e or infinite) of closed subspaces, stable with respect to U and such that the restriction of U to each H , is topologically cyclic.

348

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

Let ( x , ) , ~be~ a dense sequence in H. We define the H, inductively, as follows. HI is the closure of the vector subspace of H generated by the U(s) . xl, as s runs through A; we have x1 E H I , because A has a unit element. If H,, ... , H,-l have been constructed, it may be the case that H is equal to the (direct) sum L of the H i , and the construction stops. If not, let L' be the orthogonal supplement of L in H, and let p(n) be the smallest integer such that, if xi is the orthogonal projection of x,(,) on L', the vector subspace HL of L' generated by the U(s) xi is not equal to (0): there exists such an index because the representation U is nondegenerate, and we take H, to be the closure of H i . Since x; E H, , the subspaces H, satisfy the required conditions, by virtue of (6.4.2). (15.5.7) If A is a Banach algebra with involution, having a unit element, then every representation SH U(s) of A in a Hilbert space satisjies IIU(s)ll 5 llsll (and in particular U is a continuous mapping of A into Y(H)).

We have IlU(s)l12 = IIU(s)*U(s)ll = p(U(s)*U(s))in the star algebra 9 ( H ) (15.4.14.1). Since U(s)*U(s)= U(s*s),it follows from (1 5.2.8(i)) that P(W*S))

5 P(S*4 5 IIs*sII 2 IIs*II . llsll = llS1l2.

In particular, we recover in this way (15.3.1(ii)),

6. POSITIVE LINEAR FORMS, POSITIVE HILBERT FORMS AND REPRESEN T A T l O NS

Let A be an algebra with involution (not necessarily normed, and not necessarily having a unit element). A linear form f : A + C on A is said to be positive if (15.6.1 )

f(x*x) 20

for all x E A. (15.6.2) Let f be a positive linear form on an algebra A with involution. (i) The mapping ( x , y ) H g ( x , y ) =f ( y * x ) is a positive hermitian form on A x A: in other words, (1 5.6.2.1)

for all x , y in A.

f(x*r) =f(v*x)

6 POSITIVE LINEAR FORMS, POSITIVE HILBERT FORMS

349

(ii) For all x, y in A we have

(iii) If A has a unit element e , then (1 5.6.2.3) (1 5.6.2.4)

To prove (i), express that g ( x + Y , x + Y ) = g(x9 4

+ g(x9 Y ) + 9(Y, 4 + 9(Y, Y )

is real: by virtue of the hypothesis (15.6.1), we obtain

m ( Y , 4)= -JW,39); changing x into ix, this becomes %?(g(x,y ) ) = &(g(y, x)), hence g ( y , x) = g(x, y ) . Assertion (ii) is the Cauchy-Schwarz inequality (6.2.1) applied to g . The relations (iii) are obtained by replacing y by e in (15.6.2.1) and (15.6.2.2), and using (15.4.2). The hermitian form g so obtained from f is not arbitrary, because it clearly satisfies the relation

for all x, y , z in A. We therefore make the following definition: A positive Hilbert form on the algebra with involution A is a positive hermitian form on A x A which satisfies the relation (15.6.3). If A has a unit element, every positive Hilbert form g comes as above from a positive linear f o r m 8 we have only to definef(x) = g(x, e). But we shall see later that this is no longer the case when A has no unit element (15.7.4). We shall now show that there are remarkable relationships between Hilbert forms on A x A and representations of A. In the first place, every representation SH U(s) of A in a Hilbert space H gives rise to positive linear forms (and hence to positive Hilbert forms) in the following way: for each xo E H, if we put

350

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

then it is clear thatf,, is a linear form on A, and we have .fx,(s*s) = (W*)W) * xo I xo) = ( U s ) xo I U(S>

XO) =

II W ) xo112 > 0 *

by virtue of (15.5.1). The corresponding Hilbert form is

For example, if A = Y(H) where H is a Hilbert space offinite dimension n, and 1, : TH T is the identity representation of A in H, then the form f,, is calculated explicitly as follows: if ( e J l s i 5 n is an orthonormal basis of H, and

(cij) the matrix of T with respect to this basis, then for x , = 1A,ei we have i= 1

(15.6.6) When studying positive linear forms of the type (15.6.4) we may always assume that xo is a totalizer for U (15.5), because f , , is unchanged when we replace U by its restriction to the stable subspace of H which is the closure of the stable subspace generated by x o and the U ( s ) .xo (s E A). Under this additional assumption, the form f,, determines the representation U up to equivalence : (15.6.7) Let U, U p be two representations of A in Hilbert spaces H, H’, respectively, and suppose that U (resp. U ’) has a totalizer xo (resp. xb). Then if (U(s). xo I x o ) = (U’(s). xb I xb) for all s E A, the representations U and U ‘ are equivalent . For all s. t in A we have (Lys) . xo I U ( t ) . xo) = (U(t*s) * xo I xo)

(15.6.7.1)

= ( U ’ (t*s) ’ x; I xb)

= ( U ’ ( s ) .xb I U ’ ( t )* xb).

Since the vectors U ( t ) . x o (resp. U ’ ( t ) . x b ) form a dense subspace H, (resp. Hb) of H (resp. H’), this proves already that U(s). xo = 0 if and only if U’(s) . xb = 0 . It follows that, for each z E H, and each s E A such that U ( s ) . x o= z , the vector U ’ ( s ) * x bis constant and equal to say z‘ = T z E Hb . By (15.6.7.1) the mapping T is an isomorphism of the prehilbert space H, onto the prehilbert space Hb , which extends uniquely to an isomorphism (also denoted by T ) of the Hilbert space H onto the Hilbert space H’ (by virtue of (5.5.4) and the principle of extension of identities). It

POSITIVE LINEAR FORMS, POSITIVE HILBERT FORMS

6

351

remains to be shown that T . (U(s) . z ) = U ’ ( s ) . ( T . z), and by the principle of extension of identities we may assume that z is of the form U ( t ) * x, . But then U(s). ( U ( t ) . xo) = U(st ) x, , and therefore 1

T . ( U ( s t ) .x,)

=

U ‘ ( s t ) .X,

=

U ’ ( S ) (. U ’ ( t ) .x,)

=

U ’ ( S )* ( T . z).

This completes the proof. Furthermore, the prehilbert space H, can be deJned directly in terms of the form f x o, and even in terms of the Hilbert form gxo. We have the following general proposition : (15.6.8) Let g be a positive Hilbert form on A. Then the set n of elements s E A such that g(s, s ) = 0 is a left ideal of A, and is equal to the set of s E A such that g(s, t ) = 0 for all t E A. If II : A + A/n is the canonical linear mapping, then there exists a unique structure of prehilbert space on A/n such that ( I I ( S ) I rc(t)) = g(s, t)for all s, t E A.

The Cauchy-Schwarz inequality Ig(s, t)I2 5 g(s, s)g(t, t ) (6.2.1) shows that n is the set of all s E A such that g(s, t ) = 0 for all t E A. The relation (1 5.6.3) then proves that n is a left ideal in A. By virtue of the relation g(t, s) = g(s, t), we have g(t, s) = 0 for s E n and t E A ; it follows that if s - s ‘E n and t - f’ E n (that is, if z(s) = n(s’) and n(t) = n(t’)) then g(s, t ) = g(s’, t’), because g(s, t ) - g(s , t’) = g(s, t - t’) + g(s - s’,t’). Since g(s, t ) depends only on I I ( S ) and n(t), we may write g(s, t ) = (n(s)In(t)),and it is straightforward to check that the function (x I y ) so defined on (A/n) x (A/n) is a non-degenerate positive hermitian form. In the case where g = gxo, we have gxo(s,s) = 11 U(s) . xO(l2,hence n is the kernel of the linear mapping u : SH U ( s ) .x, of A onto H,. The formula (15.6.5) shows that the bijective mapping S : A/n + H, induced by u is a prehilbert space isomorphism. This shows, as stated earlier, that the Hilbert space H is determined by gxo up to isomorphism. Further, the representation S H U(s) can be reconstructed from the algebra structure of A and the form gxo, because

-

-

U(s) (U(t) x,)

=

U(S) (S . n(t))= S . n(st);

and once U(s) is known in H,, it extends uniquely by continuity to H (5.5.4).

Nevertheless, positive Hilbert forms of the type gxo are not the most general ones, for they satisfy an additional condition which expresses that each

352

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

U(s)is a continuous operator on Ho . In view of (15.6.5) and (5.5.1), this continuity is equivalent to the following condition on the form g = gxo : (U) For each s E A there exists a real number M, 2 0 such that

for all

tE

A.

Conversely : (15.6.10) Let g be a positioe Hilbert form on A, satisfying the condition (U), and suppose that (with the notation of (15.6.8)) the prehilbert space A/n is separable, so that (6.6.2) it can be identijied with a dense vector subspace of a Hilbert space H. Then for each S E A the endomorphism n ( t ) H n ( s t )of A/n extends to a continuous endomorphism x H V(s) . x of H, and s H V(s)is a representation of A in H. If n(t) = n(t’), then n(st) = n(st)’) because n is a left ideal. Hence the endomorphism of H, under consideration is well-defined, and the definition of the scalar product on A/n, together with (15.6.9), shows that this endomorphism is continuous (5.5.1). Hence the existence of the continuous operator V(s) (5.5.4). Since n((ss’)t) = z(s(s’t)), we have V(ss’) = V(s)V(s’). Also, by virtue of (1 5.6.3)

(V(s*)* n(t)I n(t’)) = g(s*t, t’) = g(t, s t ’ ) = g(st’,

t ) = (V(s). n(t‘) I n(t)) = (n(t)I V ( S ). n(r’))

which shows that V(s*)= (V(s))*. Finally, if A has a unit element e, then evidently V(e)= 1, , and therefore S H V(s) is a representation of A in H. It is useful to know when the representation SH V(s)so defined is nondegenerate (15.5.5). This is equivalent to the following condition on g : (N) The elements n(st)form a total set in the prehilbert space A/n. This condition is trivially satisfied when A has a unit element. Under the conditions of (15.6.10),there does not in general exist an element

xo E H such that g(s, t ) = (V(s) . xo I V(t) . xo) (cf. Section 15.9, Problem 3).

However, such a vector always exists when A has a unit element e: we may take xo = n(e).

6

POSITIVE LINEAR FORMS, POSITIVE HILBERT FORMS

353

When A has a unit element and is a Banach algebra with involution, not only does the theory of positive Hilbert forms reduce to that of positive linear forms, and the condition (N) is automatically satisfied, but also the condition (U) is satisfied. Precisely, (15.6.11) Let A be a Banach algebra with involution, having a unit element e # 0. Let f be a positive linear form on A. Then (i) f i s continuous and 11 f II = f ( e ) . ( 4 If(Y*XY)I IIlXllf(Y*Y).

We shall use the following lemma: (15.6.11.1)

adjoint y

E

If X E A is self-adjoint and llxll < 1, then there exists a selfA such that y 2 = e + x.

If t is real and It1 < 1, Taylor’s formula (8.14.3) gives

where

(-1)” 1 . 3 - . . . ( 2 n - 1) J: (%)” r,(t) = 2 2 . 4 . ~2n . 1+ s

ds

(1

+ s)1’2’

This leads immediately to the estimate

Irn(t)l 6

and therefore we have (15.6.11.2)

(1

+ t)”’

=1

Ill”

+ + t + ( 4 1 ’ + ... + (;)P +

. . a ,

the series on the right being absolutely convergent for I tl < 1 (9.1.2). It follows that in the Banach algebra A the series

converges absolutely for llxll < 1, and its sum y is self-adjoint when x is selfadjoint. Furthermore, since the square of the power series on the right-hand side of (1 5.6.1 1.2) is 1 + t, it follows that y 2 = e + x, by virtue of (5.5.3). This lemma shows that, if x E A is self-adjoint and llxll < 1, there exists such that y*y = e - x, so thatf(e - x) 2 0, or equivalentlyf(x) S f ( e ) . If now z E A is such that llzll < 1, then also 11z*z11 < 1 and therefore, using

y

EA

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XV

NORMED ALGEBRAS A N D SPECTRAL THEORY

(15.6.2.4), we have If(z)l’ S f ( e ) f ( z * z )sf(e)’. This proves that f is continuous and that llfll s f ( e ) (5.7.1). Since also I(el(= I , we havef(e) 5 Ilfll, and assertion (i) is proved.

To prove (ii), observe that for y E A the linear form x ~ f ( y * x yon ) A is positive, because f ( y * x * x y ) = f ( ( x y ) * ( x y ) )2 0; by virtue of (i), the norm of this linear form isf(y*y). This completes the proof.

PROBLEMS 1. Let A be an algebra with involution and let U be a representation of A on a Hilbert

space H. In order that U should be irreducible, it is necessary and sufficient that the subalgebra B of Y(H) consisting of the endomorphisms V such that VU(s)= U(s)V for all s E A should be equal to C . I , ( . (To show that the condition is necessary, observe that B is an’involutive closed subalgebra of Y(H), hence is a star algebra. If S is a hermitian operator belonging to B, consider the closed commutative subalgebra C of B generated by S, which is separable and therefore isomorphic to ‘c,(X) for some compact metrizable space X. Show that C has no divisors of zero, other than 0, and conclude that X consists of a single point.) 2.

Let A be a Banach algebra with unit element e, and let x ~ x be * a (not necessarily continuous) involution on A . (a) Let f be a linear form on A such that f ( a z ) 2 0 for all self-adjoint elements a E A. Show that if also p ( a ) < 1, then f ( a ) is real and I < f ( e ) (use Section 15.4, Problem 17). Show that, for each self-adjoint element a E A , f ( a ) is real, If(a)i f(e)p(a) and f(a)’ 5J(e)f(a’). (Consider f ( ( a -i-&)’) where 6 E R.) Deduce that If(x)l 0 such that /lull 5 p . p(a) for all hermitian elements a E A. (To show that (6) implies (a),show that (6) implies that /IxlI 5 2pp(x), by writing x =a ib with a, b hermitian, and using Section 15.4, Problem 18(d) and 18(g). To show that (p) implies (S), show first that if a is hermitian, then Allall < ( a ) , by using (15.2.7); deduce that if x is normal, then h2llxn1l~ !l(x*)”II 1, and use Section 15.4, Problem 19 to show that A is hermitian. Finally, to show that ( y ) implies (6), observe that for each hermitian element a such that p(a) i1 there exists in A a hermitian element b, commuting with a, such that bZ = e - a’ (Section 15.4, Problem 17); consequently u = a ib is unitary, and a = t ( u f u*).)

+

+

5.

Let A be a Banach algebra with unit element e, endowed with a hermitian involution XHX*.

(a) Let x E A be such that p(x) < 1 . Show that x(e - x*x)’/’ = ( e - x x * ) ’ / ’ x (cf. Section 15.2, Problem ll(f)). (b) Under the same hypotheses, show that the function F( 0, and that F( 0; also it is clear that IFq, sI 2 (IlFll - &)qV s , so that 1 1 1*

1H8'

+ U&>Il

=

1

NZ(FVV n S ) 2 (/IF// - ~ ) N z ( q vn s ) and consequently IIM(F)I(2 llFll - E . Since E > 0 was arbitrary, this proves our assertion. Hence the isomorphism 1 1", + V,.(x)H I + 2 extends by continuity to an isometry of d ; .onto WC(S'). Having regard to the fact that every character of %,(Sf) is of the form F H F(x') for some x' E S' (15.3.7), we conclude that Sit = S' and hence that Sn,= S. It remains to show that m,. = m, or equivalently that JF(x) dm&) = JF(x) dm(x) for all functions F E X&). But by virtue of (i) we have n

n

and since the functions I , as z runs through A, form a total set in %g(S), this establishes the equality of the bounded measures 2F. m,. and f j m for ail x, y in A. On the other hand, we have seen earlier that there exists a function 0 = ai< (where the x i belong to A) which does not vanish on Supp(F), so i

that F = GG with G E Xc(S).Consequently we have from above

n

This completes the proof of the Plancherel-Godement theorem.

9 THE PLANCHEREL-GODEMENT THEOREM

381

The Plancherel-Godement theorem applies in particular when the bitrace g is of the form ( x , y ) Hf (y*x), where f is a positive linear form on A (and therefore a trace since A is commutative). When g comes from a tracef, the formula (15.9.2.1) leads us to ask whether we also have (15.9.3)

A partial answer to this question is provided by the following theorem: (15.9.4) (Bochner-Godement theorem) (i) Let .f be a positive linear form on a commutative algebra A with involution, such that the bitrace g(x, y ) = f ( x y * ) satisjes the hypotheses of (15.9.2). Then, if the formula (15.9.3) is true and if the measure m, is bounded, f satisfies the condition

(B) There exists a real number M > 0 such that If (x)I2S M . f ( x x * ) f o r all x

E

A.

(ii) Conversely, let f be a positive linear form on A which satisfies the condition (B), and suppose that the corresponding bitrace g(x, y ) =f ( x y * ) satisfies (U) and is such that the prehilbert space A/n, and the star algebra d , are separable. Then g also satisfies (N),the measure m, is bounded, and the formula (15.9.3) is valid. (i) If m, is bounded and if (15.9.3) holds, then the Cauchy-Schwarz inequality (13.1 1.2.2) gives us

and hence the inequality of (B), with M = m,(S,). (ii) Recall that the definition of the Hilbert space H, does not presuppose that the condition (N) is satisfied. The inequality in (B) can be put in the form If(x)12 5 M \ \ n , ( ~ ) 1 \ ~ and , shows that f vanishes on n,; hence we may write f = f ‘ n,, where f ’ is a linear form on A/n,. Also I f’(ng(x>)l25 M l\ng(x)l12,so that f ‘ is continuous on the prehilbert space A/n, (5.5.1) and therefore extends to a continuous linear form on the Hilbert space H, (5.5.4). Hence, by (6.4.2), there exists a well-determined vector a E H, such that 0

From this it follows that, for all x E A, we have (15.9.4.2)

U,(x) a = ng(x).

382

XV

For if y

NORMED ALGEBRAS AND SPECTRAL THEORY E A,

then

(n,(y) I U,(x) a) = (u,(x*> * n,(y)

I4 = (n,(x*v>I4 = f ( x * y ) = (n,(y) I n,(x)),

and (15.9.4.2) follows because A/n, is dense in H,. We are now in a position to show that g satisfies the condition (N). Let b E H, be a vector belonging to the orthogonal supplement of the subspace generated by the elements n,(xy) for all x,y in A ; then we have (n,(xy) I b) = 0, that is (U,(x) * n,(y) 1 b) = 0, or again (n,(y) 1 U,(x*) . b) = 0 for all x and y in A. Since the elements n,(y) form a dense subspace of H,, it follows that ( a I U,(x*) 6) = 0, hence (U,(x) * a I b) = 0, hence finally (n,(x) I b) = 0 and therefore b = 0, because A/n, is dense in H,. Now consider the positive linear form f"(V ) = ( V * a 1 a) on the algebra di . We have

I~"(v) I I 11 v

*

all

IbII 5 11 v /I . llaliZ;

by virtue of the Gelfand-Neumark theorem (15.4.14), we may writef"(V) = h ( 9 V ) where h is a linear form on V c ( X ( d ; ) ) and , since l19VlI = IIVII, it follows that h is an (a priori complex) measure on the compact space X(&i). Since any continuous function G 2 0 on X ( d J is of the form F F, hence of the form 9 V - 9 V * = 9 ( V V * ) , we have h(G) = I/ V . all2 2 0, so that h is a positive measure. Taking into account (15.9.4.2), (15.9.4.1 ) and the canonical homeomorphism w of X ( d i ) onto Si, we see therefore that there exists a positive measure v on S, (induced by the measure o(h) on S i , and therefore bounded) such that f(x)

= (U,(x>

for all x E A. Thus it remains to show that v

a I a) =

s

$(XI 4 X )

= m,. Since

it is sufficient, by virtue of (15.9.2(11)) to show that (1) the functions ?. (where x E A) belong to Y$(S,,v ) ; ( 2 ) the support of v is S,. The first assution is trivial, because the functions 2 are bounded and continuous and the measure

s

s

v is bounded. To prove (2), sLppose that there exists a function F 2 0 belonging to Xx,(S,)such that F(x) d v ( ~=) 0. Then F(z)R(x)E(X) dv(x) = 0 for all x, y E A. Since F w is a continuous function on X ( d i ) , it is of the form 9 V where V E di,and the relation above now takes the form 0

9 THE PLANCHEREL-GODEMENT THEOREM

383

or

( V U , ( x ) . a ( u , ( y ) - a )= O .

-

But, by virtue of (15.9.4.2), ( V ng(x)1 n,(y)) = 0 for all x, y in A. Since Afn, is dense in H,, it follows that V = 0 and hence F = 0. Q.E.D.

Examples (15.9.5) We have seen (15.6.2.4) that a positive linear form f on an algebra with involution A having a unit element always satisfies the condition (B). We recall that if A is in addition a Banach algebra, then the corresponding Hilbert form g also satisfies (U) (15.6.11). If A is a separable Banach algebra with unit element e, then the prehilbert space A/n, and the star algebra d,are also separable. The first assertion follows from the continuity off (1 5.6.1I),which implies that Iln,(x)ljZ = f ( x * x ) 2 llfll Ilx*xil 6 llfll llxll’ and shows therefore that the image under ng of a denumerable dense set in A is dense in the prehilbert space Afn,. The separability of d,follows from the fact that the representation U, is continuous (1 5.5.7) and therefore transforms a denumerable dense set in A into a denumerable dense set in d,. Hence the BochnerGodement theorem can be applied to every positive linear form on A.

-

-

(15.9.6) If S H U(s) is a representation of A on a Hilbert space H, then for each xo E H the form fx,(s) = ( U ( s ). x0 I x0) satisfies the condition (B), because by (6.2.4)

Ifxo(.~>12 5 It U ( s > .xO1l2. IIxo/iz= ~ I X ~ ~ ~ % ~ ( ~ * ~ ) ~ By (15.5.6) this shows that knowledge of the hermitian characters of a commutative algebra A with involution which has a unit element determines all the representations of A. In addition, we have seen in the course of the proof of (15.9.4) that, iff is a positive linear form satisfying the conditions of (15.9.4(ii)), then the corresponding representation X H U,(x) admits a totalizer a (15.9.4.2).

Remarks It can happen that the formula (15.9.3) (and the conditions of (15.9.2)) are satisfied bdt that the measure m yis not bounded (and therefore f does not satisfy (B)) (Problem 2). On the other hand, there are examples in which the conditions (U) and ( N ) are satisfied but (B) is not (Problem 4), and examples in which (B) and (N) are satisfied but (U) is not (Problem 5).

(15.9.7)

384

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

PROBLEMS

1.

Let A be the algebra with involution WZ(R) of continuous bounded complex-valued functions on R, and let p be a bounded positive measure on R,with support equal t o R. Show that d x ,Y)= jx(r)Zl ddt)

is a bitrace on A for which the prehilbert space A / n , = A is separable, but that the star algebra d , is not separable (Section 7.4, Problem 4). 2.

s

Let A be Lebesgue measure on R and let A be the subalgebra with involution of Vg(R) consisting of square-A-integrable functions. Then g ( x , y ) = x(f)y(r) dh(t) is a bitrace

on A satisfying (U) and (N), but the measure m, is not bounded, and the right-hand ':(R) consisting side of (15.9.3) is not defined for all x E A. If B c A is the subalgebra of % of A-integrable functions, then J

is a positive linear form on B satisfying (15.9.3) but the measure m, is not bounded. 3.

Let A be the subalgebra with involution of %,-(I), where I = [0, I], consisting of functions x with continuous second-order derivative on I and such that x ( 0 ) = 0. Show that A x ) = Jol x ( t ) dt

+~ " ( 0 )

is a positive linear form on A, such that the corresponding bitrace g satisfies the condition (U) but not the condition (N). (If (f.) is a sequence of functions belonging to Wc(I), with values in [O, 11 and continuous second derivatives, and such that f . ( t ) = 1 in a neighborhood of 0 and fn(t) = 0 for t 2 l/n, consider in the Hilbert space H, the sequence of functions x. E A such that x.(t) = tf(r).) 4.

Let A be the subalgebra with involution of %',-(I) consisting of continuously differentiable functions on I which vanish at 0. Show that the formf(x) = x ' ( 0 ) on I is a positive linear form for which the corresponding bitrace g is zero (and therefore satisfies (U) and (N))but does not satisfy (B).

5.

Let A be the algebra with involution (the involution being X H ~ )of complex-valued functions on [0, 11 of the form P(r, log t ) , where P is a polynomial in two variables with complex coefficients. Then the linear form f ( x )

=

Jol

x(t)

dt is positive, and satisfies

(B) because A has a unit element; the corresponding bitrace g satisfies (N) but not (U).

r be a set endowed with an associative law of composition ( x , y ) w x y and a neutral element e, and let X H X * be an involution on r (i.e., a bijection of r onto r such that e* = e , (x*)* = x and (xy)* = y*x*). A representation of r in a Hilbert

6. Let

9 THE PLANCHEREL-GODEMENT THEOREM

385

space H is a mapping x H U(x) of r into Y(H) such that U ( e )= l H , U(xy) = U(x)U(y) and U(x*) = U(x)*. Let E be a Hilbert space and x ~ T ( x a) mapping of r into Y(E). In order that there should exist a Hilbert space H which is the Hilbert sum of E and another Hilbert space F, and a representation x H U(x) of r on H such that T(x) = PU(x) I E for all x E r, where P is the orthogonal projection of H on E, it is necessary and sufficient that T should satisfy the following three conditions: (1) T(e)= I E and T(x*) = T(x)* for all x E E. (2) If g : I' + E is any mapping such that g(x) = 0 for all but a finite set of elements x of r, then (T(X*Y) .9(Y) I g(x)) L 0.

x

(I,Y ) E

x

E

i-xr

(3) Ifg : r -+ E is any mapping such thatg(x) = 0 for all but a finite set of elements F, then for each z E I- there exists a constant M, > 0 such that

Moreover, if U and H satisfy these conditions and are such that the elements U(x) ,f(x E r and f~ H) form a total set in H, then the representation U is determined up to equivalence (equivalence of representations being defined as in (15.5)). (To show that these conditions are sufficient, let G be the subspace of E' consisting of mappings g : r -+ E such that g(x) = 0 for all but a finite set of elements x E r, and consider the form on G x G

which is a positive hermitian form. If N is the set of all g E G such that B(g, g) = 0, then B induces a nondegenerate positive hermitian form (0, h ) w ( 4 I / I ) = B(Q, h) on G/N which makes G / N a prehilbert space. Assume that G/N is a dense subspace of a Hilbert space Ho . Define an injection j : G/N + Er as follows: for each g E G , the image j ( g ) of .4 is the mapping

X H C W * Y ) .g(y). p e r

By transporting the scalar product on G/N by means o f j , we have a structure of prehilbert space on j ( G / N ) and hence, extending by continuity, an isomorphism of Ho onto a Hilbert space H. Then define U(x) by the condition that U(x) . j ( g ) is the mapping z T ( Z * X Y ) . g(y).)

Hx

Y E r

7. Let A be a separable commutative Banach algebra with involution, having a unit

element e . Let P be the set of positive linear forms on A (or of traces on A), which is a subset of the dual A' of the Banach space A (15.6.11). Iff;,f, E P, we writef, to mean that f2 -fi E P. (a) If f , f o are traces on A such that fo gf,show that there exists a sequence (y,) of elements of A such thatfo(x) = limf(y,tx) for all x E A. (If g is the bitrace correspond-

sf2

"+ m

ing t o f,note that fo may be written in the form fo = I I 0 mxg,where u is a continuous linear form on H, .) (b) Let P, be the set of tracesf€ P such that ilfii = f ( e ) 5 I ; then PI is convex and weakly compact. Show that the set of extremal points of P, (Section 12.15, Problem 5) is equal to (P n X(A)) u {O}. (Use (a) to show that if a character is a trace it must

386

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

be an extremal point of P,. To prove the converse, show that i f f € PI is such that I/(e)l then there exists z E A such that lltll < 1 and f(z*z) # 0. Using 1 (15.6.11.1), show that the linear forms f i ( x )= f ( z * z x ) / f ( z * z ) and -:

:

fdx)

- z*z>x)/(f(e) -ffz*z))

belong to P,, and that i f f is an extremal point then f=f, = f 2 . Hence we have f ( z * r x ) - z f ( x ) f ( z * z ) ; replacing z by r(e y ) where y is small, deduce that f is a character.) (c) Show that for each f~ P, there exists a unique positive measure pf on P I , with niass 1, such that for each x E A we have

+

f ( x ) = Jpnx(*:(x)

&Ax).

(Use Section 13.10, Problem 2(b). For the uniqueness of pr, use the Stone-Weierstrass theorem.)

10. R E P R E S E N T A T I O N S O F ALGEBRAS O F C O N T I N U O U S F U N C T I O N S

Let K be a coinpact nzetrizable space. The application of the results of Scction 15.9 to the case where A = %‘,-(K) allows us to describe very simply all the representations of this algebra. We consider first the topologically cjdic. representations u H T(u). (15.10.1) Let K be a compact metrizable space and A = %,(K). Then every topologically cyclic representation of the commutative algebra with involution A, in a sc>jiarabie Hilbert space E, is equivalent to one of the representations UH M,,(u) (lefined as,follows: let p be a positive measure on K, and for each u E A, let M J u ) dcnote the continuous operator on Lz(K, p) induced by multiplication by u : that is, .for each f E 2’t(K, p), M,(u) * f is the class of uf in

G ( K >10.

Let a be a totalizing vector for a representation U H T(u) of A in a separable Hilbert space E. The representation Tis determined up to equivalence by the positive linear form f,(u) = ( T ( u ) a I a) on A (15.6). Since A is separable (7.4.4),the Bochner-Godement theorem applies ; all the characters of A are hermitian, and the spectrum X(A) can be canonically identified with K (15.3.7).Hence the proposition is an immediate consequence of (1 5.9.qiv)). It follows from the definition (15.10.1) of M , that

10 REPRESENTATIONS OF ALGEBRAS OF CONTINUOUS FUNCTIONS

387

(with respect to the measure p). For it follows from (13.12.2) that //M,(u)/l2 N,(u); also, for every positive real number a < Nm(u), there exists a nonnegligible integrable subset P of K such that lu(t) I 2 c( for all t E P; hence N,(ucp,) 2 ci * N2(qP),from which (15.10.2) follows. The measure p is not uniquely determined. We shall come back to this point a little later (15.10.7). The definition of M,(u) given in (15.10.1), and the formula (15.10.2), still make sense if u is not necessarily continuous on K, but simply p-nieasurable and bounded in measure (by virtue of (13.12.5)). The mapping u w M J u ) extended in this manner is a representation of the algebra with involution Y,"(K, p) on the Hilbert space Lf(K, p). Clearly, if u1 and u2 are equal almost etlerywhere with respect to p, we have M,(u,) = M,(u,). Usually we shall restrict the representation M , to a self-adjoint subalgebra of Y","K, p) which does not depend on the choice of p, namely the algebra diC(K) of universally measurabfe, bounded complex-cafued ,functions on K (13.9). By (13.9.8.1) this is indeed a C-algebra with involution, and it is a Banach subalgebra of BC(K) by virtue of Egoroff's theorem (13.9.1 0). (15.10.3) Again let u- T(u) be a representation of %',-(K) in a separable Hilbert space E, admitting a totalizing vector a. (These hypotheses will be in force up to and including (15.10.7).) The preceding remarks show that the representation T may be extended to the algebra with involution @c(K). The extended representation (also denoted by T ) does not depend on the choice of totalizing vector a. To prove this assertion, let x and y be two vectors belonging to E. For each u E Vc(K), we have I/ T(u)I/ 5 /Iull (15.5.7) and therefore

consequently the linear form U H ( T ( u ). x I y ) is a measure p x , yon K such that llPx,yll 5 llxll . llull. From this definition it follows immediately that (15.10.3.2)

-

Also, if x = T(v) a and y = T(w) a, where 2) and each function u E q C ( K ) we have

-

14'

are in q C ( K ) , then for

( T(u) x I y ) = (T(uo). a I T(w) * a) = ( T(M'ur)* a I a) =

388

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

and therefore

The independence asserted above will be a consequence of the following more precise result: (15.10.4) (i) For all x, y E E and all u E %,-(K) we have

s

. x IY ) = 4 0 d P x , y ( o .

(15.10.4.1)

(ii) Zf(u,) is a uniformly bounded sequence offunctions in %,(K) which conoerges to u, then

for all x , y in E. It is enough to prove (15.10.4.2) for x, y lying in a rota1 subspace of E: the sesquilinear functions (x,y ) H(T(u,) x I y ) form an equicontinuous set, because

(7.5.5). Take x by definition,

=

-

T(s,) a and y = T(s2) a, with s1 and s2 in %,-(K); then,

(T(u,) . x I Y ) = (T(3,unsI) * a I a ) =

s

~2

u,si d p ?

and it is enough to apply the dominated convergence theorem (13.8.4) to the measure p. As to (15.10.4.1), it is valid by definition when u E %,(K), and in general it follows by applying the dominated convergence theorem twice to the measure p L x , yand , using (13.7.1). (15.10.5) For the operator T(u) to be hermitian (resp. positive hermitian (11.5), resp. zero, resp. unitary) it is neces’saryand suficient that u(x) should be real (resp. u(x) 2 0, resp. u(x) = 0, resp. lu(x)l = 1) almost euerywhere with

respect to the measure p .

389

10 REPRESENTATIONS OF ALGEBRAS OF CONTINUOUS FUNCTIONS

It is immediately verified that if U is a hermitian (resp. positive hermitian, resp. unitary) operator on a Hilbert space E, and if S is an isomorphism of E onto a Hilbert space E', then the operator SUS-' on E' has the same property (since these properties involve only the Hilbert structure). Hence we may take T(u) = M,(u), and in this case the sufficiency of the conditions stated is clear. On the other hand, if for example there exists a measurable subset X of K such that p ( X ) = a > 0 and 9 ( u ( x ) )2 fl > 0 for all x E X, then we have 9(( T(u) . qXI qX)) = u dp) 2 ctfl and therefore T(u) is not hermitian. The other cases are dealt with similarly.

9(Ix

(15.10.6) (i) The orthogonal projectors belonging to the algebra T ( @ J K ) ) are the operators of the form T(cpx),where X is a universally measurable subset of K . (ii) T(@,(K)) is a maximal commutative subalgebra of,ri4(E). (iii) A closed vector subspace F of E is stable under T ifand only if it is of the form T ( q x ) ( E ) ,where X is a universally measurable subset of K .

(i) By virtue of the characterization of orthogonal projectors on E (15.5.3.1) and by (15.10.5),it follows that T(u)is an orthogonal projector if and only if u is almost everywhere equal to a p-measurable bounded real-valued function u such that u2 = u. Hence v = cpx, where X is a p-measurable subset of K, and the result follows. (ii) It is clearly enough to prove that a continuous operator V E 9 ( E ) which commutes with T(u) for all u E @&) is of the form T(u). If a is a totalizing vector for T, then for each universally measurable subset X of K we have

5 IIVII . llT('px). all2 = IIVII

*

s

' p x dP.,a = 1 I VII.

.s

dP

because T('px)is an orthogonal projector which commutes with V. The Lebesgue-Nikodym theorem (13.15.5(c')) therefore shows that p v . o , r r= h * p for some p-integrable function h. Since also it follows from the inequality above that

we conclude that the linear form for every step function WE&&), WH h w d p defined on the vector space of these step functions extends by

s

390

XV

NORMED ALGEBRAS A N D SPECTRAL THEORY

continuity to L?A(K, p) ((13.9.12) and (13.9.13)). Hence, by (13.17.1), h is bounded in measure with respect to p, and we may assume that 12 E %,-(K). For all s, t E @c(K) we have (b‘T(s). u I T(t). a )

= (T(s)V * u

=

pl

1 T ( t ) .U ) =

s

si dp“..,.

dp = ( T ( h ) T ( s )u. I T ( t ) .a ) ;

since the vectors T ( s ). a form a total set in E, it follows that V = T ( h ) . (iii) To say that F i s stable with respect to T means that the orthogonal projector f’, commutes with T(u) for all u E %,(K) (15.5.3); hence P, E T(J2/c(K))by virtue of (ii), and is of the form T(cp,) by virtue of (i). We can also characterize all the totalizing vectors of the given representation: (15.10.7) For a cector Lj E Li(K, p) to be totalizing f o r the representation M,, of ‘e,(K), it is necessary and sufficient that g ( t ) # 0 alyost everywhere with respect to p . In order that the classes (fg)-,where f runs through V,(K), should generate a subcpace which I S not dense i n the Hilbert space LE(K, p), it is necessary and sufiicienl that there should exist a nonnegligibIe h E 9’$(K, p) such

i

that fgh dp = 0 for all f E %‘,(K) (6.3.1). Also we have gh E P,!.(K, p) (1 3.11.7) ; hence the measure ( g h ) * p is zero, which implies that g ( t ) h ( t )= 0 almost everywhere with respect to p (13.14.4). But since by hypothesis the integrable set A of points t E K at which h ( t ) # 0 is not negligible, we must have g ( t ) = 0 almost everywhere in A. This proves the proposition. To such a totalizing vector Lj for M,, corresponds by definition (15.10.1) the positive measure 1gI2 . p on K. Since 1gI2 is p-integrable and # O almost everywhere, we obtain in this way all the positive measures on K which are equivalent to 11 (13.15.6). In other words, the measure p of the statement of (1 5.10.1) is determined on/^ up to equicalence by the representation T.

(1 5.10.8) Now let us consider an arbitrary nondegenerate representation Tof ‘ec(K) in a separable Hilbert space E. From (15.5.6) we know that E is the Hilbert sum of a sequence (En)of closed subspaces stable with respect to T , and such that the restriction T,, of T to En admits a totalizing vector a,,, for each n. The definition of the measures p x , ygiven in (15.10.3) applies without

10 REPRESENTATIONS OF ALGEBRAS OF CONTINUOUS FUNCTIONS

391

change. Also, for each function u E %&), we define T,(u) on En (15.10.1), and by virtue of (15.10.2) we have llT,,(u)~lI llull for each n. Now we have the following lemma: (15.10.8.1) Let E be a Hilbert space which is the Hilbert sum of a sequcnce (En) of closed subspaces. For each n let U , be a continuous operator on En, and suppose that the sequence of norms (11 U J ) is bounded. Then there exists a unique continuous operator U on E whose restriction to En is U,, for euch Also, the restriction of U * to En is U,* . Suppose that /I U,ll all n and llxll2 =

5 a for all n. For each x

c llx,l12 (6.4), we have

=

n.

1x, E E, where x, E En for n

n

C IIUn

xn/125

n

a2 C IIxnI12 =a211xI12, n

which shows that the series 1U , * x, converges in E. If U . x denotes its sum, n

it is clear that U is linear and that, from above, 11 U . xi1 5 allxll, so that U is a continuous operator (5.5.1). The uniqueness of U follows from the fact that the union of the subspaces En is a total set in E (6.4). Finally, we have 11 U,*l/ = /I U J 5 a for all n, and therefore there exists a continuous operator Y on E whose restriction to Enis U,* for all n . If y = y,, with y , E En for all n

and

c n

1

llynllZ= lly(I2,then (6.4) we have

(u . x I Y >= Cn ( u n

*

xn I yn) =

C (xn I UT . yn> = (X I I/ n

Y),

which proves that V = U *. Applying (15.10.8.1), we see that there exists a unique normal continuous operator T(u)on E whose restriction to Enis T,(u), for each n. It is immediate that UI+ T(u) is a representation of @&) in E which extends the representation T of qC(K). Next, the proposition (15.10.4) generalizes without any change in the proof: we have only to observe that we can take as a total subset of E the set of all T,(s) a,, where s E VC(K) and n is arbitrary. In general there exist infinitely many decompositions of E as a Hilbert sum of subspaces with the properties of (15.10.8). However, there is the following result : (15.10.9) There exists a decomposition of E as a Hilbert sum of a (finite or infinite) sequence (En) of closed subspaces, stable with respect to T , such that the

392

XV

NORMED ALGEBRAS A N D SPECTRAL THEORY

restriction of T to Enadmits a totalizing vector a, and such that, if p, is the positive measure on K corresponding to a, (15.10.1), then p, + is a measure with base p,, (13.1 3) f o r each n. We begin with a decomposition of E as the Hilbert sum of any sequence (F,) as described above. Let b, be a totalizing vector for the restriction of T to F, , and let v, be the corresponding positive measure on K. Since the measure v, is determined only up to equivalence (15.10.7), we may multiply it by a strictly positive constant so as to ensure that the series of norms ~ ~ v ,con,~~ verges. By induction on n 1 2 we define two sequences (v;), (v:) of positive measures on K, as follows. The measures v; and vl; are those which appear in the Lebesgue canonical decomposition of v 2 relative to v,, v; being a measure with base vl, and vl; disjoint from v, (13.8.4). For k > 2, the measures v; and v; are likewise such that vk = v; + v;, where v; is a measure with base (v, vl; v;- 1) and v; is disjoint from v, vl; + * * . + v;To each of these decompositions there corresponds a partition of K consisting of two universally measurable sets B; and B; such that v; is concentrated on B; and v: is concentrated on B; . Let Fk= F; 0 FL be the corresponding decomposition of F, as a Hilbert sum of mutually orthogonal subspaces. If F, is identified with Lg(K, vk) (15.10.1) then F; and F; admitb; = @,.,and 6; = @s..,astotalizing vectors, respectively (15.10.6). The sequence of measures v1 + v'; + . . + v; is increasing, and the norms of these measures are bounded above by

+ + +

,.

+

-

m n= 1

I(v,ll. Hence (13.4.4) this sequence has a least upper bound p1 in M,(K),

and p, is also the vague limit of the sequence. The preceding construction allows us to assume that, if v1 is concentrated on B,, then the sets B,, Bl;, . . . , B;, . . . are pairwise disjoint, and vg is identified with q,.., * p,. If El is the Hilbert sum of F, and the F; for k 12, then E is the Hilbert sum of El and the F; ( k 1 2). The subspace El is identified with Lg(K, p J ; since + 6; 11b~1I2= vk(B;) I IIVkll, the series 6, + bl; + converges in El, and its sum a, is identified with @*,,where A, is the union of B, and the B; (k 2 2). Clearly a, is a totalizing vector for the restriction of T to El. Thus, starting with the given decomposition (F,) of E, we have constructed a decomposition of E into El and the F;, where for each k 2 2 the measure v; corresponding to F; is a measure with base pl.Repeating the construction, we define for each n a decomposition of E as the Hilbert sum of subspaces El, . . . , En,F!,'J , . . . , F$!,, . . . , all stable under T, such that the restriction of T to each subspace admits a totalizing vector, and such that if the corresponding measure is pk for E, (1 5 k j n) and $1,' for F?jk, then pk+l is a measure with base pk for 1 5 k 5 n - 1, and v!,'jk is a measure with base p, for all k 2 1. To achieve this we have only to apply the previous reasoning to

+

,

10

REPRESENTATIONS OF ALGEBRAS OF CONTINUOUS FUNCTIONS

the Hilbert sum E,, @ FP! @ . . * @ FPJ, @ . that

F, 0 F,

0

s

.

.

. + .

393

It is clear from the construction

0 F, c El 0 E, @ ... @ E n

for each n ; since E is the Hilbert sum of the Fk,it follows that E is also the Hilbert sum of the E, (6.4.2), and the subspaces E, and measures C(k therefore satisfy the required conditions. Put pk = gk p,, and let Sk be the set of points t E K such that gk(t) > 0. Evidently we may assume that the sequence ( s k ) is decreasing and that each s k is universally measurable (13.9.3). Put M, = K - S, and M, = s k - s k + , fGr k 2 2; also put M, = S,. For 1 5 i S k, let Hi, = T(qMk)(Ei). Then

n

kgl

it follows from (15.10.6) that the restrictions of T to the k subspaces Hi, (1 5 i 5 k ) are equivalent, and we put Gk= H,, 0 - . . @ H,, . Clearly Gk is the Hilbert sum of the subspaces H i , (1 2 i 5 k ) . Similarly, put Hi, = T(qM,)(Ei)for each i 2 1. The restrictions of T to all the subspaces Him( i 2 1) are equivalent, and we denote by G , the Hilbert sum of the H i m for all i 2 1. Then it is clear that E is the Hilbert sum of the G, ( k 2 1) and G, . It can be shown (Problem 5) that the measures are determined up to equivalence, and hence that the S, (and consequently the M,) are determined up to a p1-negligible set. The subspaces G, = T(q,,)(E) are uniquely determined by T. The restriction of T to Gk(resp. to G,) is said to have multiplicity k (resp. injinite multiplicity). A representation of multiplicity 1 is therefore topologically cyclic.

(15.10.10) Since the measures pk are determined only up to equivalence (15.10.7), we may suppose that pk = 'psk p 1 . It follows that, up to equivalence, the most general representation u ~ T ( u of ) %,-(K) on a separable Hilbert space E may be described as follows: Consider a positive measure v on K, and a decreasing sequence (Sj)15j). Now, if M is the intersection of Sp(H,,) with the complement 1- co,O [ of R, in R, then the relation M # @ would imply p,,(M) > 0 (15.1.14). Since ]-co:O[ is the union of the intervals I-m, - l / n ] , there would exist m > 0 such that

/

n 1- 00, - ( l / m ) I )= ~1

> 0,

and consequently [qH([) dp,,(() 5 -u/m < 0, contrary to hypothesis. Finally, the relation (15.11.7.2) follows from (1 5.11.7.1) and (1 5.4.14.1), because the spectral radius of H is equal to the larger of linf(Sp(H))(, Isup(Sp(H))I. (15.11.8) (i) For each ,function f E @,-(Sp(N)), the spectrum of f ( N ) is contained inf(Sp(N)) (closure in C), and (1 5.11.8.1 )

400

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

For every eigenvalue a of N, f (a) is an eigenvalue o f f ( N ) , and the eigenspace E ( a ; N ) is contained in E(f(a);f(N)). Iff is continuous, then Sp(f(N)) =f(Sp(N)). (ii) More precisely, with the notation of (15.11.3), i f f E 4Yc(Sp(N,,)), then the spectrum o f f (N,,) consists of the complex numbers fi such that

ess inf 1p -f([)l = 0 c E Sp",) (relative to the measure p,,). (iii) I f f € 4Yc(Sp(N)) and i f g is a continuous mapping off(Sp(N)) into C , then g(f(" = (9 o f ) ( N ) . (iv) If a sequence (fk) offunctions belonging to @,-(Sp(N)) is uniformly bounded and converges simply t o f , then for every x E E the sequence (f k ( N ) * x) converges in E to f ( N ) x.

-

We shall start by proving (ii). We have seen that f(NJ may be identified with multiplication by the class of the function f in L;(Sp(N,), p,,). Hence P # Sp(f(N,,)) if and only if there exists a real number a > 0 such that N2((fi- f ) u ) 2 a N,(u) for all functions u E Y;(Sp(N,,), p,,) (551). If ess inf Ip -f([)I > 0, then we may take a to be equal to this number, by

< E SP(Nn)

virtue of (13.12.2), and therefore

p # Sp(f(N,)).

ess inf

< E Sp(N,)

Ifi - f ( ( ) l

Conversely, if

= 0,

then for each E > O the set M of complex numbers [ E S ~ ( N , , )such that Ifi -f(T)l 5 E is not p,,-negligible, and we have N,((P - f ) q M ) 6 &N,(q,), hence P E SP(f(Nn))* It follows from (ii) that Sp(f(N,,)) cfISp(N,,)). If, moreover, .f is continuous, then f(Sp(N,,)) is compact (3.17.9), and for each fi = f ( a ) , where CL E Sp(N,,), every compact neighborhood of c1 has pn-measure > O (15.11.4), so thatf(Sp(N,,)) = Sp(f(N,,)). To show that Sp(f(N)) cf(Sp(N)) in general, and that Sp(f(N)) =f(Sp(N)) when f is continuous, we have only to use (15.11.5) and the fact that f(Sp(N)) is compact i f f is continuous. Finally, the assertions about eigenvalues are evident, again by reducing to the case of simple operators and using (15.11.6). To prove (iii), note first that g o f is a universally measurable mapping of Sp(N) into C (13.9.6). The relation 9(f"

= (9 o f ) ( N )

cPc4

3

follows from (15.11.1) in the case where g ( [ ) = ( p , q being in gers 2 0 ) . Now, for each E > 0, there exists a polynomial h in [ and 4 such that Ig(0 - &)I 5 E for all C € f ( S p ( W (7.3.2).Since Ig(f(T)>- h(f(l))I S E for

11 THE SPECTRAL THEORY OF HILBERT

401

iE Sp(N), it follows from (15.11.8.1) that IIg(f(N)) - h(f(N))I) 5 E and that 11(g f)(N) - (h f)(N)II 5 E . Since E was arbitrary, this gives the required 0

0

result. Finally, to prove (iv), note that because the sequence of norms 11 fk(N)II is bounded, by virtue of (1 5.11.8.1),it is enough to prove the convergence of the sequence (fk(N) - x ) for x belonging to a total subset of E ((12.15.7.1) and (7.5.5)). With the notation of (15.11.3), we may therefore restrict ourselves to proving the assertion for each N,,. But since fk(N,,) may be identified with multiplication by the class of the restriction of fk to Sp(N,,) in the space Li(Sp(N,,), p,,), the result follows from (13.1 1.4(iii)). (15.11.9) For each universally measurable subset M of Sp(N), let E(M) be the closed subspace of E, stable with respect to N and N*, which is the image of E under the orthogonalprojector PE(M) = qM(N) (15.10.6). Then the spectrum of the restriction of N to E(M) is contained in (closure in C ) .

For each n, let E,,(M) be the image of En under the orthogonal projector cp,(N,,). It is immediately seen that E(M) is the Hilbert sum of the E,,(M), hence (1 5.11.5) it is enough to prove the proposition for each N,, . If CI $ there exists a continuous function g on Sp(N,,) such that g([)(cc - [)(pM([) = qM([)for all E Sp(N,) (4.5.1). It follows that, if Ni is the restriction of N,, to E,,(M), then C L I ~ , (-~ N; ) has an inverse equal to the restriction ofg(N,,) to En(M).

a,

Remark (15.11.lo) Note that the argument which proves (ii) in (15.11.8) shows that, for each positive measure p on C with compact support IC, multiplication by the class of 1, in Li(K, p ) is a normal continuous operator N such that Sp(N) = K (converse of (15.11.3)). 1) Let f be a homeomorphism of a closed subset M of C onto a closed (15.11.I subset N of C containing Sp(N). Then there exists a unique normal continuous operator N' on E whose spectrum is contained in M and which is such that f(N') = N.

If h is the homeomorphism of N onto M which is the inverse off, then by virtue of (15.11.8(iii)) we must have N' = h(f(N')) = h ( N ) ; and since conversely the spectrum of h ( N ) is h ( S p ( N ) ) c M, it follows thatf(h(N)) = N , and the result follows.

402

XV

NORMED ALGEBRAS A N D SPECTRAL THEORY

In particular: (15.11.12) If H is any positive, self-adjoint operator then there exists a unique-positive self-adjoint operator H ’ such that H = H. l2

Apply (15.11.11) with M = N

= R,

andf(()

=

t2.

The unique positive self-adjoint operator H ’ defined in (15.11.12) is denoted by HI/’. Example (1 5.11 .I 3) Let E be the Hilbert space Lg(R; A), where 1is Lebesgue measure. Since the function f ( t ) = e-l‘l is I-integrable, the convolution g H f * g defines, on passing to the equivalence classes, a continuous operator H on E (14.10.6) with norm N , ( f ) = 2. As in (11.6.1), it is immediately seen that H is self-adjoint. It can be shown directly (Problem 5) that the interval [0, 21 in R is equal to Sp(H); this also follows from the general theorems of harmonic analysis (Chapter XXIl). Note that, for each a E R, the function g,(t) = eintis such that the convolution f * ga is defined and equal to 29,/(1 + a2). However, it is not the case that the g, are “eigenfunctions” of H , because they do not belong to Yg(R, A). In Chapter XXIII we shall obtain a generalization and an interpretation of this phenomenon. (15.11.14) The case of normal operators whose spectrum contains no nonisolated point # 0. In this case (which is that of compact normal operators (11.4.1)) let (A,,) be the (finite or infinite) sequence of points of Sp(N), other than 0. These are the eigenvalues of N (15.11.6). The eigenspace E(An;N ) corresponding to A,, is just the space E({A,,})defined in (15.11.9), and these closed subspaces are therefore pairwise orthogonal. Moreover, we have E({O}) = Ker(N), the spectrum of the restriction of N to this subspace being reduced to 0 (15.1 1.9). Finally, E is the Hilbert sum of E( (0)) and the E( {A,,}). For it is enough to apply (15.11.8(iv)) to the sequence of functions (f,), where f,(() = 1 for ( = 1, with k 5 n , f,(() = 0 for ( = lk and k > n, and f,(O) = 1 ; this shows in particular that every x E E is the limit of the sum of its projections on E({O}) and the E({A,}) with k 5 n. Hence the result (6.4). In particular, if E isfinite-dimensional, a normal operator on E may be defined to be an operator whose matrix is diagonal, with respect to a suitably chosen orthonormal basis of E.

11 THE SPECTRAL THEORY OF HILBERT

403

PROBLEMS 1.

Show that a continuous operator N on a Hilbert space E is normal if and only if /IN. X I / = /IN* . xi1 for all x t: E.

2. Let N be a continuous normal operator on a Hilbert space E. (a) For each x E E, consider the open sets W c C such that there exists a continuous

mapping f w of W into E satisfying the relation

(A 1 - N) fw(h) =x for all h E W. Show that there exists a largest open set n ( x ) with this property, that all the functions f w are restrictions of a unique mapping f of n ( x ) into E, and that f i s analytic in n ( x ) . (Use (15.5.6) to reduce to the case where the normal operator N is simple. In this case, E being identified with L&(Sp(U),p) and x with the class of a function g, the set n(x) is the interior of the set of all h E C such that the function - C)F1g([) belongs to -%SP(N), p).) (b) Put @(x) = C - n ( x ) . Show that, for each closed subset M of Sp(U), the space E(M) = vM(U)(E) is the set of all x E E such that @(x) c M. (Again reduce to the case of a simple normal operator.) (c) Show that every continuous operator V EL(E) which commutes with N also commutes with all the operators g(N), where g E *&p(N)), and in particular commutes with N* (Fuglede's theorem). (First show that Q ( V .x) 3 Q(x) for all x E E, by considering the function f defined in (a) and the function h H V . f ( h ) . Using (b), deduce that, for each closed subset M of E, the operator Vcommutes with = &N), and conclude that V commutes with g(N) for all conthe projector PE(M) tinuous functions g on Sp(N).) (d) Deduce from (c) that, if NI and N2 are commuting normal operators, then N,Nz is normal. 3. Let p be normalized Haar measure on the group U : IzI = 1 (so that d p ( @ = (27r)-l do) and let E = L&(p). Then the operator M,,(lc) = U is a unitary operator on E. Every closed subspace of E which is stable under N is either of the form vM(N)(E),where M c U is of measure < 1 ; or else is of the form @ .H2(p), where 141 = 1 (Section 15.3, Problem 15) (Beurling's theorem). Deduce that there exist closed subspaces F of E which are stable under U but not under U* = U-',and are such that the orthogonal projector PF does not commute with N (compare with (15.15.3)). 4.

Without using Riesz theory (11.4), prove that the spectrum of a compact normal operator has only isolated points, except for the point 0 (reduce to the case of a simple normal operator).

5.

Show that the spectrum of the self-adjoint operator H considered in (15.11.13) is the interval [0,2] in R. (Observe that llHI1 5 2; to prove that H 2 0, that is to say

( H . u I u) 2 0 for all u E E, consider first the functions

r+

u(t) =

elrXdx

and their

linear combinations. To show that every number of the form 2/(1 a') belongs to the spectrum of H , approximate g. by functions u,g,, where u. E Y 2is 2 0 and the sequence (u,) is increasing and tends t o 1.)

404 6.

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

Let E be a separable Hilbert space and T a continuous operator on E. Let R and L be the positive hermitian operators which are the square roots (1511.12) of T*T and TT*,respectively. We write R = abs(T), and call it the "absolute value" of T. Then L = abs(T*). (a) . . Show that Ker(T) = Ker(R) and that L(E) = T ? ) . There exists a unique isometry Vof R(E) onto T(E) such that T = YR. If we extend V by continuity to R(E),and then to an operator U E P ( E ) by putting U(x) = 0 on the orthogonal supplement of R(E), then we have also T = UR (polar decomposition of T ) . Show that R = U*T= U*UR = RU'U,

L

=

URU*,

T = LU*.

(b) For T to be invertible it is necessary and sufficient that R = abs(T) and L = abs(T*) are invertible. (To prove necessity, consider the spectra of R and L. To prove sufficiency, use the closed graph theorem.) (c) Nis normal if and only if abs(N) = abs(N*), and if this condition is satisfied there exists a unitary operator Wsuch that N = W . abs(N).

7. A compact operator Ton a separable Hilbert space E is said to be nuclear if, denoting by (A,) the full sequence of eigenvalues of abs(T) (Section 11.5, Problem 8), we have ( a j Use polar decomposition (Problem 6) to show that the product SISz of two Hilbert-Schmidt operators is nuclear. Conversely, if T is nuclear, then abs(T)"* is a self-adjoint Hilbert-Schmidt operator, and T is the product of two HilbertSchmidt operators. Consequently T* is also nuclear. If A is any continuous operator on E, then A T and TA are nuclear. (b) If A and B are two Hilbert-Schmidt operators and if (em) is a Hilbert basis of E, then the seriesx (AB . en I en) a n d C (BA . en I en) are absolutely convergent, and

"

"

their sums are equal. (Write B . en =C ( B . en I e,)e, .) Consequently, for every unitary m

operator U and every nuclear operator T, we have

Deduce that, for a nuclear operator T, the sum C ( T . enI en) is independent of the n

Hilbert basis (en) chosen. This sum is called the trace of T and is denoted by Tr(T). If A , B are two Hilbert-Schmidt operators, then Tr(AB) = Tr(BA) = (A I B*). (c) If T is nuclear, show that

where the supremum is taken over all pairs of Hilbert bases (a"), (b.) of E (use the polar decomposition of E). If we put IITlll = Tr(abs(T)), then the set 6L01(E) of nucIear operators on E is a vector space on which ljTlll is a norm, such that

ll~ll5 z IITllx.

(d) If ( T J is a sequence of nuclear (resp. Hilbert-Schmidt) operators on E which converges weakly (Section 12.15, Problem 9) to an operator T, and which is such that the sequence of norms (llTvlll)(resp. (liTvilz))is bounded, then T is a nuclear (resp. Hilbert-Schmidt) operator. (e) Show that 6L01(E) is a Banach space with respect to the norm llTlji.

11 THE SPECTRAL THEORY OF HILBERT

405

(f) Let (h.) be the sequence of eigenvalues of a nuclear operator T, each counted IA,l 2 IITlll. (For each according to its algebraic multiplicity (11.4.1). Show that integer p , consider the sum V of the p spaces N ( p X ;T ) (1 5 k < p ) , where p,, . . . , pr are the first p distinct eigenvalues in the sequence (A,,). Take a Hilbert basis of V with respect to which the matrix of TI V is triangular, and use (c).) Deduce that if T is a Hilbert-Schmidt operator and if (A,) is the sequence of its eigenvalues, each counted with its algebraic multiplicity, t h e n x lh,I2 5 11 Tll:. n

(9) If a continuous operator T E Y ( E ) is such that, for each pair of Hilbert bases (a"), (b.) of E, the series ( T . a, I b,) is convergent, then T is nuclear. (Write

"

T = LU* (Problem 6(a)), and by choosing (an)and (b,) suitably show that L' / z is a

Hilbert-Schmidt operator.) (h) For a continuous operator T E Y(E) to be nuclear, it is necessary and sufficient IIT. enll should converge. that, for a t least one Hilbert basis (en) of E, the series (Write T = UR (Problem 6(a)) and note that (R . e, I en)2 11 T . enll. This proves that the condition is sufficient. Conversely, take for (en) a basis consisting of eigenvectors of R.) (i) In the space E = /&, let (en) be the canonical Hilbert basis, and let a (l/n)e.. If F is the subspace C . a of dimension 1, show that the projector PF

=c n

is nuclear, but that the seriesx lipF.enll does not converge. 8.

Show that, for every normal operator N on a separable Hilbert space E, there exists = N.Give examples where there exist infinitely many such operators.

a normal operator N' such that N''

9. Let T be a continuous operator on a separable Hilbert space E. (a) For T to be a topological left zero-divisor (Section 15.2, Problem 3) in the algebra Y ( E ) , it is necessary and sufficient that there should exist a sequence (xn)of vectors

in E such that 1 ~ ~ = ~ 1 1 1for all n and such that ( T .x,) tends to 0 . The complex numbers 5 such that T - 5 . 1 is a topological left zero-divisor in 9 ( E ) form what is called the approximative point-spectrum Spa(T). Thus 5 $ Spa(T) means that T - 5 . 1 is injective and a homeomorphism of E onto a closed subspace of E. Show that Spa(?") is closed in C,and contains the frontier of Sp(T). If P is any polynomial, show that Spa(P(T))

= P(Spa(T)).

1) (b) Let T E Y(E). For each E C , let m(T, A) denote the dimension of Ker(T* (so that m(T, A) is either a nonnegative integer or w),equalalso to thecodimension of ( T - A . 1)(E). Let To E Y(E) and let A, be a complex number not belonging to Spa(To). Show that there exists a number E > 0 such that m(T, A) = m(To,A,) whenever I1T- Toll 5 E and Ih - hol 5 E . (c) Deduce from (b) that if K is a compact subset of C which does not intersect Spa(To), then there exists E > 0 such that m(T, h) = rn(To,h) for all E K and all T E Y(E) such that IlT- Toll 5.s. (d) Let T E Y(E) and let K be a compact subset of Sp(T) - Spa(T) with the following properties: (1) 0 $ K ; (2) the inverse image K' of K under the mapping 5 ~ 5 is convex; (3) m(T, h) = 1 for all A E K. Then there exists no operator T'E Y(E) such that T'' = T . (Supposing the contrary, let L = K n Sp(T'); then we have L C Sp(T') - Spa(T'), and L u (-L) = K'. Show that h E L implies that -A $ L

+

'

406

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

xz

a,

so that L n (-L) = which is a contradiction. Observe that if h E L, then is an eigenvalue of T* and m(T, A') = 1.) Show also that, if T is invertible, there exists E > 0 such that every operator T I E P ( E ) satisfying liTl - TI1 5 E is invertible and such that there exists no operator T'E Y(E) for which T" = TI (use (c).) 10. Let R be a bounded open subset of C, and let H be the Hilbert space of analytic

functions f o n R such that

(cf. Problem in Section 9.13). Let T be the operator which maps each f~ H to the function [++[f([). Show that, for each h E R and each function g E H such that g(h) = 0, there exists a unique f~ H such that ( T - h1) .f= 9. Also, if the disk I[ - hI < 6 is contained in R, then /1g1/2 2 & 8 z ~ ~ Deduce f ~ ~ z that . Sp(T) is the closure of R in C,and that i2 is contained in Sp(T) - Spa(T). Deduce from these results that if R is taken to be the open annulus rl < IzI < r2 (where r l > 0), then the operator T is invertible and has no square root in Y(H) (cf. Problem 9). 11. Let E be a separable Hilbert space, K a compact subset of C, and (x,y)~m,,,

a continuous sesquilinear mapping of E x E into the space M,(K) of complex measures on K. Suppose that

and that the measure mx, is positiue for all x on E such that

E

E. Let T be the continuous operator

for all x, y in E. For each functionfE 9,(K), letf(T) denote the operator defined by

(f(T). x I Y ) = If([)dm,,A) for all x , y in E (Section 15.10, Problem 1). The mapping T + f ( T ) of %,-(K) into Y(E) is linear and such that T*= c(T), where c([) = but this mapping is not in general an algebra homomorphism. Prove that there exists a separable Hilbert space H, the Hilbert sum of E and another Hilbert space F, and a representation f++ V(f) of Q,(K) on P ( H ) such that, if P is the orthogonal projection of H on E, we have f(T)= P V(f)I E. (Apply Problem 6 of Section 15.9, by taking r to be the set of all finite products of characteristic functions va,(n E N) of universally measurable sets in K, chosen in such a way that these functions form a total set in each of the spaces Yc(K, mx,.,,), where (x,) is a dense sequence in E.) ("Neumark's theorem.")

I,

12.

Let E be a separable Hilbert space. Let H be a self-adjoint operator on E such that 0 5 H 5 l E. Show that there exists a separable Hilbert space G which is the Hilbert sum of E and a Hilbert space F, and an orthogonal projector Q on G such that H = PQ I E, where P is the orthogonal projection of G on E. (For x , y in E, define mx, t o be the measure carried by the set of two points (0, 1) such that mX.,({O)) = ((IE - H ) . x I y ) and mx,,({l)) = (H* x I y), and apply Problem 11.)

11 THE SPECTRAL THEORY OF HILBERT

407

13. Let E be a separable Hilbert space, and (H.) a sequence of self-adjoint operators

with the following property: there exists an interval [-M, MI in R such that, if P(X) = a. a l X . . . a.X" is any polynomial with real coefficients and P ( 0 2 0 for - M 5 6 5 M, then also aoI a l H l . . . a. H. 2 0 (which implies inter alia that -M . I =< H, =< M . I for all n). Show that there exists a separable Hilbert space G which is the Hilbert sum of E and a Hilbert space F, and a self-adjoint operator H on G such that H. = PH" I E for all n 2 1, where P is the orthogonal projection of G on E. (Using Problem 5 of Section 13.20, prove that for each pair (x, y ) of elements of E there exists a real measure inx, on [-M, MI such that

+

+ +

+

(H. . x I Y ) =

for all n

+ +

j5"dm,, y(t),

( x I Y ) = / d m x , y(6)

2 1.) Deduce that Ht

5 Hz,

H;n+i 5 IIH2nIIHzn+z.

14. Let E be a separable Hilbert space and T a continuous operator on E such that /)TI15 1. Then there exists a separable Hilbert space H , the Hilbert sum of E and a

Hilbert space F, and a unitary operator U on H such that, if P is the orthogonal projection of H on E, then T" = PU" 1 E for all n 2 1. (Apply Problem 6 of Section 15.9, taking r = Z, the involution on I' being n w -n, and the representation of r in H such that U(n)= T" for all n 2 0. Observe that, for all y E E and all 5 E C such that 151 < 1, we have

W(1

(*)

+

. Y I (1 - @) . Y ) 2 0,

and note that every x e E can be written in the form (I- [ T ) . y for some ~ E E . +m

In particular, this is so for every linear combination x =

-m

are zero except for finitely many indices n, and where express C (rI"-'"'U(n- m) x. 1 x,)

x,e-"''+', where the x. E E

5 = re'@ with

r > 0. Then

m. n

in terms of the left-hand side of (*), and let r tend to 1.) Deduce that, if

m

"=O

Iz( = 1, and if u(T)

cn z" is a power series which converges absolutely on the circle

=cc,T", then the relation m

"=O

lu(z)l

51

(resp. Wu(z) LO) for

Irl 5 1 implies that ilu(T)II 2 1 (resp. u(T) + u(T*) 2 0).(Note that u(T) = Pu(U) I E.)

15. If N is a normal continuous operator on a separable Hilbert space E, and K a compact

subset of C containing Sp(N), then we have a (nonfaithful) representation f ~ + f ( N ) of V,(K) in E; the measures m x , ycan be considered as measures on K satisfying (15.11.2) and (15.11.2.1). In particular, if U is a unitary operator on E, then f.-.f(U) is a representation of U,(U) in E. Let (en) be a Hilbert basis of E indexed by Z, and let U be the unitary operator on E such that U .en= en+ Show that the representationf++f(U) of U,-(U) on E is topologically cyclic and that eo is a totalizing vector; and that p, = me,,, eo is the normalized Haar measure on U (cf. (7.4.2)). Deduce that Sp(U) is the whole of the circle U, and also give a direct proof of this fact. Give examples of closed vector subspaces of E which are stable under U but not under U*= U-'.

408

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

16. Let X be a locally compact space, p a bounded positive measure on X with total mass 1, and u a p-measurable mapping of X into X such that p is inuariunt with respect to u (Section 13.9, Problem 24). Let U be the unitary operator on L;(X, p) such that U . f = (fou)" (Section 13.11, Problem 10). (a) The mapping u is said to be mixing (resp. weakly mixing) with respect to if,

for each pair of p-measurable subsets A, B of X we have

lim (cL(u-"(A)n B)) = p(A)p(B)

n-. m

(resp. 1 "-1

Every mixing mapping is weakly mixing. Every weakly mixing mapping is ergodic (Section 13.9, Problem 13(d)). (b) Show that u is mixing (resp. weakly mixing) if and only if, for each pair of functions f,g in Y&(X, p), we have I i m( u n . f l O) = ( f l

I-. m

i)(ip)

(resp. lim

n-m

n

5' ~(un.fI8) - (fI i)(i1 8) 1 k=O

= 0).

An equivalent condition is that, for each f E U;(X, p) such that ( f l y ) (i.e.,/fdp

= 0),

=0

we have Iim (un

n+m

=0

(resp.

(Replace f by f+ g, and remark that if a sequence (an) satisfies the condition -1 n - 1 1 n-1 lim lak[* = 0,then also Iim lakl = 0, by the Cauchy-Schwarz inequality.)

n-.m Tl k = o

m+m

k=O

(c) For u to be ergodic with respect to p, it is necessary and sufficient that 1 should be an eigenvalue of U with multiplicity 1. If this is so, then all the eigenvalues of U have multiplicity 1 and form a subgroup of the group U of complex numbers of absolute value 1. For each eigenvector / E L&(X,p) of U,the function If\ is constant almost everywhere. (Remark that if U ./= hf and U .8 = hfi, then u .(Slf)" = (Sf)".) (d) Show that the following properties are equivalent: (a) u is weakly mixing with respect to f ~ . (B) u x u is an ergodic mapping of X x X into X x X with respect to the measure p, @ p. ( y ) The only eigenvalue of U is I. (To show that (a) implies (p), consider subsets M x N of X x X, where M and N are p-measurable subsets of X. To prove that (p) implies (y), observe that i f f i s an

11 THE SPECTRAL THEORY OF HILBERT

409

eigenvector of U,then (f@f)- is an eigenvector of the unitary operator corresponding to u x u, for the eigenvalue 1. To prove that ( y ) implies (a), use the last criterion of (b); if (f If) = 0, introduce the measure v = m7.7 (15.11.1), and observe that this measure on U is diffuse (15.11.6). Then we are reduced to proving that lim

n-rm

Write this relation in the form

1"-1

-

C

nk=o

1."

tkdv([)

1

= 0.

and remark that the diagonal of U x U is (v @ +negligible.) (e) With the notation of Section 13.9, Problem 13(c), show that if 0 is irrational, the mapping z w e Z n f Sof z U onto U is not weakly mixing. (Calculate the spectrum of the corresponding unitary operator U.) (f) Suppose that the space Li(X, p) is the Hilbert sum of the subspace C . 1 (the classes of the constant functions) and an at most denumerable family (HI),., of Hilbert spaces, where each H, has a Hilbert basis (en,)nEZ such that U .en, = en+ for all n E Z (cf. Problem 15). Then the mapping u is mixing. In particular, if. (X, p, p) is the Bernoulli scheme B(), 4) (Section 13.21, Problem 18) then u is mixing. (If, for each n E Z, f. is the function on Iz such that h(x) = -1 if pr.x = 0, and f.(x) = 1 if pr.x = 1, then the classes of the finite productsf.,f,, . . .Lk,in which all the indices are distinct, form together with the class of 1 an orthonormal basis of L&(X,p).) Likewise show that, if X is the torus T2,7r : R + T the canonical homomorphism and p the normalized Haar measure on X, then u defined by

444, T(YN = ( 4 x + Y ) , d x + 2Y)) is mixing. (9) Suppose that u is ergodic with respect to p, so that (by virtue of (c)) the eigenvalues of U form an at most denumerable subgroup G of U,the eigenspace corresponding to an eigenvalue a E G being a line D(a) in L:(X, p). Show that there exists a family of eigenvectorsf, E D(a) of U such that . f I I = 1 almost everywhere in X and such thatf., = f a f a almost everywhere,for all pairs (a, @)ofpoints of G. (Let ha E D(a) be such that (h.1 = 1 almost everywhere, for all a E G ; for each pair of points a, /3 E G we may write h., = r(a, @)h.h, almost everywhere, where r(a, p) E U is a constant. Denote by A the subgroup of Ux generated by the ha for a E G and the group of constant functions from X to U (which may be identified with U). Show that there exists a homomorphism 0 : A -+ U such that e([) = 6 for all 5 E U. For this purpose, arrange the ha in a sequence (h.) and proceed by induction: if 0 is already defined on the subgroup A. generated by U and the h, such that j < n, distinguish two cases according as h; is not constant almost everywhere for any nonzero k E Z, or on the contrary that there exists a smallest positive integer k such that hi is constant almost everywhere; use the fact that for all [ E U and all integers k > 0, there exists 7) E U such that 7' = 5. Then take f . = 8(h.)ha.) (h) Let v : X + X be another mapping which is ergodic with respect to p, and let V be the corresponding unitary operator. Show that if u satisfies the hypotheses of (g) and if U, V have the same eigenvalues, then u and v are conjugate (Section 13.12, Problem 11).

410

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

17. Let E be an infinite-dimensional separable Hilbert space and N a normal continuous

operator on E. (a) Show that E is the Hilbert sum El E2 of two infinite-dimensional subspaces, each of which is stable under N and N*. (Reduce to the case of a simple normal operator M,,(lc); with the notation of (15.11.9), observe that if M is a closed subset of Sp(N), then E(M) can be finite-dimensional only if M is a finite set for which each point has p-measure f-0. Then distinguish two cases, according as there exist infinitely many points of measure f O or not; in the second case, use Problem 3(b) of Section

+

13.18.)

(b) Deduce from (a) that there exists a decomposition of E as the Hilbert sum of a n infinite sequence (E,) of infinite-dimensional subspaces, each of which is stable under N a n d N*. Let E be a separable Hilbert space which is the Hilbert sum of an infinite family of infinite-dimensional subspaces. There exists a unitary operator S on E such that S(E,) = E n + ,for all n E Z. Show that if P (resp. Q ) is the operator which is equal to S1-2"(resp. S-2") on Enfor each n, then P 2 = Q 2 = I E , and S = P Q . (b) Deduce from (a) and from Problem 17 that every unitary operator on an infinite-dimensional separable Hilbert space is the product of four involutoryunitary operators. (c) Jxt w be a complex cube root of unity and let U be the homothety with ratio w on E, which is a unitary operator. Show that U is not the product of three involutory unitary operators. (In general, in a group G, if t is in the center of G , andifthereexist x , y , z in G such that t = x y z and x2 = y 2 = z2 = 1, then also t = y z x , t = xyxy and t 3 = xzy = rl.)

18. (a)

Let E be a n infinite-dimensional separable Hilbert space and let (e,),*o be a Hilbert basis of E. The continuous operator V such that V . en= enfl for all n 2 0 is called the one-sided shqt operator; it is an isometry of E onto the hyperplane orthogonal to eo; its spectrum is the disk 151 6 1 and contains no eigenvalue of V (Section 11.1, Problem 4); its approximative point-spectrum (Problem 9) is the circle U : 161 = 1. The spectrum of the adjoint operator V* is also the disk151 5 1, and every 1such that 151 < 1 is an eigenvalue of V*. (b) Let T be a continuous operator on E which is an isometry of E onto a (necessarily closed) subspace T(E). Show that there exists a decomposition of E as the Hilbert sum of subspace L and an at most denumerable family (F,), I of subspaces, where L and each F, is stable under Tand are such that (1) TI L is unitary and (2) each F, is infinitedimensional and TI FLis the one-sided shift operator, for a suitably chosen orthonormal basis. (Consider the orthogonal supplement N of T(E), and show that E is the Hilbert sum of the T"(N) (n >= 0) and L = T"(E).)

19. (a)

n

(c) Deduce from (a) and (b) that if T is any nonunitary isometry of E onto a subspace of E, then Sp(T) is the unit disk 151 5 1, and that IIT- UII = 2 for all unitary operators U.(Observe that IlT- Uil = / / U * T - lE/I and that U*T is not unitary, so that the point 5 = - 1 belongs to its spectrum.) 20.

(a) Let E be a separable Hilbert space, T a continuous operator, and C a compact operator on E. Show that the points of Sp(T+ C ) which do not belong to Sp(T) are eigenvalues of T C. (Reduce to the case where 5 = 0 is such a point and observe that, if T i s bijective, we may write T + C = T(IE T - l C ) , where T-lCis compact; if - 1 E Sp(T-'C), it follows that - 1 is an eigenvalue of T-IC.)

+

+

11 T H E SPECTRAL T H E O R Y OF HILBERT

411

(b) With the notation of Problem 15, let C be the operator of rank 1 defined by C . x = - ( x 1 e-&o. Show that Sp(U+ C ) is the disk 151 5 1 while Sp(U) is the circle 151 = 1. (Consider separately the restrictions of U C to the subspace generated by thee. with n 2 0 and to its orthogonal supplement.) (c) Let N be a normal operator on E and C a compact operator on E. If Sp(N) is nondenumerable, show that the same is trueof Sp((N C)*(N C))(use(a)aboveand (15.11.8(i)). Deduce that the one-sided shift operator V (Problem 19(a)) cannot be of the form N C (observe that V* V = lE).

+

+

+

+

21.

Let E be an infinite-dimensionalseparable Hilbert space. (a) Show that every nonzero two-sided ideal 3 of the ring P ( E ) contains the ideal 6 of operators of finite rank. (If T # 0 belongs to 3, show that every operator of rank 1 can be written in the form BTCfor suitably chosen operators Band C.) (b) Show that the only closed two-sided ideal of the Banach algebra P(E), other than P ( E ) and {O},is the ideal B of compact operators. (First observe that Q is the closure of 6,and then that if a two-sided ideal contains a noncompact operator, then it also contains a noncompact positive hermitian operator H (Problem 6). For such an operator, show that there exists an interval M = [a, 03 [ with a > 0 such that thespace E(M) (in the notation of (15.11.9)) is infinite-dimensional. If Vis an isometry of E onto E(M), show that V*HV is invertible.)

+

22.

Let E be an infinite-dimensional separable Hilbert space. An operator with index on E is a continuous operator T such that (1) T(E) is closed and of finite codimension; (2) T-I(O) is finite-dimensional. (a) If T is an index operator, show that there exists a continuous operator A such that l E- ATand l a - TA are of finite rank. (Show that Tis a homeomorphism of the orthogonal supplement F of T-'(O) onto T(E), and take A to be the inverse homeomorphism on F and zero on the orthogonal supplement of F.) (b) Conversely, suppose that Tis a continuous operator on E, for which there exists a continuous operator A such that l E- AT and l E - TA are compact. Show that T is an index operator. (Using (11.3.2), show first that the kernels of Tand T* are finitedimensional, and hence that T(E) has finite codimension. Then use the fact that the restriction of AT to the orthogonal supplement F of the kernel of AT is a homeomorphism onto its image, and finally use (12.13.2(iii)).)

23. Let E be an infinite-dimensional separable Hilbert space. (a) Let T be a continuous operator on E such that T-'(O)is of infinite dimension. Then E is the Hilbert sum of an infinite sequence (En)"%,of infinite-dimensional subspaces, such that the En with n 2 1 are contained in T-'(O). For each n 2 1, let S. denote an isometry of E, onto En. Let A be the continuous operator which on Eo is equal to S1, and on En is equal to S. + S;', for all n 2 1. Also let V be the operator ; I on En for all which is zero on Eo , is equal to Si ' on El,and is equal to Sn- S n 2 2. Let Todenote the restriction of P,,T to Eo , and let B be the continuous operator which is equal to VT on Eo , to -To 5'1' on E l , and to -Sn- ,TOX'on E. for all n 2 2. Prove that T = AB - BA. (b) Deduce from (a) that for any continuous operator T on E there exist four operators A, B, C, B such that

T = (AB - BA)

+ (CD -DC).

(Write T as the sum of two continuous operators, each of which has an infinitedimensional kernel.)

412

NORMED ALGEBRAS AND SPECTRAL THEORY

XV

24.

Let E be a separable Hilbert space and H a positive self-adjoint operator; then Z+ AH is invertible for all > 0. For x E E and > 0 let F,(x) = (h(1-t A H ) - ’ . x I x). Show that FA(x)is increasing as a function of A, and that it is bounded if and only if x E H”’(E). (Reduce to the case where H i s a simple operator (15.11.3))

25.

Let Eo be a real Hilbert space and E the Hilbert space obtained by extending the field of scalars of Eo to C, so that every element of E is uniquely of the form x iy with x, y E Eo, and

+

(x‘ -tiy‘ I x”

+ i f ’ ) = (x’ I x ” ) + (y’ I y ” ) + i(y‘ I x”) - i(x’ I y”).

Show that every self-adjoint operator HO on Eo extends uniquely to a self-adjoint operator H on E, having the same spectrum. 26. With the notation of Problem 2 of Section 13.13, suppose that for each compact subset K of X there exists a constant bL2 0 such that

for all u E 2.This condition implies condition (A) of Problem 2, Section 13.13, but is not equivalent to (A). (a) In Problem 5(b) of Section 6.6, suppose that X is compact and that the functions f. are real-valued, bounded and measurable with respect to a positive measure p on X and that they satisfy the condition

where

lI.f1

=

sup lfn(x)l. Show that the space .%? of functions which are p-equivalent

X E X

to the functions belonging to the space denoted by E in the problem referred to, is such that H = .%?/N is a Hilbert space isomorphic to E, and satisfying condition (B) above. (b) For every functionfe 9 ; ( X , p) with compact support K, there exists a function

U’

E

.@ such that

s

(U’ I u) = uf dp for all u E

.@. The class of

U’ is uniquely deter-

mined by the class off, and we have lUfl I b:l’N2(f). Then the set 9 defined in Section 13.13, Problem 2(b) is also the closure of .%? in the set of the Uf for which f is 20, compactly supported and belongs to 9:. Generalize the result of part (e) of this problem to the case where f E 9: is compactly supported and 2 0 almost everywhere. Likewise, generalize part (f) of the same problem. (c) Suppose that X is compact. Then Uf is defined for all functions f E -Ep:(X, p), and we have NZ(U’) b , N , ( f ) . If G . f i s the class of U’, then G is a continuous positive selfadjoint operator on L:(X, p). If F is the closure in L i of G’”(L:) (which is the orthogonal supplement of Ker(G’”) = Ker(G)), then the restriction of G’/’ to F is an isometry of the subspace F of L : onto the Hilbert space H (equipped with the norm lfil). Hence H = G’’*(Li). (d) Suppose that X is compact and that the “domination principle” is satisfied, in the form of (b): that is to say, iff f 9: is 2 0 almost everywhere, and if u E 9’ is such that Uf(x) 5 u(x) almost everywhere in the set of points x where f ( x ) > 0, then Uf(x) 5 u(x) almost everywhere in X. For each A > 0, put R1= G(I hG)-’. I f f € 9: is 20 almost everywhere, and if g is a function whose class is equal to

+

12 UNBOUNDED NORMAL OPERATORS

413

RA.f : then g(x) >= 0 almost everywhere. (Observe that

AUP+(X)2 U’(X) + hus-(x)

almost everywhere in the set of x such that g + ( x ) > 0.) Deduce that, for all U E H , we have lul E H and I(lul)”l 5 I&/. (If F,(V^) = (h(1 -t hG)-’ . B 16) for tr E 9’:, show that FA(lul-) 5 I i i I *, and use (c) above and Problem 24.) (e) Generalize the results of (d) to the case where X is locally compact. (Let (K.) be a sequence of compact subsets which cover X, and such that each is contained in the interior of the next. For each n, consider the space X nof restrictions to K, of functions of the form U’, where f E Y : ( X , p) and Supp(f) C K.; apply (d) to each of these spaces.)

12. U N B O U N D E D N O R M A L OPERATORS

(15.12.1) Let E be a separable Hilbert space, and let Z denote the identity mapping of E. A (not necessarily continuous) linear mapping T of a subspace dom(T) of E (the “domain” of T, which need not be closed) into E will be called, by abuse of language, a not necessarily bounded operator on E, or simply an unbounded operator on E. The graph I-( T ) (1.4) is a vector subspace of E x E, and T is said to be a closed operator if I-( T ) is closed in the product space E x E. The kernel Ker( T ) of a closed operator is closed in E, because it may be identified with the intersection of T(T) and E x (0) in E x E. Throughout, it is to be understood that an equality T I = T , between two unbounded operators on E implies the equality

dom(T,) = dom(T,). (15.12.2) Let T be an unbounded operator on a Hilbert space E. Then, of the

three properties ; (i) dom(T) is closed in E; (ii) T is closed; (iii) T is continuous; any two imply the third.

If T is continuous, then I-(T)is closed in dom(T) x E and therefore also in E x E if dom(T) is closed in E. Next, if T is continuous on dom(T), then it extends by continuity to a linear operator T‘ which is continuous on dom( T’) = dom( T ) (5.5.4), and I-( T‘) is the closure of I-( T ) in E x E. Hence if Tis closed we have I-(T’) = I-(T),so that dom(T) is closed in E. Finally, if dom( T ) is closed in E and if I-( T )is closed in E x E, then T is continuous by the closed graph theorem (12.16.11). (15.12.3) In what follows we shall be concerned with unbounded operators T such that dom( T ) is dense in E. Let F be the set of all y E E such that the

414

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

linear form X H (T * x I y ) is continuous on dom( T ) ;when this is so, this linear form extends by continuity to the whole of E (5.5.4) and therefore can be written as xt-i(xI T * ' y ) for a uniquely determined vector T* y , because dom( T ) is dense in E (6.3.2). This uniqueness shows that T* is a linear mapping of F into E, and therefore an unbounded operator, called the adjoint of T . (When dom(T) = E and T is continuous, this definition clearly agrees with that of (11.5).) Hence we have

-

(1 5.1 2.3.1)

(T-xly) = ( X I

T* ' y )

for all x E dom( T ) and y E dom(T*). If T, is an unbounded operator such that dom(Tl) 3 dom(T), then dom( TT) c dom( T*). In what follows we shall endow E x E with the structure of a Hilbert space such that

so that E x E is the Hilbert sum of its two subspaces E x (0) and (0) x E, each of which is isomorphic to E. Also we denote by J the continuous operator ( x , y ) ~ ( y- x, ) ; clearly J is a unitary operator on E x E, and 5' = - I . (1 5.12.4) Let T be an unbounded operator on E, such that dom( T ) is dense in E. (i) The adjoint -operator T* is closed, and its graph I-( T*) is the orthogonal supplement of J ( r ( T ) )in the Hilbert space E x E. (ii) The following properties are equivalent: (a) T can be extended to a closed operator. (b) dom(T*) is dense in E. If these conditions are satisfied, the graph of any closed operator extending T contains the graph of T**, and r(T**) = r(T). (Thus T** is the smallest closed operator which extends T, and in particular, T** = T if T is closed.) Moreover, (T**)*= T*.

(i) If a sequence (y,) of points in dom(T*) converges to y E E and is such that the sequence (T* y,) converges to z E E, then the sequence of continuous linear forms xt-i(xI T* * y,) converges for all x E E to the continuous linear form X H ( X ~ Z ) .But if x ~ d o m ( T ) , we have ( x I z ) = l i m ( T . x ( y , ) = ( T . x ( y ) ,hence yEdom(T*) and z = T * - y by n-, m

definition, which shows that T* is closed. On the other hand, to say that ( y , z ) E E x E is orthogonal to all the vectors (T * x, -x) with x E dom( T ) signifies that ( T * x I y ) = (x I z), i.e., that x I+ (T x I y ) is continuous, hence y E dom( T*)and z = T* * y .

12 UNBOUNDED NORMAL OPERATORS

415

(ii) A closed vector subspace G of E x E is not the graph of a closed operator if and only if, for some x E prl(G), there exist at least two distinct points ( x , yl) and (x, y 2 ) belonging to G, or equivalently (since G is a vector subspace) that (0,y1 - y 2 ) E G. But to say that dom(T*) is not dense in E means that there exists z # 0 in E orthogonal to dom(T*) (6.3.1), or equivalently - that (z, 0) is orthogonal to r(T*),or again that (0, - z ) belongs to T(T). It follows that (0, - z ) cannot be contained in the graph of any closed operator which extends T. Conversely, if dom(T*) is dense in E, then T** is defined, and r(T**)is the orthogonal supplement of J(T(T*));but this orthogonal supplement is also equal to

(6.3.1).

We say then that T** is the closure of T.

(15.12.5) Given two not necessarily bounded operators U, V on E, the vector U * x V * x is defined for all x E dom( U ) n dom( V ) , and we denote by U V the linear mapping X H U * x V x of dom(U) n dom( V ) into E. In particular, if V is everywhere defined, then dom(U V ) = dom(U), and the graph T(U + V ) is the image of r(U ) under the linear mapping ( x , y ) H(x, y + V x ) of E x E into E x E. If V is continuous and U is closed, it follows therefore that U + V is closed, since the mapping

+

+

+ -

+

-

(X,Y)H(X,Y

+ Y.x)

and its inverse ( x , y ) H ( x , y - V * x) are continuous. Again, the vector U ( V x ) is defined for the set of all x E E such that x ~ d o m ( V )and V e x ~ d o m ( U ) This . set is a vector subspace which we denote by dom(UV), and UV denotes the linear mapping XH U ( V . x ) of dom(UV) into E. If T is a not necessarily bounded operator which is an injective mapping of dom( T ) into E, we denote by T-' the inverse mapping of T(dom(T)) = dom(T-') into E. The graph T ( T - ' ) is the image of T ( T ) under the mapping (x, y ) ~ ( yx)., Hence T-' is closed if T is closed (and injective).

-

(15.12.6) (von Neumann) Let T be a closed operator on E such that dom(T) is dense in E. Then dom(T*T) is dense in E; the operator T*T is closed; and the operator Z + T*T (defined on dom(T*T)) is a bijection of dom(T*T) onto E. The operator B = ( I + T*T)-' is defined on E, continuous, self-adjoint and injective, and its spectrum is contained in the interval [0, 11 of R.Furthermore, the hermitian form ( x , y ) +-+ (B . x I y ) is positive and nondegenerate, and C = TB is a continuous operator defined on E, such that C(E) c dom(T*). Finally, (T*T)* = T*T.

416

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

We have seen in (15.12.4) that T(T) and J(T(T*)) are orthogonal supplements of each other in E x E. Hence for each X E E there exists a unique y E dom( T) and a unique z E dom( T*) such that (15.12.6.1)

(x, 0) = ( y , T y )

+ (T*

*

Z,

- z).

-

Put y = B x and z = C x. Clearly Band Care linear operators defined on the whole of E, and we have B(E) c dom(T) and C(E) c dom(T*). Also, by (15.12.6.1),

-

11xIl2= /lYl12 + l1T.YIl2 + lIz1l2 + IlT* - zl12,

- s

so that IIB xII llxll and IIC xII IIxII. Hence Band Care continuous. The relation (15.12.6.1) is equivalent to x=B*x+T*C*x

and

O=-C.x+TB.x,

so that C = TB and T(B(E))c dom(T*), hence B(E) c dom(T*T). Consequently T*TB is defined on all of E, and we have Z = B + T*TB=(Z+ T*T)B, which shows that B is injective and I + T*T surjective. For each w E dom(T*T) we have (15.12.6.2)

(W

+ T*T. w 1 W ) = ((w((’+ ( T * T *w 1 W ) =

llW1l2 3. lIT* WII’

because T = T**; this shows that the relation w + T*T. w = 0 implies that w = 0, and hence that Z + T*T is a bijective mapping of dom( T*T) onto E. Also, since T ( B ) is closed in E x E (1 5.1 2.2), the same is true of T(I + T* T) (15.12.5), and it follows immediately (15.12.5) that T*T is closed. We next remark that, for all, u, u in E, we have

(B* U I

U)

= (B*

UI

B u

+ T*TB - U )

( B * u I T* TB * u) = ( B u l B * u) ( T B . uI TB. u) = (B u I B V ) + ( T*TB * u I B * U ) = ((I T* T)B * u I B * U) = (U I B * u). = ( B * u IB

-

+

*

-

V) f

+

-

Hence B is self-adjoint. Also, replacing w by B x in (15.12.6.2) we obtain, for each x E E, (XI B X) = IIB * ~ 1 1 ’ llTB * xllz 2 0;

-

+

12 UNBOUNDED NORMAL OPERATORS

-

417

since I1B xII 5 IIxII, it follows from the above and from (15.11.7) that Sp(B) is contained in [0, 11. Moreover, the relation ( x I B x) = 0 implies B * x = 0, hence x = 0, and therefore the hermitian form ( x , y ) I+ ( B x Iy ) is nondegenerate. We shall prove next that dom(T*T) is dense in E. If T‘is the restriction of T to dom(T*T), it will be enough to show that r(T’)is dense in r ( T ) ,since dom(T*T) is the first projection of r(T’),and dom(T) is dense in E. To see that the subspace r(T’) of the Hilbert space r(T )is dense in r(T ) , it is enough to show that if a vector (u, T * u) E r(T )is orthogonal to r(T’),then it is zero. Now this condition is ((u, T * U ) I (v, T v)) = 0

-

-

-

for all u E dom(T*T), or equivalently ( u [ u) + ( T . u 1 T * u) = 0; or, since T * u E dom( T*),(u I u) ( u I T* T . v ) = 0, that is to say,

+

(u I ( I

But I

+ T*T )

*

u) = 0.

+ T*T maps dom( T* T ) onto E. Hence u = 0, as required.

Finally, since B is self-adjoint, T(B) is the orthogonal supplement of J(T(B)).Since T ( B ) is the image of T ( I + T* T )under the symmetry operator S : ( x , y ) ~ + ( yx), , and since JS = -SJ, it follows that T(Z+ T * T ) is the orthogonal supplement of J(T(Z + T*T)); in other words (15.12.4) ( I + T* T)* = I + T* T, or equivalently (T*T)* = T* T. (15.12.7) A not necessarily bounded operator T is said to be normal if it is closed, if dom(T) is dense in E and and if T*T= TT* (which, we recall, implies by definition that dom(T*T) = dom(TT*)). We say that T is seyadjoint if dom(T) is dense in E and if T* = T (which implies that T is closed (15.12.4)). Clearly a self-adjoint operator is normal. It follows from (15.12.6) that, if T is any closed operator such that dom( T )is dense in E, then T* T and TT*are self-adjoint. The following theorem reduces the problem of the structure of (unbounded) normal operators to that of continuous normal operators:

Let E be a separable Hilbert space. I f N is a not necessarily bounded normal operator, then dom(N) = (i) dom(N*) and IIN.xll = IIN* ex11 for all X E dom(N). The space E is the Hilbert sum of a family (En) of closed subspaces, such that En c dom(N) and En ir;stable under N and N * for all n, so that the restriction N, of N to En is a continuous normal operator. (ii) Conversely, let Enbe a sequence ojclosed subspaces of E , such that E is the Hilbert sum of the En.For each n, let N, be a continuous normal operator on (15.12.8)

418

XV

NORMED ALGEBRAS AND SPECTRAL THEORY

En. Then there exists a unique normal operator N on E such that En c dom(N) and N , = N I En, ,for all n. The set F = dom(N) is the set of all x = x ,

1 n

(where x , E Enfor all n) mch that

c IIN,, . n

x,Il2 < +a,

and we have

N .x

= n

N , .x,

N* .x =

and

n

N , * .x,.

We shall begin by proving the first assertion of (i). We have seen in the course of the proof of (15.12.6) that the graph of the restriction of N to dom(N*N) is dense in T(N); hence, for each x ~ d o m ( N ) there , exists a sequence (y,) in dom(N*N) such that lim y , = x and lim N * y , = N . x . But for each z

E dom(N*N)

fl-

we have

n-. m

30

( I N .zllZ = ( z I N * N * z ) = ( z I N N *

*

Z)

=

I/N*

ZI~',

because N*N = NN*. Applying this result with z = y , - y m , it follows that the sequence (N* * y,) is a Cauchy sequence and therefore converges in E. Since N* is closed (15.12.4), we conclude that x E dom(N*) and N* x = lim N* * y, , whence IIN XI\ = [IN* * xII. We have therefore proved

-

n- m

that dom(N) c dom(N*). Since N** = N , the operator N * is normal and dom(N*) c dom(N). Now consider (ii). We shall first show that, if N satisfies the conditions of (ii), then N*(E,) c En. For all m # n, all y, E E, and all x , E Em, we have (x, I N* . y,) = ( N x, I y,) = 0, because N(Em)c Em.Hence N * * y , is orthogonal to all the Em( m # n), hence must lie in En.Next we shall show that, if x E dom(N), then P," N x = N P E , * x = N , ' ( P E , ' X). For all y,

E

(PE,

En, we have *

and since N * * y , (PE,N

'

I yn> = ( N = PEn N*

I yn> = ( P E , .

*

'

x I pEn

'

Yn)

=(N

'

I yn) = (x I N * ' Yn)?

y , from above, N*

'

Yn) =(NPE,

*

I Yn)

=(Nn

'

(PE,

*

x)l Y n ) ;

-

since this holds for all y , E En, it follows that the two elements P E n N x and N , (PE, x) of E are equal.

-

We shall now prove the following lemma, which generalizes (15.10.8.1):

12 UNBOUNDED NORMAL OPERATORS

419

(15.12.8.1) Let (En) be a sequence of closed subspaces of E, such that E is the Hilbert sum of the En.For each n, let T , be a continuous operator on En. Then there exists a unique closed operator T o n E such that, for all n,

En c dom(T), TI En = T, , PE,T x = TPE, x (x E dom( T ) ) . Moreover, dom(T) i$ the set F of all x = x, (with x, E E,,for all n) such that

c IlT,

< +coy and we have T * x

*

It

Since we must have

1

*

n

=

1 T, n

x,.

(7’. x)ll’ = 11 T . x1I2
= V,((Sn - {ed) = R" - {0},

this proves our assertion, since lly1)2=

n

i=1

(q')'

(Sn - {-',I))

is a polynomial. Since

S, - {eo>and S, - { - e,} cover S,, , we have defined an analytic atlas 2l= {cl, c2} on S,, . In future, whenever we speak of S, as a manifold (analytic

2 EXAMPLES OF DIFFERENTIAL MANIFOLDS: DIFFEOMORPHISMS

9

or differential), it is always the structure defined by the equivalence class of the atlas % that is meant (cf. (16.2.7)), unless the contrary is expressly stated. The differential manifold S, so defined is compact, since it is a bounded closed subset of R"" ((3.1 5.1), (3.17.3), (3.17.6), and (3.20.16)) and connected if n 2 1 by virtue of (3.19.2). We remark that, since a nonempty open set in R"cannot be compact if n 2 1, an atlas of a nondiscrete compact manifold contains at least two charts (cf. Problem 5). (16.2.4) Let X be a differential manifold, % an atlas of X, and Y an open set in X. We have seen (16.1) that the restrictions to Y of any two charts c, c' belonging to % are compatible. Hence, as c runs through %, the restrictions clY form an atlas on Y, called the restriction of % to Y, and written %IY. Moreover, the equivalence class of %lY depends only on that of %, and therefore defines on Y a structure of a differential manifold depending only on that of X. This structure on Y is said to be induced by that on X. Here again, whenever we consider an open subset Y of a differential manifold as a differential manifold, it is always the induced structure that is meant. If Y is an open set in R", then (Y, ,1 ,n) is a chart on Y, called the canonical chart. (16.2.5) Let X be a separable metrizable space and an open covering of X.Suppose we are given on each X, a structure of a differential manifold in such a way that for each pair (01, jl) the structures of differential manifold induced on the open set Xu n X, by those on X, and X, (16.2.4) are the same. It then follows immediately from the definitions that if %, is an atlas of Xuand % the union of the aU, then % is an atlas of X, whose equivalence class depends only on those of the aU. The structure of differential manifold on X defined by is said to be obtained by patching together the differential manifolds X,;it is clear that it induces on each Xu the given structure of differential manifold. (16.2.6) Let X be a differential manifold, X' a topological space, u : X + X' a homeomorphism of X onto X . For each chart c=(U,cp,n) of X, the triplet (u(U), cp u - ' , n) is a chart on X , which we denote by u(c). If c and c' = (U', cp', n') are compatible, then so are u(c) and u(c'), because (cp' o u - ' ) (cp u-')-l = cp' cp-'. As c runs through the saturated atlas of charts on X, the u(c) from a saturated atlas of X',defining a structure of differential manifold, which is said to be obtained by transporting the structure on X by means of the homeomorphism u. If X, Y are two differential manifolds, a mapping u :X + Y is a direomorphism (or an isomorphism of diflerential manifolds) if u is a homeomorphism and if the structure of differential manifold on Y is the Same as 0

0

0

0

10 XVI

DIFFERENTIAL MANIFOLDS

that obtained by transporting the structure on X by means of u. Two differential manifolds X, Y are said to be diyeomorphic if there exists a diffeomorphism of X onto Y. Remarks

Consider the real line R, endowed with its canonical structure of differential manifold (16.2.2), and let u be the real-valued function such that u(t) = t for t 5 0, and u(t) = 2t for t >= 0. It is clear that u is a homeomorphism of R onto itself (4.2.2), and we may therefore endow R with the structure of differential manifold defined by the single chart u(c), where c = (R,I,, 1) is the single chart defining the canonical structure. Since u is not differentiable at the point t = 0, the charts c and u(c) are not compatible. If XIand X2are the differential manifolds defined on the underlying space R by c and u(c), respectively, then u is a diffeomorphism of XI onto X,, and we have therefore defined on R two distinct (but isomorphic) structures of differential manifold. In other words, the identity mapping 1, is not a diffeomorphism of XI onto X, . It can be shown that, for certain values of n 7, there exist on the topological space S, several nonisomorphic structures of differential manifold, having the same underlying topology. It can also be shown that the only connected differential manifolds of dimension 1 are (up to diffeomorphism) R and S, (Problem 6).

(16.2.7)

Let X be a differential manifold, X' a set, u: X + X' a bijection of X onto X . We can begin by transporting the topology of X to X' by means of u, by defining the open sets in X' to be the images under u of the open sets in X. Since u then becomes a homeomorphism of X onto X' we can transport to X' (again by means of u) the structure of differential manifold on X,as explained in (16.2.6). (16.2.8)

PROBLEMS 1. Show that the space T"= R"/Z" (12.11) is endowed with a structure of real-analytic manifold for which there exists an atlas of n 1 charts whose images are translationsin R" of the open cube I", where 1 = 10, 1[ (cf. (16.10.6)).

+

2. (a) Let K be a compact subset of R" and B a closed ball whose interior contains K. Show that for each E > 0 there exists a homeomorphismf o f R" onto itself, such that f ( x ) = x for all x 6 B and such that the diameter off(K) is 6 E. (We may assume that 0 is the center of B; takefto be of the formf(x) = xtp( Ilxll), where tp is a suitably chosen real-valued function.)

2 EXAMPLES OF DIFFERENTIAL MANIFOLDS: DIFFEOMORPHISMS

11

(b) Let W be an open set in Rn and a a point of W. Show that, for each open ball B with center a contained in W, there exists a homeomorphism of W onto an open neighborhood of a contained in B, which coincides with the identity on a neighborhood of a (same method).

3. Let X be a metrizable space and A a compact subset of X such that there exists a fundamental system ( v k ) of relatively compact open neighborhoods of A which are all homeomorphic to R". In these conditions, the space X/A (Section 12.5, Problem 10) is homeomorphic to X; moreover, for each relatively compact neighborhood U of A, there exists a homeomorphism h of X/A onto X such that, if w :X + X/A is the canonical mapping, h w coincides with lx on X - U . (We may restrict ourselves to the case where U = V1,and Vk+' C v k . Using Problem 2(a) and (3.16.5), show that there exists a sequence ( g k ) of homeomorphisms of X onto itself with the following properties: (i) g1 = l X ;(ii) g k + l agrees with g k on a neighborhood of X - v k ; (iii) the diameter of g k ( v k ) is s l / k . Deduce that the sequence ( g k ) converges uniformly to a continuous mapping g of X into itself such that g(A) is a single point, and show that g factorizes into h w ; h is the required mapping.) 0

0

4. Let A be a compact subset of R", and suppose that there exists a homeomorphism h of Rn/A onto an open set W in R" such that, if r : R"?-. R"/A is the canonical mapping, we have h(w(A))= (a}. Show that there exists a fundamental system ( v k ) of relatively compact open neighborhoods of A which are homeomorphic to R". (Using Problem 2(b), show that there exists a fundamental sequence ( u k ) of relatively compact open neighborhoods of a in R",and for each k a homeomorphism A of W onto u k , which coincides with 1 R n on a neighborhood of a. Take v k = n - - ' ( h - ' ( u k ) ) , and show that there exists a homeomorphism g k of R" onto V kwhich coincides with 1,. on a neighborhood of A and is such that h o w o gk = A o h w . ) 0

5. Let M be a compact connected metrizable space which has an open covering consisring of two subspaces X, Y each homeomorphic to R". Then M is homeomorphic to S. (Morton Brown's theorem). (If A = M - Y c X, observe that M/A is homeomorphic to S., hence X/A is homeomorphic to an open set in R"; then apply Problems 3 and 4.) 6. Let X be a connected differential manifold of dimension 1. Then there exists an atlas ((Uk,v k ,l)), where the vk(Uk)are open intervals in R, such that ( u k ) is an at most denumerable locally finite covering of X by relatively compact open sets, and such that v k extends to a homeomorphism of an open neighborhood of onto an open interval in R. but that neither of (a) Suppose that u h n u k # Show that there are just two possibilities:

u h

, Ur is contained in the other.

(a) ph(Uhn U,) is an interval, one of whose endpoints is also an endpoint of v h ( u h ) , and vk(Uhn U,) is an interval, one of whose endpoints is also an endpoint of vk(uk);

(8)

vh(Uhn u k ) is the union of two disjoint intervals, each of which has an endpoint which is also an endpoint of v h ( u h ) , and likewise for P k ( u h n u k ) and P k ( u k ) . In this case (/?),show that every other U, is contained in u h U u k and that X is diffeornorphic to S1.

12

XVI DIFFERENTIAL MANIFOLDS

(b) Deduce from (a) that X is diffeomorphic to either R or S1.(Assume that X is not Mwmorphic to S1,that each uk+1intersects U1u . u u k and that neither of these two sets is contained in the other. Using (a) and induction, construct a diffeomorphism fc of Ul u u U, onto an open set in R,such thatft+lextendsx.) +

3. DIFFERENTIABLE MAPPINGS

Let X, Y be two differential manifolds. For each integer p 2 0, a mapping f :X + Y is said to be p times continuously differentiable (resp. indefinitely dzferentiable) iff is continuous on X and satisfies the following condition: for each pair of charts (U, cp, n) and (V, JI, m) of X and Y, respectively, such that f(U) c V, the mapping F=JI0(flU)ocpp-' : cp(u)-+$(v) (which is called the local expression off for the charts under consideration) is p times continuously differentiable (resp. indefinitely differentiable) (8.12). (Forp = 0 we make the convention that the derivative of order 0 of F is F itself.) (16.3.1) For a mapping f :X -+ Y to be p times continuously diflerentiable (resp. indefinitely differentiable) it is necessary and suflcient that for each xo E X there exists a chart (U, cp, n) of X, a chart (V, $, m) of Y and a mapping F :cp(U) + $(v) which is p times continuously diflerentiable (resp. indefinitely diflerentiable), such that xo E U, f(xo) E V and such that f I U = JI-' o F 0 cp.

The condition is clearly necessary. Conversely, suppose that it is satisfied. Clearly it implies that f is continuous on X. Let ( U , cp', n') and (V', $', m') be charts of X and Y, respectively, such thatf(U')c V'. We have to show that $' (f1 U') 0 q'-' satisfies the appropriate differentiability condition. For each xo E U', let (U, cp, n), (V, JI, m), and F be two charts and a mapping satisfying the conditions of the proposition. Then first of all we have n = n' and m = m', because xo E U n U and&) E V n V'. Replacing U and U' by U n U', and V and V' by V n V', we may assume that U = U' and V = V'. However, then we have 0

$' o(f

lu)

o

q'-' =($'

o $-I)

o

F o (cp o cp'-')

and the result follows from the definition of compatible charts (16.1) and from (8.12.10). In the notation of (16.3.1), if z = (I;i)lsisn F off is of the form F(z) = F(C1,. .., = (F'(C', . . ,

r)

E

cp(U), the local expression

. r),..., FYI;', ..., r))

DIFFERENTIABLE MAPPINGS

3

13

sj

where the Fj (1 p m) are scalar functions defined on cp(U); to say that F is p times continuously differentiable (resp. indefinitely differentiable) means that the Fj have this property (8.12.6). If (cp'), ($j) are coordinates in U, V, respectively (16.1), we have (16.3.1.1)

+j(f(x))= Fj(cpl(x), ...,cp,(x))

(1 S j 5 m)

sj s

for all x E U. The Fj (1 m) are said to constitute the local expression off for the given charts. A p times continuously differentiable (resp. indefinitely differentiable) mapping is also called a mapping of class C p (resp. a mapping of class C", or a morphism of differential manifolds). A mapping of class C p (resp. C") into R is also called (if there is no risk of confusion) a function of class C p (resp. of class C") defined on X. It is clear that a mapping of class CP (where p is an integer or co) is also of class Cqfor all q < p.

The sum andproduct of two functions of class C p on X are functions of class CP. I f f is a function of class CP such that f ( x )# 0 for all x E X , then Ilfis a function of class C p .

(16.3.2)

This follows from (8.12.9), (8.12.10), and (8.12.1 1). (16.3.3) (i) Let X, Y, Z be three differential manifolds and f:X -,Y , g :Y -,Z two mappings. I f f and g are of class C p( p an integer or a),then so is g of. (ii) For a mapping f : X + Y to be a diffeomorphism of X onto Y it is necessary and sufficient that f be bijective and that f and f - be of class C".

(i) Let x E X and let (U, a, a) and (V,j?, b) be charts on X and Y, respectively, such that x E U, f ( x ) E V and f I U = j?-' o f i 0 a, where fi is of class Cp. Likewise, let (V', j?', 6') and (W, y. c) be charts on Y and Z, respectively, such that f ( x ) E V', g ( f ( x ) )E W and g1 V' = y-' 0 g1 0 b', where g1 is of class C p . Replacing V and V' by V n V', and U by f -'(V n V'), we may assume that V' = V, from which it follows that b' = b. We have then

u =Y-l

( 9 0f)l

O

(Sl

O

(8' j?-9oh) O

O

a,

and the result now follows from the definition of compatible charts and from (8.12.10). (ii) The necessity of the condition is an immediate consequence of the definitions. To prove the sufficiency it is enough to show, for each chart c = (U, cp, n) on X, if we put f l =f I U, that f(c) = (f(U), cp 0 f T1,n) is a chart on Y. Since f is a homeomorphism, it is clear first of all that f ( c ) is a

14

XVI

DIFFERENTIAL MANIFOLDS

chart of the topoIogicalspace Y, and it is enough to show that it is compatible with every chart c' = (V, $, rn) of the manifold Y (16.1). We may assume that V =f(U), and then it follows from the definition of morphisms and from the hypotheses that $ (cp of;')-' = $ o f i cp-' and (cp of;') $-I are indefinitely differentiable. Hence the result. 0

0

0

Examples (i) When X and Y are open subsets of finite-dimensional real vector spaces, the definition of a mapping of class C p (p an integer or m) agrees with that of (8.12), by virtue of (16.2.2). If X is any differential manifold and (U, cp, n) is any chart on X,then cp is a diffeomorphism of U onto the open set cp(U) in R". Conversely, every diffeomorphism cp of an open set U in X onto an open set cp(U) in R"defines a chart (U, cp, n) on X. If Y is an open subset of a differential manifold X,the canonical injection of Y into X i s a mapping of class C". (ii) The mapping (16.3.4)

(16.3.4.1)

f : X W

2x 1-

llXllZ

is a diffeomorphism ofthe open ball B: llxll < 1 in R" (Ilxll being the Euclidean norm (16.2.3)) onto R".The inverse diffeomorphism is (16.3.4.2)

The mapping (16.3.4.3)

g :X H -

X

lIxIlZ

is a diffeomorphism of the exterior llxll > I of B onto the complement of (0)in B. The cornpositionfo g is therefore a diffeomorphism of the exterior of B onto R" - (0). (iii) For the two manifolds XI, X, defined in (16.2.7), both of which have

R as underlying space, the identity map 1, is not of class C ' , whether considered as a mapping from X, to X, or from X, to X,. If u is the real-valued function t~ t 3 (which is a homeomorphism of R onto itself) and if we endow R with the structure of differential manifold defined by the single chart u(c), we obtain a differential manifold X, again having R as underlying

3

DIFFERENTIABLE MAPPINGS

15

space and distinct from both Xi and X2.This time, the mapping l,, considered as a mapping from X, to X, , is of class C" but is not a diffeomorphism, since the inverse mapping is not even of class C'.

Remark (16.3.5) If in the definition of a mapping of class C" we replace differential manifolds by real-analytic (resp. complex-analytic) manifolds, and indefinitely differentiable mappings of open sets of R" (resp. C")into R"'(resp. Cm) by analytic mappings (9.3), we arrive at the definition of an analytic mapping of one real-analytic (resp. complex-analytic) manifold into another. In the complex case, such mappings are also called holomorphic. We leave to the reader the task of formulating for such mappings the analogues of the propositions of this section.

PROBLEMS

Let X be a pure differential manifold (resp. a real-analytic manifold, r a p . a complexanalytic manifold) of dimension n. For each open set U c X, let F(U) be the set of C"-mappings of U into R (resp. real-analytic mappings of U into R, resp. complexanalytic mappings of U into C). Show that the sets F ( U ) have the following properties: (a) For each open set V C U, the restrictions to V of the functions f~ F ( U ) belong to FW). (b) For each open set U c X and each covering (U.) of U by open sets contained in U, if a function fdefined on U is such that f l U. E F(U.) for each a,then fe F(U). (c) For each point x E X, there exists a homeomorphism u of an open neighborhood U of x onto an open set in R"(resp. R",resp. C") such that, for each open set V c U, FW)is the set of all functions of the form g 0 u where g runs through the set of C mmappings of u(U) into R (resp. real-analytic mappings of u(U) into R, resp. complexanalytic mappings of u(U) into C). Conversely, let X be a separable metrizable space and suppose we are given, for each open set U in X, a set F ( U ) with the above properties. Show that there exists a unique structure of differential manifold (resp. real-analytic manifold, r a p . complex-analytic manifold) on X for which F ( U ) is the set of C"-mappings of U into R (resp. realanalytic mappings of U into R, resp. complex-analytic mappings of u into C) for each open set U in x. (Observe that, if u = (u', . ,uJ), the functions UJ belong to F(U).)

..

In C, considered as a complex-analytic manifold, every nonempty simply connected open set other than C is isomorphic to the unit disk 1 z 1 < 1, and the latter is not isomorphic to C. Hence there are two classes of simply connected nonempty open sets with respect to the relation of isomorphism (Section 10.3, Problem 4). Deduce that in the plane R2 any two simply connected nonempty open sets are diffeomorphic. Give an example of two nonisomorphic complex-analytic manifolds having the same underlying structure of differential manifold.

16

XVI

DIFFERENTIAL MANIFOLDS

3. (a) Let X,Y be two connected real-analytic(resp.complex-analytic)manifoldsandf,g two analytic mappings of X into Y.Show that if there exists a nonempty open set U c X on which f and g agree, then f= g (9.4.2). (b) Let X be a connected complex analytic manifold and let f be a holomorphic complex-valued function on X,not identically zero. Show that the set of points x E X such thatf(x) # 0 is a connected dense open set. (If a, b are two points of X,show that there exists a sequence ( c ~ of )points ~ of ~ X~such ~ that ~ co = a, c. = b and such that ,n - 1, the points ct and ct+l both belong to the domain of definifor each i = 0, 1, tion U of a chart (U, 'p, n), where cp O

t-'D"h(t) = 0 by (16.4.1.1). This

proves (16.4.1.2) by induction on n.

(16.4.1.4) Let I be the interval [ - I ,

+

I ] in R. There exists a function g of class C" on R" which is > O in the interior of K = I", zero on the exterior of K, and such that

S-S

g(tl,

. . . , t,)

dt,

. . . dt,

= 1.

Put h,(t) = h( 1 + t)h( 1 - t ) , where h is the function defined by (16.4.1.3). Then we may take g ( t l , .. . , t,) = ch,(t,) . . . ho(t,) with a suitable constant c. ( 16.4. I .5) .End ofthe proof

Let M be a compact subset of X, and N a closed subset of X such that M n N = /zr. For each x E M , there exists a chart ( U , , cp,, n,) such that x E U,, U, n N = /zr, cp,(U,) 2 I" and cp,(x) = 0. The real-valued function f, which is equal tog cp, on U, and 0 on the complement of U, is of class C" and is >O on V, = cp-'(i"), which is an open neighborhood of x . We can cover M by a finite number of such neighborhoods V X i ; the function is of class C", vanishes everywhere on N and is >O every0

x,fxi I

where on M. If f = d-'

c(

=

inf c , f x i ( x ) ,we have a > 0 (3.17.10), and the function

xsM

i

Ii f x satisfies i condition ( I )

of (12.6.5).

(16.4.2) Let X be a differential manifold, K a compact subset of X, and (Ak)lsksm a j n i t e covering of K by open subsets of X. Then there exist m functions fk of class C" on X with values in the interval [0, I ] such that Supp(f,) c A, for I 5 k 5 m , I . f k ( x ) = 1 for all x E K and x f k ( x ) 5 I for

all x E X.

k

k

This follows from the preceding result and from (12.6.5) and (12.6.4). (16.4.3) Let X be a differential manifold, F a closed subset of X, and g a mapping of F into R. Suppose that,,for each x E F, there exists an open neighborhood V, of x in X and a function,f, of class C' ( r an integer or co) on Vx which is equal to g on V, n F. Then for each open neighborhood U of F there exists a,functionf of class C' which is zero on the complement of U and equal to g on F.

+

4 DIFFERENTIABLE PARTITIONS OF UNITY

19

For each X E CF, let V, be an open neighborhood of x which does not intersect F. For each x E F, on the other hand, we may assume that V, c U (by replacing V, by V, n U). Let (A,,) be a denumerable open covering of X which is locally finite and finer than the covering (V.J,.x(12.6.1), For each n, choose an x such that A,, c V,. If x # F, let f,, denote the zero function on A,,, and if x E F, letf. denote the restriction off, to A,,. If (h,,) is a C" partition of unity on X subordinate to the covering (A,) (16.4.1),then the function gnwhich is equal to h,,f,, on A,, and is zero on the complement of A, is of class c' on X, and the functionf = g,, satisfies the required conn ditions.

(16.4.4) For brevity we shall say that a function g with the property stated in (16.4.3) is of class C' on F (although in general F is not a differential manifold); equivalently, g is the restriction to F of a function of class C' on X (cf. Problem 6).

PROBLEMS

Let KO,K1be disjoint closed subsets of the sphere S.. Show that there exists a C "function f o n R"+'- (0) which is equal to 0 on KOand to 1 onK, ,satisfiesf(tx) = f ( x ) for all real numbers t > 0 and is such that for each multi-index a,Ilxlll"Wf(x) remains bounded as x + O (Ilxll denotes the Euclidean norm). For each C"-function g on Rn+' such that D"g(0)= 0 for each multi-index a,the function gfextends to a C"-function on R"+l. Let (xk)kk be a sequence of distinct points of R" tending to 0. For each k let akbe a multi-index such that I akI + m, and let ( c k , v ) v Nn be a multiple sequence of numbers such that ck,v = 0 for 1 v1 5 I I and v # a,. Show that there exists a c "-function f on R"with the following properties: (a) DVf(xt) = ct," for all v E N"; (b) D'f(0) = 0 for all Y. (Use the method of Problem 4 of Section 8.14 to construct by induction a sequence of C"-functionsfk, whose supports are pairwise disjoint and do not contain 0, such that (i) D"ji(xk)= ck,v for each multi-index v, and (ii) llD"hiI 5 T kfor all v such that (YI < l a k ( . Then takef=xfk.)

+

k

Let X be a differential manifold. Show that for each x E X there exists a chart (U, v, n) at the point x such that 9 is the restriction to U of a C"-mapping of X into R". Let F be a closed subset of R"and U its complement. Let h, k, 1) be three numbers in the interval 10, I[. Show that there exists a denumerable covering of U by open Euclidean balls B(Q, r l ) with the following properties:

20

X V I DIFFERENTIAL MANIFOLDS

(a) the balls B(ui, kri) cover U; (b) rl = hd(ul, F) for each i; (c) there exists an integer N(h, k, 9) depending only on h, k, 7 such that for each x E U the closed ball with center x and radius qd(x, F) meets at most N closed balls B'(uI,rd. (Let e be a real number >O. For each m E Z, let F,,, be the set of points x E U such that d(x, F) = (1 E ) ~ and , let T, be an at most denumerable subset of F, consisting of points whose mutual distances are 2 e(1 e)'", and such that the open balls with centers at these points and radii equal to &(I 6)"' cover F, . Let (u,) be the sequence T,, arranged in any order. Show that if we take consisting of the points of

+

u

+ +

INEZ

ri = hd(ui, F), the required conditions are satisfied provided that E < &hk.If x E U and 8 = d(x, F), observe that there exists rn f Z such that (1 + E)"' 8 < (1 + E)"+', and deduce that d(x, T,) 5 2 4 1 + e)'". Then show that there exist two constants c > 0 and C > 0, depending only on h and 7, and such that (i) if B'(ui, r,) meets the ball with center x and radius 78,then d(x, ul)5 C8, and (ii) if j # i is another index with the same property, then d(ui, u,) 2 c8.)

5. With the notation of Problem 4, put B1= B(ui, r,). Show that there exists a Cmpartition of unity (U,)subordinate to the covering (B,), and for each a E N"a constant C. such that IID"ur(x)ll5 Cu(d(x,F))-Iuifor all x E R",i and m. 6. Let F be a closed subset of R".Generalizing the definition of a mapping of class C', a mapping f : F -+R" is said to be of class C' (resp. C ") if for each multi-index a E N" such that I a I r (resp. for each multi-index a) there exists a mapping f.:F -+ R" with fo = A such that the following conditions are satisfied: if for each integer s 5 r (resp. each integer s => 0) we write

where x E F, z E F, and I a I 5 s, then for each xo E F, each E > 0, and each pair (a, s) with 1 a ( 9 s, there exists p > 0 such that [ I R J x , z)ll e(Ix - z((J-l'l for all x, z E F such that IIx - xo II < p and I I z - xo II < p. These conditions imply that thef. are continuous on F. When F = RR,this definition is equivalent to the previous definition of functions of class C (resp. C "). (a) Show that if the mapping f:F 4 R" is of class C' then f can be extended to a mapping h :R"-+ R" of class C' on R" and of class C" on U = CF, and hence the definition above agrees with (1 6 4.4). (The Taylor polynomial of order s 5 r off at the point z E F is the polynomial in xl, ...,x"

With the notation of Problem 5 , show that the function h defined by

where b, E F is such that d(ui, b,) = &ai, F), satisfies the required conditions. For this purpose, show that if 1 a I r , then D"h(x)- D"T:. f ( x ) -+ 0 as x 4 u E Fr(F), where

DIFFERENTIABLE PARTITIONS OF UNITY

4

21

xo E F is such that d(x, xo) = d(x, F). Using Leibniz's formula, this reduces to majorizing the norm IlD6T;,f(x) - DBT:,f(x)ll for fi 5 a, by using the results of Problems 4 and 5.) (b) Show that iffis of class C" there exists an extension h offto R"which is of class C (Whitney's extension theorem). As a consequence, the definition above agrees with that of (16.4.4). (For each integer r, show that there exists a number d, such that the relations z E F, x E R",IJx- zll 5 d,, s 5 r, and I a I 5 s imply IID"T:f(x) - D"T:f(x)ll I_Ilx - ZIP-'". If V, is the neighborhood of F consisting of the points x such that d(x, F) 5 id,, let rl denote the largest r such that a, E V, (we can always suppose that the sequence (d,) tends to 0). Then define f( x ) if X E F , h(x) = u,(x)z:f(x) if x E v,

{

the points b, being defined as in (a). Show that h has the required properties by arguing as in (a).) With the notation of Problems 4 and 5, suppose that F is compact. Show that for each p > 0 one can define a function up 2 0 of class C? on R",such that u,(x) = 1 whenever d(x, F) 5 p , u,(x) = 0 whenever d(x, F) 2 2p, and such that for every functionf: Rn+R of class C' which vanishes on F together with all its partial derivatives D9f of order 1 a I 5 r, the functions u , f , and their partial derivatives D"(u,f) of order I a I 5 r tend uniformly to 0 on R" as p +0. (Take u, to be a sum of certain of the functions u, defined in Problem 5.) Let F be a closed subset of R" and let f b e a real-valued function of class C' on F, and

g a real-valued function of class C' on the open set &F.For each z E R" let P, be the polynomial in x ' , . . .,x" which is equal to T: fif z E F, and is equal to T:g if z $ F, in

the notation of Problem 6. Show that there exists a function h of class C' on R"such that P, = T: h for all z E R"if and only if the coefficients of P, are continuous functions of z. (Reduce to the case r = 1 by induction, then to the case n = 1 by using (8.9). Then we have P,(x) = a(z) ( x - a)b(z) where a and b are continuous functions of z E R. Reduce to the case b = 0 and then show that the function a(z) has zero derivative at each z E R.)

+

(a) Let f be a real-valued function 20 of class C2 on a neighborhood of 0 in RN. Suppose that f a n d its derivatives of order (2 all vanish at 0, and that there exist positive real numbers c, M such that IDID I f ( x )I 5 M for all pairs of indices (i,j) and all x E RN such that IxJl 5 2c (1 N). Show that if 1x11

we have (1 )

+

sj

1x21

+...+

lXNl

sc,

I D,f(x)I 5 2 M f M

(1 S j I_ N). 5 c implies that I D J f ( x ) l (= Mc for all j . Then J by supposing that at some point x satisfying I x'l 5 c the

(Observe first that the relationx 1 xJl argue by contradiction,

I

inequality (1) is false for some index j , withf(x) > 0; use Taylor's formula to conclude 91 2c (1 < j 5 N).) thatf(y) < 0 for some pointy such that 1 (b) Let f b e a real-valued function 20 of class C2 on an open set U in R",such that


0. Finally let (E.) be a sequence of strictly positive numbers tending to 0. Put s, = 1

+ 2(a1+ ... + an-1)+ an,

Show that the function g(x) = ~ g . ( x ) is 2 0 and of class C" on R, and that all its n

derivatives vanish at the zeros of g. For each p E 10, A[, show that it is possible to g is not of class C ' ; also that it is possible choose the sequence (E.) so that the function ' to choose the sequence (E.) so that the function g1'2(which is of class C1 by virtue of (b) above) is not of class C2. 10. Let E be a finite-dimensional real vector space and let M, ( 1 sj 5 r ) be vector sub-

spaces of E. Show that the following conditions are equivalent:

(a) If mj=codimMj, then for each subset H of [ l , r ] in N the codimension of (I M,is m,. JEH

J€H

(b) The sum of the annihilators M,O C E* of the M, in the dual E* of E is direct. (c) There exists a direct sum decomposition of E of the form P @ N1@ . . . @ N, such that each M j is the direct sum of P and the N kwith k # j .

(d) I f P =

n

15jCr

M j , t h e n c o d i m P = x m,. I =I

(To show that (a) implies (c), observe that if P =

n

lSJSI

MI and

Qj

=

n Mk, then

k f l

P has codimension m, in Q j .) A family of vector subspaces MIsatisfying these conditions is said to be ingeneral position in E. Let V be the union of the M, and letfbe a real-valued function on V such that the restriction offto MI is of class C' for 1 5 j 5 r . Show thatfis the restriction to V of a function of class C' on E. (Proceed by induction on r.) 11. Give an example of a C"-mapping f: R+R2 such that f(R) is the square sup( Ix 1 I, 1x2 I ) = 1.

5. T A N G E N T SPACES, T A N G E N T LINEAR MAPPINGS, RANK

(16.5.1) Let X, Y be two differential manifolds, x a point of X. Let f,,fi be two C'-functions, each defined on an open neighborhood of x, with values in Y. The functionsf,,fi are said to be tangent at the point x if

5 TANGENT SPACES, TANGENT LINEAR MAPPINGS, RANK

23

f i ( x ) =fz(x) and if the following condition is satisfied: If (U, cp, n) is a chart of X at x such that U is contained in the domains of definition of f l and f i

and if (V, $, m) is a chart of Y at the pointfl(x) such thatfi(U) and f z ( U ) are contained in V, then the functions o (fi I U) cp-' and $ o (fzI U) o cp-' (that is to say, the local expressions of fl and f i ) are tangent at the point cp(x) (8.1), i.e., they have the same derivative at this point. If this condition is satisfied for one choice of the charts (U, cp, n) and (V, $, m),then it is satisfied for any other pair of charts (U',cp', n) and (V', $', rn) satisfying the same conditions. For we may assume without loss of generality that U = U' and V = V', and then we have 0

$'o(fi~U)ocp'-'=(II/'~II/-')~($~(f;~u)~cp-').(cp'"cp-')-' for i = 1, 2, and the assertion therefore follows from (8.2.1). Furthermore, it follows immediately from this definition that the relation ''fland f 2 are tangent at the point x " is an equivalence relation.

(16.5.1.1) Consider in particular the real line R, a differential manifold X, a point x E X, and the relation " f l and f 2 are tangent at the point 0" between two functions f l and f z of class C', defined on an open neighborhood of 0 in R, with values in X and such that f l ( 0 ) =f2(0)= x. The equivalence classes for this relation are called the tangent vectors to X a t the point x , and the set of them is denoted by TJX). Let c = (U, cp, n ) be a chart on X at the point x . Then the definition just given shows that we obtain a bijection 0, :TJX) -+ R" (also denoted by 0,. ). by mapping the equivalence class of a mapping f :V -+ X (where V is an open neighborhood of 0 in R) of class C' and such that f ( 0 ) = x, to the vector (D(q f))(O). The inverse of this bijection maps a vector h E R" to the tangent vector, belonging to T,(X), which is the equivalence class of the mapping t w p - ' ( p ( x )+ (h), where 5 belongs to a sufficiently small neighborhood of 0 in R. If c' = (U, cp', n ) is another chart of X at x (we may assume that c' and c have the same domain of definition), then the mapping 0,. 0 0;' is the bijective linear mapping 0

(1 6.5.1.2)

0,, 0 0:': hM(D(cp'0 cp-')(cp(x))* h.

It follows that we can define a structure of a real vector space of dimension n on T,(X) by transporting by means of 0,' the vector space structure of R", that is to say by defining 0,-'(h)

+ 0,-'(h')

= B,-'(h

+ h')

and

1 * 0,-'(h)

= O,-'(Lh)

24

XVI DIFFERENTIAL MANIFOLDS

for 1 E R;moreover, this vector space structure is independent of the choice of the chart c because the mapping (16.5.1.2) is linear. The set T,(X), endowed with this vector space structure, is called the tangent vector space, or simply the tangent space, to the differential manifold X at the point x. If (ei)’ is the canonical basis of R”,the tangent vectors ec;:(ei) (1 5 i 5 n) form a basis of the tangent space T,(X). This basis is said to be associated with the chart c. The reader should beware of confusing the notions of tangent vector and tangent space defined here with the elementary notions of “tangent vector” or “tangent plane” defined for ordinary “surfaces” in R3.The relationship between these notions will be made clear in (16.8.6). For each tangent vector h, E T,(X),

is called the local expression of h, , relative to the chart c.

Example (16.5.2) Let E be a real vector space of dimension n, endowed with its canonical structure of differential manifold (16.2.2). For each linear bijection cp: E + R” and each x E E, the triplet c(cp, x ) = (E, cp, n) is a chart on E, hence defines a linear bijection O,,,, x) :T,(E) + R”,and therefore by composition a linear bijection (16.5.2.1)

2,

= cp-1

eCc(,,,) : TJE)

-,E

which is independent of the linear bijection cp, by virtue of (16.5.1) and the relation D(q’ cp-’)(cp(x)) = cp‘ 0 cp-’ for two linear bijections cp, cp’ of E onto R”(8.1.3). The bijection 2, is called canonical. 0

(16.5.3) Now let X, Y be two differential manifolds, f:X -+ Y a mapping of class C1, x a point of X, and y =f(x). Let c = (U, cp, n) and c‘ = (V, $, rn) be charts of X and Y at x , y , respectively, such thatf(u) c V, and consider U)0 cp-’ off relative to c and c’. This local the local expression F = $ o expression is a C’-mapping of cp(U) into $(V), and its derivative F’(cp(x)) (8.1) is therefore a linear mapping of Rninto R”.We shall show that the linear mapping

(fl

(16.5.3.1)

T , c ~ )= e,

1

~‘(cp(x)) e, : T,(x)

-+

T,(Y)

is independent of the choice of charts c, c‘ at x, y. For if we replace c and c’ by two other charts c1 = (Ul, cpl,n) and c; = (Vl, $‘,rn) at x and y ,

5 TANGENT SPACES, TANGENT LINEAR MAPPINGS, RANK

25

respectively, we may assume that U = U, and V = V,, by replacing U and U, by U n U,, and V and V, by V n V1;then the local expression o f f relative to the charts c, and c; is ($, o$-')oF0(cpl o c p - ' ) - ' , and the assertion follows from (8.2.1) and (16.5.1.2). The mapping T,( f ) is called the tangent linear mapping to f at the point x. For a point z = (ci) E R", put F(z) = (F1(cl,..., p), ...,Fm(cl, .., p)). Since the matrix of F'(cp(x)) with respect to the canonical bases is the Jacobian matrix (Dj F'(cp'(x), . . .,cp"(x))) of type (m, n) (8.10), this Jacobian matrix is (8;'(ej)),,,,,. also thematrixofT,(f)relative to the bases(8;'(ei))l.i..and The mapping F'(cp(x)), or its matrix relative to these bases, is called the local expression of T x ( f )relative to the charts c and c'. To say that f and g are tangent at a point X E X therefore means that f(x) = g(x) and TXU) = Tx(g). The rank of the linear mapping Tx(f) is called the rank o ff at the point x and is denoted by rkxCf). The mapping xwrk,(f) of X into the discrete space N c R is lower semicontinuous on X ((10.3) and (12.7)). We have

.

rkx(f

1 5 inf(dim,(X),

dimf(*)(V).

(16.5.4) Let X, Y, Z be three differential manifolds, and f :X + Y, g : Y -t Z two mappings of class C' . For each x E X,we have (16.5.4.1)

Tx(g o

f

1 = Tf(X,(S)

O

TAf 1-

This follows immediately from the definitions and from (8.2.1). Let X beadirereenrialmanifold,Yadflerentialmanifold, andf: X + Y a mapping of class C'. For f to be locally constant on X it is necessary and suflcient that T,( f ) = 0 for all x E X (or equivalently, that rk,(f) = 0 for all x E X). (16.5.5)

It is clear that the condition is necessary. Conversely, since each point

x E X has a connected neighborhood contained in the domain of definition of a chart at x, the relation TJf) = 0 for all x implies that f is locally con-

stant (8.6.1).

If X is connected, the condition Tx(f) = 0 for all x E X therefore forces f to be constant on X, because if xo E X the set of points X E X such that f(x) =f(xo) is both open and closed (3.15.1). Let X, Y be two differentialmanifoldr,f : X + Y a mapping of class C' (r an integer >O, or co),x a point of X. Then the following conditions are equivalent : (16.5.6)

26

XVI

DIFFERENTIAL MANIFOLDS

(a) T,(f) is a bijective linear mapping; (b) rk,Cf) = dim,(X) = dimf(,)(Y); (c) There exists an open neighborhood U of x in X such that f lU is a homeomorphism of U onto an open neighborhood V of f(x), and the inverse homeomorphism is of class C'. The equivalence of (a) and (b) is linear algebra (A.4.18). For the equivalence of (a) and (c) we reduce immediately, by using charts, to the situation where X = R" and Y = R"',and then the result follows from (10.2.5). When the conditions of (16.5.6) are satisfied with r = oc), the mapping f is said to be a local direomorphism at x, or &ale at x, and X is said to be &taleover Y at thepoint x (relative to f). If X is an open subset of Y, endowed with the induced structure of differential manifold (16.2.4), then the canonical injection of X into Y is Ctale.

Remark

A bijective local diffeomorphism is clearly a diffeomorphism, but a mapping f:X -,Y can be a local diffeomorphism at each point of X without being injective, even if X is connected. An example is the analytic mapping ZHZ' of C - {0} onto itself (cf. (16.12.4)).

(16.5.6.1)

(16.5.7) Now let X be a differential manifold, E a finite-dimensional real vector space, f :X E a mapping of class C', and x a point of X. Then the linear mapping (cf. (1 6.5.2))

is an element of Hom(T,(X), E), called the dzrerential off at the point x, and denoted by d, f. In the particular case where Xis also a finite-dimensionalreal T,(f) Z; is prevector space G, it is immediate that the mapping z,,, cisely the derivative Df(x) defined in (8.1) (an element of Hom(G, E)). Hence in this case we have, if h, E T,(X), 0

(16.5.7.2)

0

d, f * h, = Df(x) * z,(h,).

If u is any linear mapping of E into another finite-dimensional real vector space F, it follows immediately from the above definition that (16.5.7.3)

d,(u o f) = u o d,f.

5 TANGENT SPACES, TANGENT LINEAR MAPPINGS, RANK

In particular, if we take a basis (bj),

27

of E, so that

where the f j are real-valued functions of class C' on X, then by taking for u in (16.5.7.3) the coordinate functions on E we obtain rn

(16.5.7.4)

d, f * h, =

C (d,f j , h,)bj

j= 1

for h, E T,(X). This is also written in the abbreviated form (16.5.7.5)

(instead of C ( d , f j ) @ b,, which is the correct form when T,(X)* @ E i

is identified with Hom(T,(X), E)). The differentials d, f j belong to the dual T,(X)* of T,(X). The elements of this dual space are called tangent covectors to X at x (or simply covectors at x). (16.5.8) Let c = (U, rp, n) be a chart on X at x . Then the bijection 0, introduced earlier is given by (16.5.8.1)

0, = d,rp,

for if we take the chart c' = (R",I,,, n ) on R",the definition (16.5.3.1) shows that T,(rp) = 0; o 0,, and our assertion follows from (1 6.5.2.1) and the definition of the differential (16.5.7.1). This shows that the covectors d,rp' form the basis dual to the basis (OF1(ei)) of T,(X). This dual basis is likewise said to be associated with the chart c. From this result and the definition (16.5.7.1) we see that if f is a C'mapping of X into a finite-dimensional real vector space E, and if F=(flU)Orp-': R"+E,

then

for h, E T,(X), where Di F(rp(x)), the partial derivative of F at the point q ( x ) E R",is identified with a vector in E (8.4). If we identify Hom(T,(X), E) canonically with E @I (T,(X))*, then the formula (16.5.8.2) takes the form

28

XVI DIFFERENTIAL MANIFOLDS

(16.5.8.3)

c DiF(CpW) 8 dx n

dx f =

I=1

'pi,

and in particular, when E = R (so that the Di F(cp(x)) are scalars) (16.5.8.4)

dxf =

iD, F ( 4 W 4

i=1

'pi.

These are the localexpressions of d, f and dxf relative to the chart c. Finally, consider a mapping n :Y + X of class C'. For each C'-mapping f :X + E, where as above E is a finite-dimensionalreal vector space, we have for each y E Y

and in particular, when E = R,

by the definition of the transpose of a linear mapping.

.

(16.5.9) Let X be a differential manifold, let f I , . .,f , , be n functions of class C" dejined on an open neighborhood V of a point x E X,and let f denote the of V into R".Then thefollowing conditions are equivalent: mapping (f')l

(a) There exists an open neighborhood U c V of x such that (U, f I U, n) is a chart on X at the point x ; (b) The dzrerentials d,f' (1 5 i S n) form a basis of (Tx(X))*. For if (W, cp, n) is a chart on X at x , and we put F'

d, f =

c Dj F'(cp(x)) d, cp',

=f'

0

cp-',

we have

n

j=1

and condition (b) signifies that the Jacobian matrix (DjF'(cp(x))) is invertible. The result therefore follows from (16.5.6). (16.5.1 0 ) Let f be a real-valuedfunction of class C' on a differential manifold X . I j f attains a relative minimum (resp. a relative maximum) at a point xo E X, that is to say iff@) 2 f ( x o )(resp. f ( x ) S f(xo))for allpoints x insomeneighborhood of xo , then d,J= 0.

We reduce immediately to the case X = R", and then it is enough to prove that the partial derivatives Dif ( x o ) are all zero, and so we reduce to

5 TANGENT SPACES, TANGENT LINEAR MAPPINGS, RANK

29

the case n = 1. However, thenf'(x,) is the limit at the point 0 of the function h w c f ( x 0 + h) -f(xo))/h, which is defined for all sufficiently small h # 0, and is 20 for h > 0 and $0 for h c 0. Hence the result, by (3.15.4). (16.5.11) The converse of the proposition (16.5.10) is false, as is already shown by the example of the function t I-+ t3 at the point t = 0. At the points x E X such that d, f = 0 we say that f is stationary, or that x is a criticalpoint off; the numberf(x) is called a critical value off. To see whether, at such a point, f has a relative minimum or maximum or neither, we introduce (assuming thatfis of class C2) a quadratic form on the vector space T,(X), as follows. Consider a C2-mapping u :V + X,where V is a neighborhood of 0 in R and u(0) = x. We shall show that the hypothesis that f is stationary at the point x implies that, for the real-valued function v =f 0 u of class C2, the value v"(0) depends only on the tangent vector h, which is the class of the function u. To see this, let c = (U, cp, n) be a chart of X at the point x, and let F = f o cp-l be the corresponding local expression off; we may write v = F w, where w = cp o u is a C2-mapping of V into R". Then we have, by (8.1.4) and (8.12.1), 0

d ( t ) = DF(w(t)) * w'(t),

-

v"(t)= D*F(w(t)) (w'(t), w'(t))

+ DF(w(t))- w"(t);

but by hypothesis DF(cp(x)) = 0, so that

the tranIf c1 = (U, cp,, n) is another chart on X at x and $ = cp 0 cp; sition homeomorphism, and if we put F, =f 0 c,p'; w, = cp, 0 u, then we have F, = F $, so that for all y E U and t E R" 0

DFl(cp,O) * t

-

= DF(cpdy)) . (D$(cp,(Y)) t).

Differentiating again, putting y we shall obtain

= x,

and remembering that DF(cp(x))= 0,

D2Fl(cpl(~))* (5, t) = D2F(cp(x))* (D$(cpl(X))* s, D$(cp,(xN * t).

-

Since on the other hand w = J/ w,, we have B,(h,) = D+(cp,(x)) OC,(h,) and this shows that v"(0) depends only on h,. The formula (16.5.11.1) shows moreover that there is a symmetric bilinear form on T,(x), called the Hessian off at thepoint x and denoted by Hess,(f), such that 0

v"(0) = Hess,(f)

- (h,,

hx).

30

XVI DIFFERENTIAL MANIFOLDS

The symmetric bilinear form D2F(cp(x))on R" is the local expression of the Hessian o f f at the critical point x relative to the chart c; its matrix with respect to the canonical basis of R" is therefore the symmetric matrix (DiDjF(cp(x))),called the Hessian matrix of F at the point q ( x ) (8.12.3). We have now the following suflcient criterion for a C2-function to have a relative minimum or maximum at a point of X : (16.5.12) Let f be a function of class C2 on a diferential manifold X. If at a point x E X we have d, f = 0 and fi Hess,( f ) is positive dejinite (resp. negative dejinitive), then f attains a relative minimum (resp. relative maximum), at the point x.

We reduce immediately to the case X = R". Suppose that the Hessian is positive definite. Then as h runs over the sphere Sn-l, the continuous function h H D 2 f ( x )* ( h , h) is always > O ; hence its greatest lower bound a is > O (3.17.10). Since the function (y, h ) H D 2 f i y ) * ( hh) , is continuous on X x Sn-l,there exists p > 0 such that D2f(y) ( h , h) 2 +a for all y such that Ily - xIJ p and all h E Sn-l. Now Taylor's formula (8.14.2) gives, for 5 E R,

-=

and the result follows.

Remark Let X be a complex-analytic manifold and let XI, denote the underlying differential manifold (16.1.6). As at the beginning of this section we can define the notion of holomorphic mappings f,,f 2 of X into a complex-analytic manifold Y which are tangent at a point. In particular, the tangent vectors to X at a point x will be the equivalence classes of holomorphic functions defined on a neighborhood of 0 in C, with values in X. A chart c = (U, cp, n ) on X at x defines a bijection 8, :T,(X) -+ C" as before, and we deduce that T,(X) is endowed intrinsically with the structure of a complex vector space of dimension n. However, since c is also a chart of X I , , there is also a bijection OclSi T,(X,,) + R2",and therefore, by identifying canonically C" with R2", a bijection 8;; o O,:T,(X) -+ T,(XIR) which is R-linear and does not depend on the choice of the chart c. Hence, by means of this canonical bijection, we may identifv T,(XIR)with the real vector

(16.5.13)

5 TANGENT SPACES, TANGENT LINEAR MAPPINGS, R A N K

31

space obtained by restricting the scalars to R in T,(X). Multiplication by i= is an R-automorphism J, : h,Hih, of the real vector space T,(X,,) such that J: = - I , , where I , is the identity automorphism. The notion of a diyerential is defined just as above for holomorphic mappings of X into a complex vector space E of finite dimension. The elements of the dual T,(X)* of the complex vector space T,(X) are called covectors at x. The dual T,(X)* = Homc(Tx(X), C) can be embedded canonically in HomR(Tx(XlR), C) = T,(XIR)* 8 iTx(XIR)* = (Tx(XlR)*)(q. To be precise, we have by transposition an automorphism ‘J, of T,(XIR)*, which extends canonically to a C-automorphism (also denoted by ‘J,) of (Tx(x\R)*)(C) :

‘J;(h:@'+({"')>'+*..+ (1")>'

(cf. Section 8.14, Problem 7). (b) Deduce from (a) that the nondegenerate critical points offare isolated. 5. Let G be a finite group of diffeomorphisms of a differential manifold X, and let XG be the set of points of X which are fixed by G.

(a) If x E XG,show that there exists a chart on X at x such that the local expressions of the dfleomorphisms s E G are finear mappings. (Reduce to the case where X is a neighborhood of 0 in R".I f f i s a positive definite quadratic form on R",consider the function g(x) = f(s .x) and use Section 8.14, Problem 7.) (Cf. Section 19.1, Probaee

lem 6.) (b) Deduce that XG is a closed submanifold of X (16.8.3). (c) Suppose that X is connected. If s E G is such that there exists a point xo E XGsuch that the tangent linear mapping T+) is the identity, show that s is the identity mapping. (Use (a) to show that the set of points x E X such that s(x) = x is both open and closed.) 6. If we fix an orjgin in a finite-dimensional real affine space E, the canonical topology (12.13.2) of the vector space so obtained does not depend on the choice of origin, and is

called the canonical topology on E.The dimension of a convex set in E is the dimension of the affine-linear variety generated by the set. A conuex body in E is by definition a closed convex set in E, of dimension equal to the dimension of E; equivalently, it is a closed convex set in E whose interior is not empty (Section 12.14, Problem ll(d)). A conuex polyhedron in E is the intersection of a finite number of closed half-spaces. Hence the intersection of two convex polyhedra is a convex polyhedron. Show that the frontier of a convex polyhedron P of dimension n is the union of a finite number of convex polyhedra of dimension n - 1 which are intersections of P with hyperplanes of support (Section 5.8, Problem 3) of P. These are well determined by this condition and are called the faces of P.

7.

Let E be a real affine space of dimension n and let f:E +R be a C2-function, bounded below.Suppose that for each x E E the symmetric bilinear form (h,k ) w D 2 f ( x ).(h, k) is positive definite. Show that f is strictly convex, and that for a > inf f(x) the x€E

set A. = { x E E :f ( x ) 5 a} is a closed strictly convex set of dimension n, whose frontier is the set Fa = {x E E :f(x) = a}. Through each point x E Fathere passes a unique hyperplane of support, whose equation is for all u E E. If a > inff(x), X.2E

then grad f(x) # O

for all

XE

F.. Put g(x) =

(gradf(x))/[lgradf(x)II and show that g is a homeomorphism of F. onto Sn-l and that

XVI DIFFERENTIAL MANIFOLDS

34

both g and the inverse homeomorphism ho are of class C'. For each z = t u in R", with t 2 0 and I/u11 = 1, let H(z) = (z I ho(u)), which is a C1-function on R" - (0).We have

H(z) =

SUP

yEAa

(Y lz)

(the f i c t i o n of support of Aa). The function H is convex and positively homogeneous. 8. Let A be a compact convex body in R", having 0 as an interior point.

(a) Prove that for each E > O there exists a convex polyhedron P such that A c P c (1 &)A.(Separate each point of the frontier of (1 &)Afrom A by a hyperplane (Section 12.15, Problem 4(d)).) (b) Let P be a compact convex polyhedron of dimension n in R", having 0 as an interior point. We may suppose that P is defined by m inequalities gJ(x) 5 1, where each gJ is a nonzero linear form on R". Let N > 0 and put

+

+

f(x) =

I

J=l

exp(N(g,(x) - 1)).

Show that the real-analytic function f satisfies the conditions of Problem 7, and that the convex set B = {x E Rn :f(x) 5 l} satisfies P c B c (1 N-' log m) P. (c) Deduce from (a) and (b) that for each E > 0 there exists a real-analytic function f on R" satisfying the conditions of Problem 7 and such that if B = { x E R":f(x) =< l}, we have A c B c (1 &)A.

+

+

9. Let X, Y be two differential manifolds andf, g two mappings of class C'(r 2 1) defined on an open neighborhood of a point x E X, with values in Y. If k is an integer such that 0 5 k 5 r, the functions f and g have contact of order 2 k at the point x e X if f(x) = g(x) and if, for each chart (U, 'p,.n) on X at x and each chart (V,I/J, m) on Y at the point f(x) =g(x), the local expressions F, G off, g are such that IIF(t)- G(t)ll/liz - tJJk tends to 0 as t E R"tends toz = ~ ( x )or,equivalently, ; if DpF(z) = DpG(z) for 1 5 p 5 k. If this condition is satisfied for one pair of charts, then it is satisfied for all pairs. Iff and g have contact of order z k for all k, they are said to have contact of infinite order at x. The relation "fand g have contact of order hk at the point x" is an equivalence relation

between Ck-mappingsdefined on a neighborhood of x with values in Y. An equivalence class for this relation is called a jet of order k from X to Y , with source x and target y (the common value of the mappings in the equivalence class). The equivalence class of f i s denoted by J:( f ) and is called the jet of order k o f f a t the point x . The set of jets of order k with source x and target y is written J:(X, Y ) y ;the set of jets of order k with source x (resp. with target y ) is written J:(X, Y ) (resp. Jk(X, Y ) y )The . union of the sets J:(X, Y), for all x E X and all y E Y is written Jk(X,Y). If Y = R, we write Pt(X) in place of J:@, Y), and P:(f) in place of J:( f). The set P:Q has a natural R-algebra structure, and we have

Then P:(X), = m is the unique maximal ideal of this algebra; we have I I I ~ +=~ 0, and m/m2is canonically isomorphic to the vector space T,(X)* of covectors at the point x. The set J$(R", R")', of jets of order k from R" to R" with source and target at the origins of these spaces is denoted by L:. ". This set carries a natural structure of a real vector space of dimension m (1

(" ")

- m,

and the jets of the monomials xt-rx" . e,

5j 5 m, 0 < Ia 1 5 k ) form a canonical basis. Every set of jets J:(X, Y),

is in one-

6

PRODUCTS OF MANIFOLDS

35

one correspondencewith L:. ,,,by means of charts at the points x, y (where dim,(X) = n and dim,(Y) = m), but if k 1 2 the vector space structure on J:(X, Y), obtained by transporting that of Lk. ,, depends on the choice of charts. When k = 1, J&R, X), is the tangent space Tx(X).

6. PRODUCTS OF MANIFOLDS

All the definitions and all the results of the next three sections (16.6)(16.8) (with the single exception of (16.8.9)) can be transposed to the contexts of real- or complex-analytic manifolds, simply by replacing C"-mappings by analytic mappings in the statements and the proofs. We shall therefore make use of them for real- and complex-analytic manifolds without further comment. Let X,, X, be two topological spaces. If c, =

w,, n,) 4p1,

and

c2 = (U,, (P2 9 n2)

are charts of X,, X,, respectively, the triple (U, x U, , (P, x 'pZ,n , + n,) is a chart of X, x X, (3.20.15 and 12.5); it is denoted by c1 x c,. If c;, c; are two other charts on X,, X, , respectively, and if ci and c; are compatible for i = 1,2, then c, x c2 and c; x c; are compatible, by (8.12.6). If a,is an atlas of X, and a,is an atlas of X2 , the set 2I of charts cl x c2 , where c, E 211 and c2 E a,,is therefore an atlas of X, x X2, and is denoted (by abuse of notation) by 2I, x %, . Moreover, if 21iand 21; are compatible atlases of X i (i = 1, 2), then the atlases a,x 21, and x 2ti are cornpatible. If X, and X, are differential manifolds, the product space X = X, x X2 is separable and metrizable (3.20.16), and the atlases x a,, where %I (resp. a,)runs through the equivalence class of atlases defining the structure of differential manifold on X, (resp. X,), are all equivalent. Hence their equivalence class defines on X a structure of differential manifold which depends only on the structures of XI and X, . The space X endowed with this structure is called the product of the differential manifolds X, and X, . It should be noted that even if 21, and %, are saturated atlases, %XIx ' i l l , will in general not be saturated. Whenever we consider XI x X, as a differential manifold, it is always the product structure as defined above that is meant, unless the contrary is expressly stated. Example (16.6.1) If El, E, are two finite-dimensional real vector spaces, each endowed with its canonical structure of differential manifold, it follows from the definitions (16.2.2) that the product manifold El x E, is the product vector space endowed with its canonical structure of differential manifold.

36

X V I DIFFERENTIAL MANIFOLDS

Let X I , X , be two direrentialmanifolds,X = X I x X, theirproduct. The projections prI :X + XI, pr, : X + X, are morphisms (16.3). For each point ( x , , xz) E X , the mapping (16.6.2)

(T(x,,xz)(Prl),T(XI,X2)(PrZ)) :T(,I, is an isomorphism of vector spaces.

x2)(x1 x X,)

-,T,,(X,)

x T,,(X,)

In view of the definition of the product manifold structure, we reduce immediately to the situation where XI and X, are open sets in R"' and R"' respectively. The first assertion is then a trivial consequence of (8.12.10), and the second follows from (8.1.5) applied to a C'-mapping of a neighborhood of 0 in R,with values in R"' x R"'. We shall identify canonically T(x,,xz)(X:x X,) with the product T,,(X,) x T,,(X,) by means of the isomorphism defined in (16.6.2). The canonical injection T,,(X,) -,T(,,, J X 1 x X,) resulting from this identification is just the tangent linear mapping at the point x , to the injection y , H (yl,xz), which is a morphism of X, into X. Likewise for the canonical injection T,,(X,) -,T,,,, ,JX1 x X,). It is clear that

(16.6.4) Let Y , X , , X , be three digerential manifolds and fl :Y -,X,, fz :Y -,X , two mappings. Then the mapping f = (5,fz): Y -,X I x X, is of class C' ( r an integer >O, or 00) if and only if f, and f, are of class c'. Moreover, for all y E Y , we have

T,((.h

9

fz))= (Ty(fi1, Ty(f2))

with the identification (16.6.2).

Once again we reduce to the case in which X I and X2 are open sets in R"' and R"', and the result then follows from (8.12.6). (16.6.5) Let X I , X , , Y,, Y, be diferential manijbids, and A : Yi + X i (i = 1,2) mappings of class C'. Thenfi x f, : Y, x Y, + X, x X 2 is a mapping of class C', and we have

T(y,,Y J f i

x f 2 ) = T,,(fl) x

TY'(fZ),

rk(Yl,YZ)(flx fz) = rky1(f1)+ rky*(f2). The second formula is a trivial consequence of the first, and that follows from (16.6.4) and (16.6.2), since fi x f, = (fl prl,f 2 prJ. 0

0

7 IMMERSIONS, SUBMERSIONS, SUBIMMERSIONS

37

(16.6.6) Let X,, X2, Z be three diyerential manifolds, f X, x X, + Z a mapping of class C' (ran integer >O, or a),and (a,, a,) apoint of X1 x X, . Let f(a,, * ) (resp. f( , a,))denote the partial mapping X,H f(a,, x,) (resp. x1w f ( x l , a,)). Then we have

-

T(4,,u2)(f)= T4,(f( *

9

ad)

0

PI

+ TaZ(f(a1, . ))

o

P2

9

where PI

= T(u1,42)(~r1) : T(ai,uz)(X~x X2)+Ta,(XA

P2

= T(41, 42)(Pr2) : T(41,4z)(x1

x2)

T4,(x,)

are the canonicalprojections (with the identijcation (16.6.2)). Once more, the proof reduces to the case where X,, X, ,Z are open sets in R"',R"', and R'", and then it follows from (8.9.1). In particular, if Z = E is a finite-dimensional real vector space, we have

(16.6.8)

With the hypotheses and notation of (16.6.6), suppose that T42f(a1,

*

) : TaZ(X2)

-+

T,(Z)

(where c =f(al, 4 )

is bijective. Then there exists an open neighborhood U , of a, in X, and an open neighborhood U , of a, in X2 with the following properties: for each x, E U , there exists a unique point u(x,) E U , such that f ( x , , u(xI)) = c, and u is a C'-mapping of U, into U, . Furthermore, we have

("implicit function theorem "). We reduce to the case where X,, Xz ,and Z are open sets in R"',R"', and R", respectively, and then the theorem is a particular case of (10.2.3).

7. IMMERSIONS, SUBMERSIONS, SUBIMMERSIONS

(16.7.1) Let X, Y be two differential manifolds, f : X + Y a mapping of class C", and x a point of X. The mapping f is said to be a subimmersion at the point x if there exists a neighborhood U of x in X such that the function x ' ~ r k , . ( f ) is constant on U. The mapping f is said to be an immersion

38

XVI

DIFFERENTIAL MANIFOLDS

(resp. a submersion) at x if the linear mapping T,cf) is injective (resp. surjective). By virtue of the lower semicontinuity of the rank of f (16.5), this implies that rk,.Cf) = dim,,(X) (resp. rk,,(f) = dimf(,.,(Y)) for all x' in some neighborhood of x, and hence f is a subimmersion at the point x. A mapping f : X Y of class C" is both an immersion and a submersion at the point x if and only iff is Ctale at x (16.5.6). It is clear that the set U of points of X at which f is a subimmersion (resp. a submersion, resp. an immersion, resp. Ctale) is open in X. The mapping f is said to be an immersion (resp. a submersion, a subimmersion, Ctale) if U = X. For example, the projections of a product manifold XI x X, onto its factors X I , X, are submersions (16.6.2). -+

(16.7.2) I f f : X + Y and g : Y Z are both submersions (resp. both immersions), then g f : X Z is a submersion (resp. an immersion). -+

0

-+

This follows immediately from the definitions (16.7.1) and from (16.5.4). We remark that the composition of two subimmersions is not necessarily a subimmersion (Section 16.8, Problem l(b)). (16.7.3) If fi : XI -+ Y, and f , : X , + Y , are both submersions (resp. immersions, resp. subimmersions), then flx f i : XI x X, -+ Y, x Y, is a submersion (resp. an immersion, resp. a subimmersion).

This follows from the definitions (1 6.7.1) and from (1 6.6.5). (16.7.4) Let f : X --+ Y be a mapping of class C". In order that f should be a subimmersion of rank r at a point x E X, it is necessary and sufficient that there should exist a chart ( U , cp, n) of X , a chart ( V , II/, m ) of Y and a C"-mapping F :q(U) $(V) such that x E U , f ( x ) E V, q ( x ) = 0, $ ( f ( x ) ) = 0, f I U = $-' F cp, and such that the local expression F o f f is the restriction to q ( U ) of the mapping -+

0

0

of R" into R"

This is an immediate consequence of the rank theorem (10.3.1). (16.7.5) r f f : X -+ Y is a submersion, the image under f of any open set U in X is open in Y .

8 SUBMANIFOLDS

39

If x E U, it follows from (16.7.4) that there exists an open neighborhood W c U o f x such that f ( W ) is open in Y; now apply axiom (0,)of topological spaces (12.1). (16.7.6) It should be remarked that if f : X -+ Y is an immersion, f ( X ) is not necessarily closed, noreven locally compact, even iff is injective (16.9.9.3). On the other hand, a C"-mapping f : X -+ Y can be injective (resp. surjective) without being an immersion (resp. a submersion), as is shown by the example of the bijective mapping l e t 3 of R onto R (cf. however Section 10.3, Problem 2). (16.7.7) (i) Let f : X -+ Y be an injective immersion, and let g : Z + X be a continuous mapping ( X , Y , Z being differential manifolds). For g to be of class c'it is necessary and suficient thatf g : Z + Y should be of class C'. (ii) Let f : X -+ Y be a surjective submersion and let g : Y 2 be a mapping ( X , Y , Z being dixerential manifolds). For g to be of class C' it is necessary and suficient that g f : X + Z should be of class C'. 0

-+

0

Only the sufficiency of these conditions requires proof, by (16.3.3). Also the questions are local on X, Y, and Z by virtue of the continuity of g

(which follows from the continuity of g f i n (ii)). In case (i), we may therefore suppose (16.7.4) thatfis the mapping (tl, . . . , t")~ ( t '. .,., t",0,. . . ,0) of R" into R" (with n 5 m). The assertion of (i) is then that a mapping z H ( g ' ( z ) , . . . ,g"(z), 0, .. . ,0) of Z into R" is of class C' provided that the gi are of class C'. In case (ii) we may likewise suppose that f is the mapping (el, . . . , 5") . .., 5") of R" onto R" (with n 2 m). The assertion of (ii) is then that the mapping (tl,.. ., t " ) ~ g ( t ' , .. ., 5") is of class C' on R" if and only if the mapping (c', ..., y)t+g(tl, ..., t") is of class C' on R" (16.6.6). 0

Remark (16.7.8) The preceding results can be extended immediately to the situation where C'-mappings (r a positive integer) replace C"-mappings, subimmersions are replaced by mappings of locally constant rank, and submersions (resp. immersions) by mappings f such that T,(f) is surjective (resp. injective). 8. SUBMANIFOLDS

(16.8.1) Let X be a separable metrizable space, Y a differential manifold, f a mapping of X into Y . In order that there should exist on X a structure of diflerential manifold for which the underlying topology is the given topology

40

XVI

DIFFERENTIAL MANIFOLDS

on X and such that the mapping f is an immersion, it is necessary and sufficient that the following condition be satisfied: (16.8.1.1) For each a E X , there exists an open neighborhood U of a in X anda chart (V, $, m )of Y such thatf(U) c V andsuch that $ (fI U) is a homeomorphism of U onto the intersection of $(V) with a linear subvariety of R". When this condition is satisfied, the structure of diferential manifold on X satisfying the conditions above is unique. 0

The necessity of the condition follows immediately from (16.7.4). To prove sufficiency, consider for each a e X a neighborhood U, of a in X and a chart (V,, $, ma) of Y satisfying (16.8.1.1), and let E, denote the linear subvariety of Rmasuch that $,(V,) n E, = $,(f(U,)). (E, is unique because E, n $,(V,) is a nonempty open subset of E, .) Let n, = dim E, and let 1, be an affine-linear bijection of E, onto R".. Finally let cp, be the composition of 1, and $, 0 (fI U,). We shall show that the charts c, = (U,,cp, n,) form an atlas of X. Suppose therefore that U, n Ub # @; then (16.8.1 .I) implies that cp,(U, n ub) = W.6 and (pb(u, n ub) = Wb, are open sets in Rnaand Rnb,respectively. If $&(resp. $6,) is the restriction of $. (resp. $b) to V, n v b , and Vab (resp. %a) the restriction of cp, (resp. (pb) to U, n ub, it is immediate that (Pb, o c p ~ = l Ab 0 (i+hbao $,;I) o ((,I;l)l w,,). Hence to show that %a ' ;pc is indefinitely differentiable, it is enough to observe that the restriction of an indefinitely differentiable mapping 0

$ba

';$ ,

0

: V, n v

b+

R"

(where m = ma = mb)

to the intersection of the open set V, n v b in R" with a linear subvariety E, of R" is indefinitely differentiable on this intersection, which is obvious by (8.12.8). The uniqueness of the structure of differential manifold on X follows from (16.7.7(i)). For by replacing X by an open neighborhood of a point of X, we reduce to the case where f is injective, and apply (16.7.7(i)) to g = 1, (considering two structures of differential manifold on X satisfying the conditions of (16.8.1)). When the condition (16.8.1.1) is satisfied, the unique structure of differential manifold defined in (16.8.1) is called the inverse image under f of the structure of differential manifold on Y. In particular: (16.8.2) Let X be a separable metrizable space, Y a differential manifold, f a mapping of X into Y with the property that for each x E X there exists an

8 SUBMANIFOLDS

41

open neighborhood U of x such that f I U is a homeomorphism of U onto an open set in Y . Then there exists a unique structure of dflerential manifold on X for whichf is an immersion, andf is then infact ktale (16.5.6). (16.8.3) Let Y be a differential manifold, X a subspace of Y. I f the canonical injection f : X -+ Y satisfies the condition (16.8.1.1), then the space X, endowed with the structure of differential manifold which is the inverse image under f of that of Y, is said to be a submanifold of Y. We also say that X is a submanifold o f Y; this abuse of language is justified by the property of uniqueness in (16.8.2). The condition (16.8.1 .I), in the present situation, is that for each x E X there exists a chart ( V , $, m ) of Y such that x E V , $(x) = 0 and such that $(V n X ) is the intersection of the open set $(V) of R" and the vector subspace of R" given by the equations = 0, . . . , = 0; and then (V n X, $1 (V n X), n ) is a chart of the submanifold X. It follows that V n X is closed in V, and hence that X is locally closed in Y (12.2.3). Moreover, there is an open neighborhood W c V of x in Y which is diffeomorphic to (W n X) x Z, where Z is a submangoldof dimension n - m of Y ,containing x. To see this we reduce to the case where Y = $(V) c R" and X = $(V) n R", and then it is obvious.

r"

cn+'

(16.8.3.1) In view of (16.7.4), the condition for X to be a submanifold of Y can be expressed as follows :for each x E X there exists an open neighborhood U o j x in Y and a submersion g : U -+ R"-" such that X n U is the set ofpoints z E U such that g(z) = 0. (16.8.3.2) In particular, when Y = R", we may always assume (by translation if necessary) that OEX, and then, by permuting the coordinates, that if g = ( g l , .. . ,g"-"), the determinant of the matrix formed by the first n - m columns o f the Jacobian matrix of g is # O at the point 0. If we identify R" with R" x R"-", it follows from the implicit function theorem (10.2.2) that there exists an open neighborhood V of 0 in R" such that U n (V x Rn-") is the graph of a C"-mappingf= (f',. . . ,f"-") of V into R"-", or in other words this submanifold is the set of points x = ( tl,. . . , 5") such that r j E V for 1 m and lm+k -f k ( < ' ,. . ., 5") = 0 for 1 k 5 n - m.

sjs

'

s

(16.8.3.3) Every open subset of a manifold Y, endowed with the induced structure (16.2.4) is a submanifold of Y. Conversely, a submanifold X of Y such that dim,(X) = dim,(Y) for all x E X is open in Y. Every discrete subspace (necessarily at most denumerable) of Y is a submanifold of dimension 0. In a pure manifold Y of dimension n, a pure submanifold of dimension n - 1 is called a hypersurface.

42

XVI DIFFERENTIAL MANIFOLDS

(16.8.3.4) Let X be a submanifold of Y, X a submanifold of Y , and j : X + Y, j ' : X + Y' the canonical injections. Let g : Y + Y be a C'mapping such that go() c X'. Then we can write g o j = j' o f , where f is a C'-mapping of X into X'. This follows from (16.7.7(i)). (16.8.4) Let X, Y be two diferential manifolds, f : X -+ Y an immersion. I f f is a homeomorphism of X into the subspace f ( X ) of Y, then f ( X ) is a submanifold of Y, and f : X -,f ( X ) is a difleomorphism.

For each x E X we apply (16.7.4) (keeping the notation used there) with r = n = dim,(X) and m = dimf(x,(Y). Since f is a homeomorphism of X onto the subspacef(X), it follows that F(p(U)) is an open neighborhood of F(cp(x)) = $(f ( x ) )in $(V) n R",and hence there exists an open neighborhood T c $(V) of ~ ( f ( x )in) Rm such that T n R" = F(p(U)). Putting W = $-'(T), the chart (W, $1 W, m) on Y satisfies the condition of (16.8.3) relative to the subspace f(X). Furthermore, it follows from (16.7.4) that F is a diffeomorphism of q(U)onto the open set F(q(U)), and the second assertion of (1 6.8.4) follows. An immersion f which satisfies the hypotheses of (16.8.4) is called an embedding of X in Y.

Remark (16.8.5) It can happen that an immersion f : X -,Y is injective and that f ( X ) is closed in Y but that f is not an embedding. For example, take X to be the open interval ] - co, 1[ in R,Y = R2,andfto be the immersion t 2 - 1 t(t2 - 1)

tH

(TiTi'7Tfl).

This immersion is not an embedding, because,f(- I)

= lim f ( t ) (cf. Problem 2). t-tl

If X is a submanifold of Y a n d j : X -+ Y is the canonical injection, it follows from the definition of an immersion (16.7.1) that for each x E X the linear mapping T,(j) :T,(X) ?-. Tx(Y) is an injection, by means of which we shall identifr canonically T,(X) with a vector subspace of T,(Y). In the particular case where Y is a vector space E of finite dimension n, we recall (1 6.5.2) that there is a canonical linear bijection 2, : T,(E) + E. The image of z,(Tx(X)) under the translation hl-, h + x is an afie-linear variety in E, passing through the point x , of dimension m = dim,(X). This is called the tangent asne-linear variety to X at the point x (or the tangent (16.8.6)

8 SUBMANIFOLDS

43

to X at x if m = 1, the tangent plane if m = 2, the tangent hyperplane if m = n - 1). It is the set of points x + z,(h,) in E, as h, runs through T,(X). It should be observed that the possibility of defining such a "tangent linear variety" as a submanifold of E depends essentially on the group-structure of E, and that there is no analogous definition when E is replaced by an arbitrary differential manifold Y. Remark

(16.8.6.1) If X is a differential manifold, a a point of X, and E a vector subspace of T,(X), then there exists a submanifold Z of X containing a and such that T,(Z) = E. T o see this, it is enough to consider the case where X is an open set in R" and a = 0, and then we may take Z to be the intersection of X with a vector subspace of R". (16.8.7) (i) Let Z be a differential manifold, Y a submanifold of Z, X a subspace of Y. Then X is a submanifold of Z if and only i f X is a submanifold OJY.

(ii) If X, (resp. X,) is a submanifold of Y, (resp. YJ, then XI x X, is a submanifold of Y, x Y, . Assertion (ii) follows immediately from the definitions of a submanifold (16.8.3) and a product manifold (16.6). As to (i), suppose first that X is a submanifold of Y. Using local charts, we reduce to the case where Z = R", Y is an open subset of R" (where n < m ) , and there exists a submersion f : Y R"-P such that X is the set of points y E Y satisfying f ( y ) = 0. We then extend f to a submersion g : Y x R"-" -+ Rm-Pby defining g ( y , t ) = ( f ( y ) ,t ) for y E Y and t E R"-". Then X is the set of points ( y , t ) satisfying g(y, t ) = 0, hence is a submanifold of Z. Conversely, suppose that X is a submanifold of Z. This time we may suppose that Z = R" and that X is an open set in RPcontaining the origin. The tangent space T,(Y) can be identified with a vector subspace E of R" containing RP. Let F be a supplement of RP in E, let G be a supplement of E in R", and let n be the projection of R" onto F parallel to G + RP.The restriction h of n to Y is a submersion, because by definition the rank of T,(h) is dim(E) - p = dim(F). The set X' of points y E Y such that h(y) = 0 is therefore a submanifold of Y , hence of Z , and of the same dimension as X. Since X' contains X and the canonical injection .j of X into X' is of class C" (16.7.7(i)),it follows that j is a local diffeomorphism (16.5.6); hence X is open in X' and therefore is a submanifold of Y. -+

(16.8.8) L e t f : X - + Y beasubimmersion (16.7.1),aapointofX,andb=f ( a ) .

44

XVI DIFFERENTIAL MANIFOLDS

(i) The subspace f -'(b) is a closed submanifold of X . The tangent space T,( f -'(b)) tof -'(b) is the kernel of T,(f), and hence we have an exact sequence

TQU)

0 T,(f -l(W--* TAX) TO). There exists an open neighborhood U of a in X such that.f(U) is a sub(ii) manifold of Y ,and we have +

(1 6.8.8.1)

dim,(X)

= dim,( f ( U ) )

+ dim,( f

-'(b)).

Furthermore, if E is any supplement of T,( f -'(b)) in T,(X), there exists u submanifold of V of U whose tangent space at the point a is E. For each submanifold V of U having this property, there exists an open neighborhood W of b in Y such that the restriction o f f to V' = V nf -'(W) is an isomorphism of V' onto W nf ( U ) , and such that T,(V) is a supplement of T,( f -'( f ( x ) ) ) in T,(X) for all x E V'. (iii) rS To(f ) is not surjective, V may also be chosen so thatf ( U ) is nowhere dense (12.16) in Y . (iv) I f f is an injective subimmersion, thenf is an immersion. I f f is a bijective submersion, then f is a diffeomorphism (cf. Problem 3). We can apply (16.7.4) to the point a E X, and then it is clear that

U nf -l(b) = (p-'((p(U) n F-'(O)). Hence we reduce to the case where X and Y are open sets in finite-dimensional real vector spaces and f is the restriction to X of a linear mapping. Parts (i)-(iii) of the proposition are now obvious ((16.5.2) and (12.16)). As to (iv), it follows from (i) that iff is injective, then To(f ) is injective, for each a E X, hence f is an immersion. If in addition f is a submersion, then f is a local diffeomorphism. Finally, iff is also bijective, then f is a diffeomorphism. (16.8.9) Let Y be a di3erential manifold and let be a sequence of real-valued Cw-functions on Y . Let X be the set of x E Y such thatfi(x) = 0 for 1 5 i 5 r. Suppose that, for each x E X , the differentials d, f i (1 5 i 5 r ) are linearly independent covectors in T,(Y)*. Then:

(i) X is a closed submanifold of Y , and for each x E X the tangent space T,(X) is the annihilator in T,(Y) of the subspace of T,(Y)* spanned by the differentials d,fi, and consequently is of dimension dim,(Y) - r. (ii) Let F be a Cw-function on Y which vanishes at all points of X . Then for each x,, E X there exists an open neighborhood U of x,, in Y and r functions Fj (1 $ j 5 r ) of class C" on U such that

8 SUBMANIFOLDS

45

for ally E U. If Y is a real- (resp. complex-) analytic mangold and thefunctions F and& are analytic, then the functions Fj may be chosen to be analytic. (i) Since the differentials dxfi are linearly independent, there exists an open neighborhood V(x) of x in Y such that the differentials d,. fiare linearly independent for all x' E V(x) (16.5.8.4). Replacing Y by the open set which is the union of the neighborhoods V ( x ) as x runs through X, we may therefore assume that the d, fiare linearly independent for all y E Y. However, then the mapping g :y ~ ( J ( y ) )s i,5 r of Y into R' is a submersion, by virtue of (16.5.7.2) and the definition (16.7.1). Since X = g-l(O), we can now apply (16.8.8).

(ii) By virtue of (16.7.4) we may limit ourselves to the case where Y is an open set in R" and f i ( x ) = xi, the j t h coordinate of x (1 S j 5 r), so that X = U n R"-' (where R"-' is identified with the subspace spanned by the last n - r vectors of the canonical basis of R"). Moreover we may take xo to be the origin. Then the assertion (for C"-functions) is a consequence of the following lemma: (16.8.9.1) Let F be a real-valuedfunction of class C" on an open cube I" c R" (where I is an open interval in R). Then in 1" we can write (16.8.9.2)

F(x', . . . ,x") = F(0,. . . ,0) + x'Fl(x', .. . , x") + x2F2(x2,. . .,x") + * - . + x"-'F,,-,(x"-', x") + x"F,(x"),

where F,, . . . , F, are P-functions on I". Assuming this lemma, since F(x', . . . ,x") = 0 whenever x1 = * * . = x'= 0 we obtain successively that F,, F,,-,, . . ., F,+, are identically zero: first we put all the x i except x" equal to zero, then all the x i except x"-' and x", and so on. To prove (16.8.9.1) we write

F(d, . . . , 2)= (F(x', x', . . . , x") - F(0, x', . . ., 2)) + F(0, x', . . . , x") so that by induction on n we are reduced to proving that the function (16.8.9.3) G(xl, . . . , x") = (xl)-'(F(xl,

. . .,x") - F(O, xz, ...,x")>

which is defined whenever x1 # 0, tends to a finite limit as x1 + 0, and that the function so extended is of class C" on I". Using Taylor's formula (8.14.2) we have, for x1 # 0 and any integer p 2 1,

46

XVI DIFFERENTIAL MANIFOLDS

(16.8.9.4) G(x',

...,X") = D,F(O, x', ...,X") + X1 D:F(O,

+-

(x1)P-l

P!

DSF(0, x2, .. . ,x")

x',

. . . ,2')+ -

* *

+ (xl)-'H(x', . ..,x"),

where (16.8.9.5)

'F(t, x2, ..., x") d t

H(xl, ..., x")=

is of class C" on I" by hypothesis. Differentiating under the integral sign (8.11.2) and replacing F by some derivative of the form DTDY...D;F, we are reduced to showing that, as x1+ 0, the derivatives (16.8.9.6)

D!((x1)-lH) =

k

j=O

(-l)j(k))J!(x')-'-'D:"H 1

(0 k S p - l), calculated by Leibniz's rule (8.13.2), tend to zero uniformly in (x', . . , ,2')on a neighborhood V of 0 in R"-'; but by (8.11.2) we have (16.8.9.7)

D;-jH(x',

.. . ,x") =

(p - k + j ) !

DP,+'F(t,x2, .. .,x") d t

and hence by the mean value theorem

where C is a constant, for all (x', . . . ,x") E V. Using this inequality in (16.8.9.6) now completes the proof of the lemma. In the case where F is (real- or complex-) analytic, the proof is much simpler, by considering the Taylor expansion of F at the origin, and it follows from (9.1.4) that the F, are analytic in a neighborhood of 0. We remark that (16.8.3.2) shows conversely that for each submanifold X of Y and each point a E X, there exists an open neighborhood V of a such that V n X is defined by equations satisfying the conditions of (16.8.9). Examples (16.8.10)

In (16.8.9) let us take Y = R"" - (0)and r = 1, and the sequence

(A) to consist of the single functionf: YH Ilyll, the Euclidean norm on Y.

8 SUBMANIFOLDS

47

Then X =f -'(l), as a topological space, is just the unit sphere S,; since Df is of rank 1, it follows that S, is endowed with a structure of a submanifold of Y. Let us show that this structure of differential manifold on S, is the same as that defined in (16.2.3). For this it is enough to observe, in view of (16.8.3), that the formulas (16.2.3.1) define a dzTeomorphism of the submanifold s,,- (eo} of R"" onto R". Now consider the mapping g : y-(y/llyll, Ilyll) of Y into S, x : R . This mapping is a bijection, whose inverse is (z, [)I+ [z; also g is a submersion, because Df # 0 and the restriction of the mapping y~-+y/llyll t o a sphere I S , (where 1 > 0) is a homothety of this sphere onto S,, hence a diffeomorphism. It follows that g is a diffeomorphism, by virtue of (16.6.4) and (16.8.8(iv)). (16.8.11) Suppose that the conditions of (16.6.8) are satisfied, so that f : X , x X , --f Z is a submersion at the point ( a l , a,). Then there exists an open neighborhood W of (a,, a,) in X, x X, such that, if Y is the set of points ( x , , x,) satisfying f ( x l , x , ) = c, then Y n W is a submanifold of X, x X,, and the restriction of pr, to Y n W is an isomorphism of this submanifold onto an open subset of X,. If X, = X2 = Z = C and i f f is holomorphic, the set Yo of points of Y where D, f ( x , , x,) # 0 is an open subset of Y; this complex-analytic submanifold of C2 is called the Riemann surface (relative to the second coordinate) defined by the holomorphicfunction f.For example, if f(x17

x2> =

- ex',

we have Y = Y o , and Y is an analytic subgroup of the complex-analytic group C* x C (16.9.10). This surface Y is called the Riemann surface of the logarithmic function (cf. Problem 12). The restriction of pr, to Y is an &ale morphism of Y onto C* = C - (0). For each f E Y we write log(t) = pr, t , so that we have log(tt') = log(t) log(t') for t , t' in Y (the law of composition in Y being ( x ,y)(x', y ' ) = (xx',y + y')), and the mapping t~ log t is a holomorphic mapping (16.3.5) of Y into C.

+

(16.8.12) Let f : X Y be a submersion,Z any submanifold of Y .Thenf - ' ( Z ) is a submanifold of X , and the restriction f - '(2) + Z o f f is a submersion (cf. Problem 17). --f

Using a chart satisfying (16.7.4), we may assume that Y is an open set in R" and X an open set in R" ( n 5 m),and that f is the restriction to X of the canonical projection ((l, . . . , trn)++(t', . . . , Y). Let x Ef -'(Z) and let y E 2 be its projection. Then by hypothesis there exists a chart (U, $, n) on Y at the point y such that $(U n 2) = $(U) n RP,where p =< n. If we

48

XVI DIFFERENTIAL MANIFOLDS

denote by cp the restriction to f -'(U) of J/ x lRm-": U x Rm-"+ R"', then

cf-'(TJ), cp, m) is a chart on X at the point x , such that cp(s-'(U) nf-'(Z)) = cp(f'(U))

n RP+"'-".

Hence the result. (16.8.1 3) Let X, Y be two diyerential manifolds and f:X + Y a mapping of class C". Then the graph rf off in X x Y is a closedsubmanifold of X x Y, the mapping g : x ~ ( xf(x)) , is an embedding (16.8.4) and the vector subspace TCx, ,(,),(Tf) of T(,, ,(,.,(X x Y) is the graph of the linear mapping T,(f).

We know that g is a homeomorphism of X onto r, (the inverse of g being the restriction to rf of the projection prl), and that rf,being the set of points z E X x Y such that pr,(z) =f(pr,(z)), is closed in X x Y. Hence (16.8.4) it is enough to prove that g is an immersion, but since T,(g) = (Tx(lx),T,(f)) by virtue of (16.6.4), it is clear that T,(g) is a linear mapping of rank equal to dim,(X). Hence the result. In particular, when Y = X , the diagonal A of X x X is a closed submanifold of X x X, and the diagonal mapping XH(X, x) is a diffeomorphism of X onto A. In R x R,the set of pairs (c, q) such that 5 # 0 and q = sin(l/c) is an (analytic) submanifold whose closure is not locally connected. Remark (16.8.14) Let Z be a submanifold of X x Y such that at a point (a, b) E Z the restriction to T(,, b)(Z) of the projection T,(X) x Tb(Y)+ T,(X) is a bijection onto T,(X). Since this restriction is equal to T(,, b)@), where p :Z -+ X is the restriction of prl : X x Y + X, it follows (16.5.6) that there exists an open neighborhood U of (a, b) such that p [U is a diffeomorphism onto an open neighborhood V of a in X. Sincep(x, y ) = x , the inverse diffeomorphism g : V + U is of the form XH (x,f(x)), where f:V 3 Y is of class C", and U is therefore the graph off in V x Y.

PROBLEMS

1. (a) Let f : X + Y be a submersion and g : Y + Z an immersion. Show that g f : X Z is a subimmersion. (b) The mapping f:t H(t, t z, t 3, of R into R3 is an immersion, and the projection g : (x, y, z)H (y, z) of R3into R2 is a submersion, but g 0 f is not a subimmersion oi R into R2,although it is injective. 0

--f

8 SUBMANIFOLDS

49

2. Let f :X +-Y be. an injective immersion which is proper (Section 12.7, Problem 2).

Show that fis an embedding (observe that the image of a closed subset of X is closed in Y). Give an example of an embedding of R into RZwhich is not proper (consider a ''spiral ").

3. (a) In R3,the union of the line z = 1,y = 0 and the complement in the plane Y :z = 0 of the line z = 0, y = 0 is a nonconnected manifold X. Show that the restriction to X of the projection (x, y, z) H(x, y) of R3onto Y is a bijective immersion of X onto Y which is not a diffeomorphism. (b) Let X be a connecteddifferential manifold, Y a differential manifold, andf :X --+ Y a bijective subimmersion of X onto Y.Show thatfis a dfleomorphism. (Observe that the set of points x E X at which f is a submersion is both open and closed in X;to show that this set is nonempty, use (16.8.8(iii)), (12.6.1), and Baire's theorem (12.16.1).) 4. Give the analogs of the results of Sections 16.3-16.8 for manifolds of class C (r 2 I), which were defined in Section 16.1, Problem 2. Show that the analog of (16.8.9(ii)) is

false.

5. Let E be a finite-dimensional real vector space, F a closed subset of E,and IIxII a norm defining the topology of E (12.13.2). If a is a nonisolated point of F,the contingenf of F at a is the union of the rays through 0 whose direction vectors of norm 1 are l i t s of sequences of the form ((x. u)/llx, - all), where (xJ is a sequence of points of F, distinct from a and with a as limit. The paratingent of F at (1 is the union of the lines

-

through 0 whose direction vectors are limits of sequences of the form

((xn-

ym)/lkn-~nll),

where (x,,) and (y,,) are two sequences of points of F, distinct from a and with a as limit, and such that xn # y,,for all n. Show that F is a submanifold of class C1 of E if and only if, for each nonisolated point a E F, (i) the paratingent of F at a is a vector subspace P of E, and (ii) if N is a supplement of P in E and p : E +- P is the projection parallel to N, then the image under p of any neighborhood of a in F is a neighborhood of p(a) in P. (Show first that there exists a compact neighborhood U of a in F such that p ] U is a bijection onto a compact neighborhood V ofp(a), by contradiction. If we identify E with P x N, then U is the graph of a continuous mapping f :V +N. Show that f is of class C' by using condition (i) above, Problem 3 of Section 8.6, and arguing by contradiction.) Give an example in which the contingent and the paratingent of F at a are each equal to the whole of E but the condition (ii) above is not satisfied. (Take E = R2.) 6. In a differential manifold X,let Y1, . . . , Y,be submanifoldswith a common point a.

Show that the union of the Y t cannot be a submanifold of X unless it has the same dimension at a as one of the Y1. (UseProblem 5.)

7. Show that a complex-analytic submanifold of C? which is compact and connected consists of a single point (seeSection 16.3, Problem 3).

50

XVI DIFFERENTIAL MANIFOLDS

8. Let X, Y be. two differential manifolds, f : X+Y a mapping of class C", and U a connected open subset of X. If r is the least upper bound of rk,(f) as x runs over U, show that r is finite and that the set of points x E U at which rk,Cf) = r is open. Deduce

that the set of points at which f is a subimmersion is a dense open subset of X (argue by contradiction). Iffis an open mapping, the set of points at whichfis a submersion is dense in X.

9. Let X, Y be two differential manifolds, f : X +Y a mapping of class C'. If Z is a sub-

manifold of Y, the mappingfis said to be trunsuersal ouer Z at x E f-'(Z) if the tangent space T,(,,(Y) is the sum of T,(,)(Z) and T,(f)(T,(X)), and f is said to be transyersul ouer Z if this condition is satisfied for all x ~ f - l ( Z ) .If so, thenf-I(Z) is a submanifold of X, and for each x ~ f - l ( Z )the tangent space T,Cf-'(Z)) is the inverse image under T,Cf) of T,(,,(Z). (Since the question is a local one, we may take Z to be a submanifold given by an equation g ( y ) = 0, where g : Y +RP is a submersion; consider the composite mapping g f.) In particular, if X and Z are submanifolds of Y, we say that X and Z are transversal at u point x E X n Z if the canonical injection of X into Y is transversal over Z at x , or equivalently if T,(Y) = T,(X) T,(Z), which is symmetrical in X and Z. The submanifolds X and Z are said to be trunsuersal if they are transversal at all points x f X n Z; in that case, X n Z is a submanifold of Y . 0

+

10. Let f :X +Z and g : Y +Z be two mappings of class C", and consider their product f X g :X X Y +Z x Z, which is also of class C". Show that f x g is transversal over the diagonal A of Z x Z if and only if, for each pair (x, y ) E X x Y such thatf(x) = g(y), we have (*)

Tz(Z) = TxW(Tx(X))

+ T&)(TAY)),

where z =f(x) = g(y). This condition is always satisfied if either f o r g is a submersion. When condition (*) is satisfied, the set of points (x, y ) E X x Y such thatf(x) = g ( y ) is a submanifold of X x Y, which is called the fiber product of X and Y over Z and is written X x = Y .The tangent space at the point (x, y ) E X x Y to the fiber product is the subspace of T,(X) x T,(Y) consisting of the pairs (h, k) such that TX(n .h = T y ( g ).k.

In this situation, f and g are said to be transversal mappings into Z. Show that if f i s a submersion (resp. an immersion, resp. a subimmersion), then so is the restriction X x z Y + Y of pr2. 11. Let Y be a differential (resp. real-analytic, resp. complex-analytic) manifold, X a Hausdorf€ topological space, and p : X +Y a mapping with the following property: For each x E X there exists an open neighborhood V of x such that p I V is a homeo-

morphism of the subspace V onto a submanifold of Y .

(a) Show that X is locally connected, that each point of x has a closed neighborhood which is homeomorphic to a closed ball in R",and that for each y E Y the fiber p - ' ( y ) is a discrete subspace of X. (b) Let b be a denumerable basis for the topology of Y . A pair (W, U ) is said to be distinguished if U E b and if W is a connected component of p - ' ( U ) such that QI' is compact and metrizable and p I is a homeomorphism of %' onto a subspace of Y .

8 SUBMANIFOLDS

51

Show that for each x E X there exists a distinguished pair (W, U) such that x E W. Show also that if (W, U) is a distinguished pair, the set of distinguished pairs ( W , U') such that W n W' # @ is denumerable (use the fact that W is separable). (c) Deduce from (b) that each connected component Xo of X is metrizable and separable. (Consider the following relation between two points x, x' in X: There exists a finite sequence of distinguished pairs (W,, U,) (1 5 i 5 r ) such that x E W, ,x' E W,, and W t n Wt+l # @ for 1 5 i 5 r - 1. Then apply (12.4.7).) (d) Show that there exists on Xo a unique structure of differential (resp. real-analytic, resp. complex-analytic) manifold such that plXo is an immersion of Xo into Y (Poincari- Volterra theorem).

h = (PA, f~),where PAis a nonempty open polydisk in C" (resp. R")andf, is a complex (resp. real) analytic function on PA.For each pair of elements A, p in L, define a set A,, as follows: A,, = 0 if PAn P, = @ or if the restrictions off, andf, to PAn P,, are distinct; A,, = PAn P, if the restrictions of f A and f , to PAn P,, are equal. Let hrrnbe the identity mapping of AA, onto itself. Show that the mappings ,I/ satisfy the patching condition (12.2.4.1), and hence that we obtain a topological space X by patching together the PA along the AA, by means of the ha,,. Let nA: P A+X be the canonical mapping and let XA= TA(PJ be its image, which is an open subset of X. IfjA : PA+ C" (resp. j , : PA+R")is the canonical injection, show that there exists a unique mapping p : X .+C" (resp. p : X -+R")such that p T , =in for all h. The restriction p I X, is a homeomorphism of X, ontop,, and n, is the inverse homeomorphism. Show that X is Hausdoff (use (9.4.1)). Deduce that the results of Problem 11 apply to X and p: If (Y,)is the family of connected components of X, then there exists on each Y. a unique structure of a complex (resp. real) analytic manifold such that p I Y . is a local isomorphism of analytic manifolds. For each index a, there exists a complex (resp. real) analytic function F, on Y . such that, for each index h for which X I c Y , , the restriction of F. to X, is equal t o h (PI X,); for each such index A, Y , is said to be the analytic manifold defined by f, (the Riemann surface of fi in the complex case when n = l), and F, is the natural continuation of f, .

12. Let L be the set of all pairs

0

0

13. (a) Let X, Y be two pure differential manifolds of the same dimension, and let f : X 4 Y be a mapping of class C". Let S C X be the closed set of points at which f is not a local diffeomorphism, and suppose that the set of nonisolated points of S is discrete. Show that, for each point xo E X, the image underfof any neighborhood of xo is a neighborhood off(xo), and hence thatfis an open mapping. (Reduce to the case X = Y = R",and show that it is impossible that on each sphere Ilx - xo j l = p in R" there should exist a point x such that f ( x ) =&), by a compactness argument. Let IJx- xoll= p be a sphere on whichf(x) # f ( x o ) , so that l/f(x)-f(xo)II 2 cc > 0; let D be the open ball Ilx - xoll < p and D' the open ball lly -f(xo)ll < cc; and let G = D n CS and H = D' n Show that f(Fr(G)) does not intersect H; deduce thatf(G) 2 H and hence thatf(D) is a neighborhood o f f ( x o ) . ) (b) Deduce from (a) that Fr(f(U)) cf(Fr(U)) for every relatively compact open set U in X. Deduce that if in addition f is proper (Section 12.7, Problem 2) and X, Y are connected, thenf(X) = Y . (c) If Y = R",show that for each a E R", inf llf(x) - a I1 is not attained a t a point IE x xo E X unless f ( x o ) = a. (d) Suppose that Y = R2and that X is an open neighborhood of the disk D : I z I 5 I in RZ.Show that iffsatisfies the conditions of (a), thenf(S,) cannot be a Bernoulli

Cfo.

52

XVI DIFFERENTIAL MANIFOLDS

+

lemniscate B (with equation (5: = 5: - fg). (Observe that the image of the intezior of D cannot contain any point of the unbounded connected component of the complement of B.) 14. Let X be a connected complex-analytic manifold of dimension n, and let Y, (1 5j 5 r) be a finite number of closed submanifolds of X,of dimensions - 1. Show that the complement in X of the union of the Y, is a connected dense open set (use (12.6.1) and Section 16.3, Problem 3(c)).

sn

15. Let X be a real-analytic manifold and Yo a differential manifold whose underlying set is contained in X. Suppose that, for each point x E YO,there exists a chart c = (U,9,n) of X at x and a neighborhood V c U of x in YO,such that Q I V maps V onto cp(U) n Rm and (V, Q 1 V,m) is a chart of Y o .In these conditions show that there exists a unique real-analytic submanifold Y of X whose underlying differential manifold is Yo. 16. Let Xo be a differential manifold, Y a real-analytic manifold, and suppose that the differential manifold Yo underlying Y is a submanifold of Xo Show that for each y E Y there exists a chart (U, 9,n) of Xo and an open neighborhood V of y in Y contained in U, such that (V,Q I V, m) is a chart of Y for which fl) = #) n R".

.

17. If j :R +Rf is the canonical injection, give examples of submanifolds Y c R2 such that j-'c y ) is not a submanifold of R.

9. LIE GROUPS

(16.9.1) Let G be a set endowed with a group structure and a structure of a differential manifold. These two structures are said to be compatible if the mappings (x, y)t+xy of G x G into G and x w x - ' of G into G are of class C". It comes to the same thing to require that the mapping (x, y)~+xy-' (or (x, y ) ~ x - ' y )should be of class C", by virtue of the relations x - l = ex-' and xy = x(y-')-'. A group endowed with a structure of differential manifold which is compatible with its group structure is called a Lie group (or a real Lie group). It is clear that the topology of a Lie group G is compatible with the group structure, and the topological group so defined is metrizable, separable, locally compact, and locally connected (16.1.3), and the set of its connected components is therefore at most denumerable. Moreover, this metrizable group is complete (12.9.5). An isomorphism of a Lie group G onto a Lie group G is by definition an isomorphism of the group G onto the group G' which is also a diffeomorphism (16.2.6). If G = G , we say automorphism in place of isomorphism. ForeachaeG, theleft andrighttranslationsy(a):xwaxand6(a-') :x w x a

9

LIE GROUPS

53

are diffeomorphisms of G onto itself (16.6) In particular, it follows that a Lie group is a pure differential manifold (16.1.3). For each a E G , the inner automorphism Int(a) :X H U X U - ' is a Lie group automorphism of G. Examples (1 6.9.2) If E is a finite-dimensional real vector space, the canonical structure of differential manifold on E (16.2.2) is compatible with the additive group structure of E. Hence E is endowed with a canonical structure of (commutative) Lie group. (16.9.3) Let A be an R-algebra of finite dimension with unit element. Then A is normable (15. I .8) by virtue of the continuity of polynomial functions on R" and hence (because R" is complete) the multiplicative group A* of invertible elements of A is a nonempty open set in A (15.2.4). The structure of differential manifold induced on A* by the canonical structure on the vector space A (16.2.2) is compatible with the group structure of A*, by the argument of (8.12.11), which applies to any Banach algebra, not merely to Y(E; E). In particular, if E is a real vector space of dimension n, the algebra A = Y(E; E) = End(E) may be identified, together with its canonical structure of differential manifold, with the vector space R"', and therefore the linear group GL(E) = A* is a Lie group of dimension n2. Likewise, if E is a vector space of dimension n over the field of complex numbers C (resp. the division ring of quaternions Ht), then GL(E) is a (real) Lie group of dimension 2nZ (resp. 4nz). When E = R" (resp. E = C", resp. E = €P' (the Zeft vector space)), we write G u n , R) (resp. GL(n, C), resp. GL(n, H)) instead of GL(E).

If G I , G,, . . ., G , are Lie groups, G = GI x G , x * - - x G, is a Lie group when endowed with the product group structure and the product manifold structure; this follows immediately from (16.6.5). The Lie group G so defined is called the product of the Lie groups G i. (16.9.4)

(16.9.5) If G is a Lie group, then the set G endowed with the same structure of differential manifold and the opposite group structure is again a Lie group, by virtue of (16.6.5); it is called the opposite of the Lie group G, and is denoted by Go.

"

t For the elementary algebraic properties of quaternions, see the author's book Linear Algebra and Geometry," Houghton, Boston, Massachusetts, 1969.

54

XVI

DIFFERENTIAL MANIFOLDS

(16.9.6) Let G be a Lie group and H a subgroup of G which is a submanifold of G (16.8.3). Then the group structure and manifold structure of H are compatible. For H x H is a submanifold of G x G , and if

and

j:HxH+GxG

j':H+G

are the canonical injections, the mapping g : ( x , y ) ~ x y - 'of H x H into H is such that j' 0 g = f o j , where f is the mapping ( x , y ) ~ x y - 'of G x G into G; our assertion now follows from (16.8.3.4). The set H, endowed with its structures of group and differential manifold, is called a Lie subgroup of G . It is closed in G by (12.9.6). Every open subgroup of a Lie group G is a Lie subgroup. In particular, the neutral component Go of G is a Lie subgroup, because G is locally connected (12.8.7). Every discrete subgroup of G is a Lie subgroup. In a product G , x . x G, of Lie groups, if Hi is a Lie subgroup of Gi for 1 i S m, then HI x . . * x H, is a Lie subgroup of GI x . * * x G, . For each n 2 1 , the Lie group GL(n, R) is a Lie subgroup of GL(n, C ) , which in turn is a Lie subgroup of GL(n, H) (if we identify C with the subfield of H generated by 1 and i ) . For each pair of integers p 2 1, q >= 1, the product group GL(P?R) x GL(9, R)

may be canonically identified with the Lie subgroup of GL(p + q, R) consisting of matrices of the form

:( :),

where S E GL(p, R) and T E GL(q, R);

likewise when R is replaced by C or H. Remark (16.9.6.1) To verify that a subgroup H of a Lie group G is a Lie subgroup

of G, it is enough to verify that at one point xo E H there exists a chart ( V , $, m) on G such that xo E H, $ ( x 0 ) = 0 and $(V n H) is the intersection of the open set $ ( V ) of R" with a vector subspace of R". For if x is any other point of H, we shall obtain a chart having analogous properties by taking ( ( x x ; ' ) ~ ,$ y(xo x - I ) , m), because (xox - l ) . ((xx;')V n H) = V n H. 0

(16.9.7) Let G , G' be two Lie groups. A mapping u : G G' is said to be a Lie group homomorphism (or simply a homomorphism) if u is a homomorphism of groups and a morphism of differential manifolds. in order that a group homomorphism u : G + G' should be a Lie group homomorphism, it is necessary and sufficient that there should exist an open neighborhood U of e in G such that ulU is of class C m ; for then u is of class C" on aU for all a E G, because u(x) = u(a)u(a-'x) for all x E aU. --f

9

LIE GROUPS

55

A Lie group homomorphisni of a Lie group G into a linear group GL(E) (where E is a finite-dimensional real vector space) is called a linear representation of G on E. (16.9.8) Let G be a Lie group and s an element of G . We have already seen that the translations y(s) : X H S X

6(s) = xl+xs-I

are diffeomorphisms of G onto itself. Hence for each x E G we have tangent linear mappings Tx(Y(s)) : TAG)

+

Tx(Ws)) : TAG) + TXs-1(G)

Tsx(G),

which'are bijections (16.5.6). The image of a vector h, E T,(G) under T,(y(s)) (resp. under T,(6(s))) is denoted by s . h, (resp. h, . s-') when there is no risk of confusion. I f s, t E G and h, ET,(G), it is clear that s * (h, . t-') = (s * h,) .t ;this is an element of T,,,- {(G),which we denote by s . h, * t-'. I f s, t E G and h, E T,(G), we have

-'

(st) * h, = s . ( t . h,), h,) = e * h, = h, . This follows from (16.5.4).

and in particular s * (s-'

(16.9.9) (i) Let i : G + G be the mapping for all x E G and h, E T,(G) we have

Tx(i)* h, = -x-'

*

on a Lie group G. Then

XWX-'

h, ' x-'.

(ii) The C" mapping m : (x,y ) ~ x oyf G x G into G is a submersion, and with the identijication (1 6.6.2) we have

T(,, y)(m). (h, h,) = x . h, + h, * Y . 2

(iii) If u : G -+ G' is a Lie group homomorphism, then u is a subimmersion of constant rank; Ker(u) is a normal Lie subgroup of G, and for all x E G, we have

-

T,(u)

. h,

= U ( X ) * (T,(u) * (x-'

. h,)),

where hxE T,(G) and e is the identity element of G. (iv) If u : G G' is a surjective Lie group homomorphism, then u is a submersion and we have (16.9.9.1)

dim(G) - dim(G')

= dim(Ker(u)).

In particular, a bijective homomorphism is an isomorphism.

56

XVI

DIFFERENTIAL MANIFOLDS

At a point (xo,yo) E G x G, the partial mappings m( . ,yo) and m ( x o , * ) are just the translations S(y0') and y(x,), so that (ii) follows from (16.6.6). Next, we have m(x, i(x)) = e for all x E G, so that from (ii) and (16.5.4) we deduce that h,

*

X-'

+x

*

(T,(i)

*

h,.)

=0

for all h, E T,(G); this establishes (i). Assertion (iii) follows from the relation u 0 y(x-') = y(u(x-')) u, which by (16.5.4) leads to 0

T,(u). (x-' . h,) = u ( x ) - ~. (T,(u) * hx)

for all h,ET,(G). It follows that T,(u) and T,(u) have the same rank, hence u is a subimmersion. The assertion about Ker(u) then follows from (16.8.8).

To prove (iv) we argue by contradiction. If u is not a submersion, then there exists a point xo E G and a compact neighborhood V ( x o ) of xo in G such that u(V(xo))is nowhere dense in G' (16.8.8). If follows that for all x E G the set u ( ( ~ x , ~ ) V ( x , = ) ) u ( x x ~ ' ) u ( V ( x o )is) nowhere dense in G'. Since there exists a denumerable open covering (A,,) of G which is finer than the covering formed by the sets (xx;')V(x,) (12.6.1), we conclude that G' is a denumerable union of nowhere dense subsets, which is absurd (12.16.1). Examples (16.9.9.2) The mapping X++det(X)is a homomorphism of the Lie group GL(n, R) (resp. GL(n, C)) onto the Lie group R* (resp. C*). The kernel,

which is denoted by SL(n, R) (resp. SL(n, C)), is a real Lie group of dimension n2 - 1 (resp. 2(n2 - I?), called the unimodular group (or special linear group) in n variables. Remarks (16.9.9.3) The image u(G) is not necessarily a Lie subgroup of G'. This is shown by the example where G = Z x Z and G' = R, the homomorphism u being defined by u(m, n) = m + n8, where 8 is a fixed irrational number (12.8.2.1).

(16.9.9.4) Let G, G' be two topological groups. A local homomorphism from G to G' is by definition a continuous mapping h of an open neighborhood U of the identity element e of G with values in G , such that h(xy) = h(x)h(y) whenever x , y and xy all lie in U. Since there exists a symmetric open neighborhood V of e such that V z c U (12.8.3), these conditions are satisfied whenever x E V and y E V . A local isomorphism from G to G' is defined to

9 LIE GROUPS

57

be a local homomorphism h which is a homeomorphism of U onto a neighborhood U' of the identity element e' of G'. If we put V' = h(V), then V' is an open neighborhood of e' in G'. Putting x' = h(x) and y' = h(y) with x , y E V , we have x'y' = h(x)h(y)= h(xy), so that x'y' E U' and

'

'

h - (x')h- ' ( y') = h - (X'Y'). This shows that h-' IV' is a local isomorphism from G to G. Two topological groups G, G are said to be locally isomorphic if there exists a local isomorphism from G to G'. This relation is an equivalence relation; for we have just shown that it is symmetric, and it is clearly reflexive and transitive. For example, if H is a discrete normal subgroup of G, then G and G/H are locally isomorphic (12.11.2(iii)). If G and G' are Lie groups and h is a local homomorphism of class C" from G to G , the argument of (16.9.9(iii)) shows that h is a subimmersion at all points of V , and the same calculation gives its tangent linear mapping T,(h) at all x E V . (16.9.10) Let G, G' be twoliegroups, u : G + G a homomorphismof (abstract) groups. In order that u should be a Lie group homomorphism, it is necessary and suficient that the graph rushould be a Lie subgroup of G x G'. The necessity of the condition follows from (16.8.13). Conversely, if the condition is satisfied, let h be the restriction of the projection pr, to the submanifold r,,. Then clearly h is a bijective Lie group homomorphism, hence an isomorphism (16.9.9(iv)). Remarks (16.9.1 1) Everything in this section remains valid, mutatis mutandis, when we replace differential manifolds by real- or complex-analytic manifolds (16.1.4) and C"-mappings by analytic mappings (16.3.5). This leads us to the notions of a real- (resp. complex-) analyticgroup. Such a group has an underlying structure of (real) Lie group (resp. real-analytic group). We shall prove later (in Chapter XXI) that every real Lie group can be endowed with a structure of a real-analytic group, such that the given Lie group structure is the underlying structure. On the other hand, a real Lie group cannot necessarily be endowed with a structure of a complex-analytic group for which the given structure is the underlying one, even when the group is of even dimension. (16.9.12) We shall prove in Chapter XIX (19.10.1) that every closed subgroup of a (real) Lie group is a Lie subgroup. The example of C and its subgroup R shows that the corresponding statement for complex-analytic groups is false.

58

XVI

DIFFERENTIAL MANIFOLDS

PROBLEMS

1. Let X, Y, Z be three differential manifolds and f : X + Y, g : Y Z two mappings of class C' (k 2 I); let x be a point of X, and put y = f ( x ) . Then the jet J:(g of) (Section 16.5, Problem 9) depends only on J:(f) and J:(g); it is called the composition of these two jets and is written J:(g) J:(f). A jet u of order k from X to Y is said to be invertible if there exists a jet v of order k of Y into X such that u u and u 0 v are defined and equal to the jets of the identity mappings 1 and 1y , respectively. A jet is invertible if and only if it is the jet of a local diffeomorphism of X into Y. The set Gk(n)C L!& of invertible jets from R" to R" with source and target at the origin is a group with respect to composition of jets, and is an open subset of the vector space L:," endowed with its canonical topology. Show that the structure of analytic manifold induced on G'(n) by that of L:," is compatible with the group structure, and that Gk(n)acts analytically on the left on L!, ", , and that Gk(m)acts analytically on the right on Li,,,,. If r =< s, the jet J:(f) of a C'-mapping f:X -+ Y depends only on the jet J:(f) of order s, so that we have canonical surjections J:(X, Y), --f J:(X, Y), and P:(X) + P;(X), the latter being an R-algebra homomorphism. In particular, we have a surjective mapping L;, +L;.,,, ,which is linear and whose kernel is the set of jets of order s from R" to R'" with source and target at the respective origins of these spaces and which have contact of order Z r with the zero mapping. In particular, by restricting the canonical mapping L;.,, C,nto G'(n), we obtain a surjective Lie group homomorphism G"(n)+ G'(n). If s = r I and r 2 1, show that the kernel of this homomorphism is an additive group RN,and calculate N ; show also that the group G1(n)is isomorphic to GL(n, R). --f

0

0

-+

2.

+

Let G be a Lie group, e its identity element, and let A be a commutative Lie group, written additively. Suppose that there exists a C"-mapping B : G x G --f A satisfying the relations

B(x, e ) = B(e, x) = 0

B(x, y)

+ B(xy, z) = B(x, yz) + B( y , z)

for all X E G ; for all x , y, z E G .

Show that on the product manifold G x A the law of composition (x, U)(X u) = (XY, t- u

+ B(x, A)

defines a Lie group structure. The identity element is (e, 0), and the set {(e,u ) : u E A} is a Lie subgroup isomorphic to A. The center of the group is N x A, where N is the set of all elements n belonging to the center of G such that B(n, x ) = B(x, n) for all x E G . The quotient group (G x A)/A is isomorphic to G. When G and A are real vector spaces (G now being written in additive notation) we may take B to be a bilinear function on G x G with values in A. This generalizes to the situation where G is an arbitrary Lie group and B is a C"-mapping such that the mappings x +P B (x, y) and x ++ B( y , x ) are homomorphisms of G into A for each y E G.

3. (a) Let M be a differential manifold, e a point of M, U an open neighborhood of e, and rn : U x U + M a mapping of class C", satisfying the following conditions:

10 ORBIT SPACES AND HOMOGENEOUS SPACES

59

(1) m(e, x ) = m(x, e) = x for all x E U; (2) there exists an open neighborhood V of e contained in U such that m(V x V) c U and such that m(m(x, y ) , z ) = m(x, m ( y , z))

for all x , y , z in V. Show that there exists an open neighborhood W of e contained in V and a diffeomorphism 0 : W + W such that &) = e, e(O(x)) = x and m(x, @)) = m(O(x), X ) = e

for all x E W. (Apply the implicit function theorem (16.6.8) to the equations m(x, y) = e and m( y, x) = e.) (b) Let M be a differential manifold and m : M x M -+ M a law of composition of class C" on M, which is associative and admits an identity element. Let G be the set of elements of M which are invertible with respect to m. Show that G is open in M and is a Lie group for the structure of differential manifold and law of composition induced from M. (Remark that if s E G, then x ~ m ( sx,) is a diffeomorphism of M onto M, and use (a) above.)

10. ORBIT SPACES A N D H O M O G E N E O U S SPACES

(16.10.1) Let G be a Lie group, X a differential manifold. We say that G acts diferentiubly on the left on X if we are given a left action (s, x ) ~ .sx of G on X (12.10) which is a C"-mapping of G x X into X. Usually we shall omit the word "differentiably" when there is no risk of ambiguity. Similarly we define a (differentiable) right action of G on X. For example, if p : G + GL(E) is a linear representation of G on a finitedimensional real vector space (16.9.7), then G acts differentiably on E by (s, X)H p(s) . x. Conversely, if G acts differentiably on E in such a way that, for each s E G , the mapping p(s) : XHS . x is linear, then if (aj), 6 j 6 m is a basis of E, the mapping SH p(s) . aj is a C"-mapping of G into E, and hence the entries in the matrix of p(s) relative to the basis ( a j )are C"-functions on G . This shows that sr-,p(s) is a Lie group homomorphism of G into GL(E). Suppose that G acts on the left on X, and put m(s, x) = s . x.We denote the tangent linear mappings

T,(m(s,

*

1) :T,(X) -,T,.,(X)

and

T,(m( . , x ) ) : T,(G) + TS.,(X),

respectively, by k,Hs

*

k,,

h,++ h, . x.

Since m(s, . ) : X H S . x is a difeomorphisnz of X onto itself for each s E G (12.10.2) it follows that k,++s. k, is a linear bijection, and by (16.5.4) we have the formulas (16.10.1.1)

(st) * k,

= s * ( t * k,),

t * (h, . X) = ( t . h,) . x,

h, . ( t . X) = (h, . t ) . x

60

XVI

DIFFERENTIAL MANIFOLDS

-

for all s, t in G and x and X: the notation t h, and h, * t was introduced in (16.9.8). Further, it follows from (16.6.6) that the tangent linear mapping to m is given by (16.10.1.2)

T(,,,)(m)

@,, k,)

=s

k,

+ h,

*

x

which proves that m is a submersion. (16.10.2) For each x E X, the mapping SHS . x of G into X is a subimmersion of constant rank. The stabilizer S , of x is a Lie subgroup of G .

The first assertion follows from the facts that u : h , H ( t s - l ) * h, is a * k ,., a bijection of bijection of T,(G) onto T,(G) (16.9.8), v : k,.,-(ts-') T,.,(X) onto Tt.,(X), and the diagram

(in which f:h,H h, * x and g : h , H h, . x are the tangent linear mappings defined above) is commutative. The second assertion follows from the first and from (16.8.8). (16.10.3) In order that there should exist on the orbit space X/G (12.10) a structure of diyerential manifold for which the underlying topological space is the topological space X/G and for which the canonical mapping n: : X -+ X/G is a submersion, it is necessary and suficient that the set R of pairs (x, y ) belonging to the same orbit (that is to say, the graph (1.3) of the equivalence relation "there exists s E G such that y = s * x") should be a closed submanifold of the product manifold X x X . The structure of diferential manifold on X/G satisfying these requirements is then unique.

If n: is a submersion, then so also is n x n: :X x X -+ (X/G) x (X/G), and we have R = (z x n:)-'(A), where A is the diagonal of (X/G) x (X/G), which is a closed submanifold of this product (16.8.13). Hence R i s a closed submanifold of X x X by (16.8.12). Conversely, suppose that R is a closed submanifold of X x X. Then (12.10.7) the space X/G i s Hausdorff. We shall prove: (16.10.3.1) For each x E X , there exists a chart ( U , cp, n) on X at the point x, such that q(x)= 0 and q(U) = V x W, where V , W are open subsets of R", R"-", respectively, and such that the relation (z, z') E R n (U x U) is equivalent to pr,(cp(z)) = pr2(q(z')).

10 ORBIT SPACES A N D HOMOGENEOUS SPACES

61

This result is a consequence of the following: (16.10.3.2) For each x E X, there exists an open neighborhood U of x in X, a submanifold S of U containing x anda submersion s : U + S such that for each z E U, s(z) is the onlypoint of S for which (z, s(z)) E R.

Let us assume (16.10.3.2) for the moment. Replacing U by a smaller open neighborhood of x if necessary, we can assume that there exists a chart (U, cp, n) on X at x for which the submersion s is of the form indicated in (16.7.4), and it is then clear that this chart (again restricted, if necessary, to the inverse image of a product of open sets in R" and R"-") satisfies the conditions of (16.10.3.1). For the proof of (16.10.3.2), let n = dim,(X); observe that since the submanifold R contains the diagonal of X x X, the dimension of R at the point ( x , x) is of the form m + n where 0 5 m 5 n, and T(,,,)(R) contains the diagonal A of T,,,,,(X x X) = T,(X) x T,(X). Hence T(,,,)(R) is the direct sum of A and a subspace (0) x E = T(,,,)(R) n ((0) x T,(X)), of dimension m. By (16.8.3) there exists an open neighborhood U, of x in X and a submersion f : U, x U, -+ R"-", such that R n (U, x U,) is the set of pairs (z, 2') E U, x U, satisfying f ( z , z') = 0. Furthermore ((16.8.7) and (16.8.8)), the intersection R n ((x} x U,) is a submanifold of { x } x X, of dimension m at the point (x, x ) , and consisting of the points ( x , z ) E { x } x U, such that f ( x , z ) = O ; the tangent space to this submanifold at the point ( x , x ) is (0) x E, the mapping Z H ~ ( X , Z ) of U, into R"-" being a submersion at the point x , because A is contained in the kernel of T,,,,,(f). Observe that the mapping Z H ~ ( X . , z) is also a submersion of U, into R"-" at the point x ; for since the relation R is symmetric, the kernel of T ( , , , ) ( f ) , which is the tangent space to R at ( x , x ) , is invariant under the mapping (u, V)H(V, u) of T,(X) x T,(X) onto itself, and hence its intersection with T,(X) x (0) is of dimension m , which proves the assertion. Replacing Uo if necessary by a smaller neighborhood of x , we can assume (16.8.8) that there exists a submersion g : U, -+ R" such that, if N is the submanifold of U, given by the equation g(z) = 0, then x E N and the tangent space at ( x , x) to {x} x N is a supplement (0) x F of (0) x E in (0) x T,(X). This being so, consider the mapping u : (z, z ' ) ~ ( f ( z , z'), g(z')) of U, x U, into R" = R"-" x R". The choices off and g show that the partial mapping z' t,( f ( x ,z'), g(z')) of U, into R" has a bijectioe tangent linear mapping at the point x . For if a vector h E (0) x T,(X) runs through (0) x E, then its image under T,(f(x, .)) is zero and its image under T,(g) runs through R"; and if, on the other hand, h runs through (0) x F, its image under T,(g) is zero and its image under T,(f(x, runs through R"-". By the implicit function theorem (16.6.8) there exist two open neighborhoods U 1 , U2 of x in X and a C"-mapping *>>

62

XVI

DIFFERENTIAL MANIFOLDS

u : U, -+ U, such that for each z E U 1 ,the only solution z' of equations

f(z,z') = 0,

E U2 of

the system

g(z') = 0

is z' = u(z). Moreover, since z ~ f ( zx), is a submersion of U, into R"-" at the point x, it follows that T,(f( . ,x)) is of rank n - m,hence (16.6.8.1) T,(u) is of rank n - m, or in other words u is a submersion, at the point x, of U, into the submanifold N of U, . Replacing U, by a smaller neighborhood we can therefore assume that u is a submersion of U, into N , so that zi(U,) c U, n N

is open in N. If S = u(Ul) n U1 and U = v - ' ( S ) n U,, then U, S and s = u I U satisfy the conditions of (16.10.3.2). Now that (16.10.3.1) is established, we shall return to the proof of the sufficiency of the condition in (16.10.3). For each x E X, let (Ux,cp, n,) be a chart of X at x satisfying the condition of (16.10.3.1), where V, W, m are replaced by V, , W, , m, . Observe first that if K, is the image under cp'; of a compact neighborhood of 0 in (0) x W,, then n(KJ is a neighborhood of n(x) in X/G and the restriction of n to K, is injective. Hence (12.10.9) X/G is metrizable, locally compact, and separable. Moreover, the open sets in X/G contained in n(U,) are the sets of the form n(q;'(V, x T)), where T runs through the open sets in W, (12.10.5); in other words, there is a homeomorphism w, : n(U,) + W, such that w;'(w) = n(q;'(O, w)). We have to show next that the charts (n(lJ,), ox, n, - m,) are mutually compatible. So let (U,, q,, n,) and (U,,, q,.. n,,) be two charts of X of the family considered above; put S = x-'(n(U,) n n(U,.)) n U,,

S' = n-'(n(u,) n ~(IJ,,)) n U,.

which are open sets, and put

Q = cp,(S)

c V, x W,,

Q'

= cp,.(S') c

V,, x W,,

.

The projections

P = pr,(Q)

c

W,,

P'

= pr,(Q') c

W,,

are then open sets such that Q = V, x P and Q' = V,, x P'. Since n(S) = n(S') = n(V,) n n(V,.) by definition, for each p E P, there exists a unique point p' =f(p) E P such that

n((P;'(v, x {PI)) = .(cp,.'(VX, x {P", and what has to be shown is that the bijectionf: P -+ P' so defined is of class C" in a neighborhood of each point p E P. Now, let q E Q (resp. q' E Q')

10

ORBIT SPACES AND HOMOGENEOUS SPACES

63

be such that pr,(q) = p (resp. pr,(q') = p ' ) ; if z = q;'(q) and z' = q;'(q'), there exists an element s E G such that z' = s z. Consider the diffeomorphism g : U H S * u of X onto itself; since it maps z to z', there exists an open neighborhood T c U, of z such that g(T) c U,, , and the composite mapping U H rp,,(g(u)) defined on T furnishes a chart on X at the point z, with domain T, which is therefore compatible with that defined by rp,. The mapping r++q,,(g(q;l(r))), defined on the neighborhood q,(T) of q, is therefore of class C". However, by its definition it is of the form r H ( e ( r ) ,f(prz(r))), which shows thatfis of class C" in a neighborhood of pr,(q) = p . Finally, the uniqueness of the structure of differential manifold on X/G in (16.10.3) is a consequence of (16.7.7(ii)) (take f = n and g = lX,Gthere). Q.E.D. When the condition of (16.10.3) is satisfied, the space X/G endowed with the structure of differential manifold defined in (16.10.3) is called the orbit manifold of the action of G on X. If n : X X/G is the canonical submersion, then we have n(s * x ) = n(x) for all s E G and all x E X; taking tangent mappings and using the notation introduced earlier. we obtain --f

(16.10.3.3)

TS.,(n) *

(S

*

h,) = T,(n)

. h,

for all h, E T,(X). (16.10.3.4) It should be noted that the condition in (16.10.3) is not always satisfied, even when G is afinite group. Consider for example the case where X = R and G is the multiplicative subgroup (1, - l} of R*,the action of G on X being multiplication (cf. Problem 1). (16.10.4) Suppose that the orbit manifold X/G exists. Then a mapping @ : X/G --* Y, where Y is a differential manifold, is of class C' (resp. a subimmersion, resp. a submersion) i f and only i f the composite mapping Q, n : X -+ Y has the same property. 0

The assertion relative to class C" is a particular case of (16.7.7(ii)). The other assertions follow from the relation rk,(@ n) = rk+)(@), which comes from the fact that n is a submersion. 0

(16.10.5) Let G (resp. G') be a Lie group acting differentiably on a differential manifold X (resp. XI). Then G x G' acts differentiably on X x X'.

64

XVI

DIFFERENTIAL MANIFOLDS

If the orbit manifolds X/G

and X'/G exist, then so does the orbit manifold (X x X')/(G x G), and the canonical mapping

(X x X')/(G x G')

--f

(X/G) x (X'/G')

is a difeomorphism. The first two assertions follow from (16.6.5), (16.10.3), and (16.8.7(ii)). The third is a consequence of (16.10.4) and (16.5.6). (16.10.6) Let H be a Lie subgroup of a Lie group G, and consider H as acting on G on the right by translation. Then the orbit manifold G/H exists, G acts differentiably on the left on G/H, and we have

dim(G/H) = dim(G) - dim(H).

If H is normal in G, the manifold structure of G/H is compatible with its group structure. To verify that the condition of (16.10.3) is satisfied in the present situation, we observe that the set R c G x G is here the set of pairs (x, y) such that x-'y E H. Now the mapping ( x ,y ) ~ x - ' yof G x G into G is a submersion (16.9.9), and H is a submanifold of G, hence (16.8.12) R is a submanifold of G x G. To show that G acts differentiably on G/H, let p denote the mapping ( X , Y ) H X ~ of G x G into G, fi the mapping ( x , j ) ~ x -ofj G x (G/H) into G/H, and n : G + G/H the canonical mapping. Then we have a comniutative diagram GxG P ' G lc,xn

I

G x (G/W

7 G/H

and 0 x (G/H) may be identified with the orbit manifold (G x G)/({e}x H) (16.10.5). The fact that fi is of class C" now follows from (16.10.4), since p and n are C"-mappings. When H is normal in G, let m denote the mapping (x, y)t+xy-' of G x G into G, and m the mapping (a, j ) w i j - ' of (G/H) x (G/H) into G/H. Then we have a commutative diagram GxG

rn

- G

10 ORBIT SPACES A N D HOMOGENEOUS SPACES

65

Identifying (G/H) x (G/H) with (G x G)/(H x H), it follows as above that m is of class C". Finally, the dimension formula follows immediately from (16.10.3.1). Examples (16.10.6.1) If H is a discrete normal subgroup of G, the canonical mapping n : G -+ G/H is a local direomorphism (16.5.6). As an example, consider the

n-dimensional torus T" = R"/Z", which is a compact connected commutative Lie group, being the canonical image of the cube [0, l]" in R". (16.10.7) Let G be a Lie group acting diflerentiably on a diflerential manifold X . Ifapoint x E X is such that the orbit G * x is a locally closedsubspace of X , then G x is a submanifold of X , and the canonical mapping f , : G/S, + G x (12.1 1.4) is an isomorphism of differential manifolds. In particular, the above condition is satisfied for all x E X wherever the orbit manifold X/G exists.

-

-

Every point of the subspace G . x of X has by hypothesis a neighborhood homeomorphic to a complete metric space (3.14.5), hence (12.16.12) f, is fx a homeomorphism. Next, since the composite mapping h, : G 3 G/S,+ G * x is a subimmersion (16.10.2), it follows from (16.10.6), (16.10.4), and (16.8.8) that f, is an immersion; hence the first two assertions follow from (16.8.4). Finally, if the manifold X/G exists, then since G . x is the section R(x) of the set R defined in (16.10.3) it follows that G . x is closed in X (3.20.12).

-

The example of the group Z acting differentiably on T by the rule n x = x + rp(ne), where rp : R + T is the canonical homomorphism and f3 is irrational, shows that the hypothesis on G . x in (16.10.7) is not always satisfied. In particular: (i) IfG is a Lie group which acts diflerentiably and transitively on a differential manifold X , thenfor each x E X the canonical mappingf , : G/S, -+ X is a diffeomorphism. (ii) If u ; G --+ G' is a surjective homomorphism of Lie groups with kernel H , then the canonical mapping G/H + G' is an isomorphism of Lie groups. (16.10.8)

The assertion (ii) follows from (i) by considering G as acting transitively on G' by means of the mapping ( x , x')Hu(x)x'.

66

XVI DIFFERENTIAL MANIFOLDS

(16.10.9) Let u : G 4G' be a homomorphism of Lie groups, H a Lie subgroup of G, and H' a Lie subgroup of G such that u(H) c H'. Then the unique mapping ii :G/H --+ G / H ' for which the diagram

G

G'

It

(a,n' being the canonical mappings) is commutative, is of class C".

We have n' 0 u = ii 0 a, and it is clear that n' 0 u is of class C"; now apply (1 6.1 0.4) and (1 6.10.6). (16.10.10) Let G (resp. G') be a Lie group acting differentiably on a differential manifold X (resp. X'). If p :G + G is a Lie group homeomorphism and f : X -+ X a C"-mapping, then G and G' are said to act equivariantly (relative to p andf) on X and X if the diagram

GxX-X

m

is commutative (where m,m' define the actions of G on X and G' on X' respectively). This leads, for each pair (s, x ) E G x X, to a commutative diagram of tangent linear mappings (16.5.4) :

Remarks (16.10.11~ We leave it to the reader to transpose the results of this section

to the context of real- (resp. complex-) analytic groups acting analytically on real- (resp. complex-) manifolds.

10 ORBIT SPACES A N D HOMOGENEOUS SPACES

67

(16.10.12) Let E be a set (not a priori equipped with a topology), G a Lie group acting transitively on E, and suppose that the stabilizer of each element of E is a Lie subgroup of G (in fact it is sufficient that this should be the case for one point of E). Then there exists on E a unique structure of differential manifold such that G acts differentiably on E: this follows from (16.10.6) and (16.10.8).

PROBLEMS

1. If a Lie group G acts differentiably and properly (Section 12.10, Problem 1) on a differential manifold X,the orbit-manifold X/G does not necessarily exist (cf. (1 6.10.3.4)). Show however that the orbit-manifold X/G does exist if, in addition to the hypotheses above, G acts freely on X. 2. Let G be a Lie group, H a Lie subgroup of G, and X a differential manifold on which H acts differentiably (on the left).

(a) The Lie group H acts differentiably on the right on G x X by the rule (s, x ) . t = (sr, t

-l

. x).

Show that with respect to this action the orbit manifold Y = H\(G x X) always exists. If H = G, the orbit manifold is diffeomorphic to X.If n : G x X -+Y is the canonical mapping, show that G acts differentiably on the left on Y by the rule s‘ . n(s, x ) = Tr(s‘s, x).

(b) Show that the mapping h : x Hn(e, x ) is a diffeomorphism of X onto a submanifold X‘of Y, that X’is stable under the action of H c G, and that H acts equivariantly (relative to h) on X and X . If s E G is such that s . X’n X # 0, then s E H. The stabilizer of n(e, x ) under the action of G is equal to the stabilizer of x under the action of H. (c) Show that the mapping (s, y ) Hs . y is a surjective submersion of G x X’onto Y (cf. 16.14.8). 3. Let G be a Lie group, H a Lie subgroup of G, and n : G +G/H the canonical submersion.

(a) For each Lie subgroup G’ of G, show that G’ n H is a Lie subgroup and that WIG’is a subimmersion of G’ into G/H. If n(G’) is closed in G/H (which will be the case if either H or G’ is compact), then n(G’) is a submanifold of G/H diffeomorphic to G‘/(G’ n H).(Use (16.10.7).) (b) Consider the Lie group G = R x T2and the Lie subgroup H = R x {0} of G. If : R +T is the canonical homomorphism, let G’ be the subgroup of G which is the , where 0 is a fixed irrational image of R under the homomorphism x t-+ ( x , ~ ( x )V(Ox)), number. Show that G’ is a Lie subgroup of G but that n(G’) is dense and not closed in G/H = T2. (c) If dim(G’) - dim(G’ n H) = dim(G) - dim(H), the restriction of n to G’ is a submersion into G/H, and hence factorizes into G’ + G’/(G’ n H)+ G/H, where u is

68

XVI

DIFFERENTIAL MANIFOLDS

a diffeomorphism of G'/(G' n H) onto an open submanifold of G/H. If either G' or H is compact and G is connected, then u is a diffeomorphism of G'/(G' n H) onto G/H. Give an example in which the image of u is a nondense open set in G/H. (Take G = SL(2, R)andG'to be the subgroupof upper triangular matrices

(0"

x?l), where x > 0,

and H the subgroup of lower unitriangular matrices

11. EXAMPLES: U N I T A R Y GROUPS, STIEFEL MANIFOLDS, GRASSMANNIANS, PROJECTIVE SPACES

(16.11 .I) Let E be a real vector space of dimension n, and let X(E) be the set of all symmetric bilinear forms on E x E, which is a real vector space of dimension in(n + 1). For each pair (p, q) of integers 50 such that p + q = n, the subset 3 f P J E ) of symmetric bilinear formsof signature (p, q) on E is open in the vector space X(E). To see this, let m0be a form belonging to X,JE); then there exists a direct sum decomposition E = P 0 N, where P and N are vector subspaces of dimensions p and q, respectively, such that Q0(x, x) > 0 for x # 0 in P, and O0(x, x) < 0 for x # 0 in N. If llxll is a norm which defines the topology on E, then there exist two real numbers a > 0 and b > 0 such that Qo(x, x) 2 ~11x11~ for all X E P and @,(x, x) 6 -bllxl12 for all x E N, because spheres are compact (3.17.10). If Y is a sufficiently small symmetrical bilinear form such that 1 Y(x, x) I I inf(a, b)IlxllZfor all x E E, and if Q, = Q 0 Y , then we shall have @(x,x) L +a11x112 for X E P and @(x,x) 5 - 3 b l l ~ 1 )for ~ x E N ; this shows that Q, has signature (p, q), by virtue of the law of inertia.

+

+

(16.11.2)

The group GL(E) acts differentiably (indeed analytically) on X ( E ) and on each of the X P J E ) . Namely, if Q, is any symmetric bilinear form and s E GL(E), then s . Q is the form (x, y)H@(s-l . x, s - '

. y).

Moreover, each of the open sets X p J E ) is an orbit of this action. Hence it follows from (16.10.2) that, for each Q E Hp, JE), the subgroup of elements s E GL(E) such that s * Q, = 0 is a Lie subgroup of GL(E) of dimension nz - 3n(n + 1) = $n(n - 1). This Lie group is called the orthogonal group of the form Q, and is denoted by O(Q,).When p = 3 and q = 1 it is called the Lorentz group. When p = n and q = 0, it is called simply the orthogonalgroup in n variables. All orthogonal groups O(Q,) with Q, of signature (n,0) are isomorphic to the group corresponding to E = R" and @ the Euclidean scalar product, namely n

Q,(x,Y)=(xIY)=

C

j= 1

tjqj,

11 EXAMPLES

69

where x = (ti),y = (qj). This group is also denoted by O(n, R), or simply O(n) if there is no risk of ambiguity. It is compact, because the matrices S = (aij) belonging to O(n) are characterized by the relation 'S . S = I and therefore in particular satisfy the relations

j= 1

at.= 1 for 1 5 i

5 n ; hence

they form a bounded closed subset of R"'. The kernel in O(n) of the homomorphism s H det(s) is a Lie subgroup of O ( n ) of index 2 (because det(s) = - 1 if s is a reflection in a hyperplane) called the rotation group or special orthogonal group in n variables, and denoted by SO@,R) or S O ( n ) ; it is an open subgroup of O(n). (16.11.3) There are analogous definitions and results when E is taken to a vector space of dimension n over the field of complex numbers C, or a left vector space of dimension n over the division ring of quaternions H. In either case 2 ( E ) now denotes the set of hermitian sesquilinear forms Q, on E x E, that is to say forms Q, satisfying

+

Q,(x x', Y) = @(x, y) + W',y), @(Ax,y) = AQ,(x, y) (A E C (resp. H)), @(Y, x) = Q,k Y).

It follows that @(x,x) is always real, and therefore X ( E ) is a real vector space of dimension n + n ( n - 1) = n2 in the complex case, and of dimension n + 2n(n - 1) = 2n2 - 17 in the quaternionic case. Just as in (16.11.1), we can show that for each signature ( p , q ) such that p q = n, the subspace Z,,,,(E) of forms of signature (p, q ) is an open subspace of X ( E ) . If Q, E X,,,&E), we see as in (16.11.2) that the subgroup of elements s E GL(E) such that s .Q, = d) is a Lie subgroup of dimension 2n2 - n2 = n2 in the complex case, and of dimension4n2 - (2nZ- n) = n(2n 1) in the quaternionic case. This subgroup is called the unitary group of the form @, and is denoted by U(Q,). When (p, q ) = (n, 0), it is called simply the unitary group in n variables. All unitary groups U(@) with @ of signature (n, 0) are isomorphic to the group corresponding to E = C" (resp. E = H") and

+

+

n

WX,Y)=(XIY)=

C

j= 1

(cj),

t j r j ,

where x = y = (qj). This group is also denoted by U(n, C) or U(n)(resp. by U(n, H)). It is conipuct, because the matrices S = ( a i j ) belonging to U(n,C) (resp. U(n, H)) are characterized by the relation S * = I and therefore in particular satisfy the relations

n

's

1 ( a i j [ ' = I for I 5 i 6 n ; consequently they

j= I

form a bounded closed subset of Cn2(resp. H"').

70

XVI

DIFFERENTIAL MANIFOLDS

The homomorphism sHdet(s) of U(n,c) onto U(1,C) = U (the unit circle in C) is surjective, because if 5 E U and if (ej)lsjs,,is an orthogonal basis of C",the automorphism s of C" defined by s(e,) = Gel, s(ej) = ej for 2 5 n is unitary and has determinant 5. Hence (16.9.9) the kernel of this homomorphism is a normal Lie subgroup of U(n) of dimension n2 - 1, called the special unitary group and denoted by SU(n).

sj

(16.11.4) Let n , p be two integers 2 1. The space RnPofsequences (Xk),Sk,, of @ vectors in R" can be identified with the set of real matrices X with n rows

and p columns, the kth column being the vector x k . The group GL(n, R) acts differentiably (indeed analytically) on the left on RnPas follows: the automorphism s E GL(n, R) transforms the sequence ( x k ) into the sequence (s * x k ) . Equivalently, if we identify s with its matrixS relative to the canonical basis of R", the action of GL(n, R) on RnPis left multiplication (S, X) -,S * X of matrices. Now let p 6 n and let S,,, (or Sn,,(R)) be the subset of RnPconsisting of sequences ( x k ) l s k $ , which are orthonormal relative to the Euclidean scalar product (6.5). This set may also be described as follows: the orthogonal group O(n, R) acts differentiably on RnPby restriction of the action of GL (n, R) defined above, and S,,, is the orbit, under this action of O(n, R), of the orthonormal sequence (ek)lsksp consisting of the first p vectors of the canonical basis ( e k ) l s k < , , of R". Since, by virtue of (12.10.5), this orbit is compact and hence closed in RnP,it follows from (16.10.7) that S,,, is a compact submanifold of RnP,called the (real) Stiefel manifold of orthonormal systems of p vectors (sometimes called p-frames) in R". It is clear that the subgroup of O(n, R) which stabilizes the p-frame (ek)l5 k s p may be canonically identified with the orthogonal group O(n - p, R), by identifying R"-, with the subspace of R" spanned by e p + l ., .., e,, (when n = p , this group consists only of the identity element). Hence S,,, is isomorphic to the homogeneous space

,

,

O(n,R>/O(n- p, R). When p = n the Stiefel manifold S,,,,(R) may be identified with O(n, R), and when p = 1 the manifold Sn,,(R) may be identified with the sphere S,,-, (16.2.3). When 1 = < p5 n - 1, S,,,(R) is also the orbit of the p-frame (ek),, k s p under the action of the rotation group SO@,R). Since the p be identified with SO(n - p , R), it follows that stabilizer of ( e k ) l g k smay S,,,,(R) may be identified with the homogeneous space SO@,R)/SO(n- p , R) forlgppn-1. (1 6.11.5)

In the considerations of (16.11.4) the field R can be replaced everywhere by C or H, the Euclidean scalar product being replaced by the Hermitian

(16.11.6)

11 EXAMPLES

71

scalar product. In this way we define complex and quaternionic Stiefel manifolds S,,,,(C) and S,,,J H ) . They are isomorphic, respectively, to the homogeneous spaces U(n, C)/U(n- p , C ) and U(n, H)/U(n- p , H); and if lspsn-I, S,,,,(C) is also isomorphic to SU(n, C)/SU(n- p , C ) . They are therefore compact manifolds. When p = 1, S,,,, ( C ) may be identified with the sphere S2,,-,,and S,,,,(H) with S4,,-,. (16.1 1.7) The groups SO@, R), SU(n, C ) , U(n, C ) , and U(n, H) are connected i f n 2 1 ;so also are the Stiefel manifolds S,,,,(C) and S,,,JH) if 1 p 5 n - 1, andSn,,(R)ij"1~pjn-1andn22.

When n = I , the groups SO(1, R) and SU(1, C ) consist of the identity element alone. The group U(1, C ) may be identified with the unit circle U = S, in C , and U(1, H) with the multiplicative group of quaternions of norm 1, which as a topological space is the sphere S,; hence these two groups are connected (16.2.3). It follows also from (16.2.3) that the Stiefel manifolds S,,,,(R) = S n - , are connected if n 2 2, and that S,,,,(C)= S2,,-]and S,,, ,(H) = S4,,-, are connected if n 2 1 . Consequently the homogeneous spaces SO(n)/SO(n- l),

W ,C)/U(n- 1 , C ) ,

SU(n)/SU(n- I), U(n, H)/U(n- H )

are connected if n 2 2. The first assertion of (16.11.7) now follows by induction on n , by virtue of (12.10.12). The second is an immediate consequence, by (3.19.7). (16.11.8) If E is a vector space over a field K, we denote by G,(E) the set of vector subspaces of dimension p in E. The set G,(R") is denoted by G,,,,(R) or simply G,,, ,. It is clear that the orthogonal group O(n, R) acts transitively on GnJR). Furthermore, if F is the subspace of R" generated by the first p vectors in the canonical basis, then the stabilizer of F under this action leaves fixed (as a whole) the orthogonal supplement of F, namely the subspace spanned by the last n - p vectors of the canonical basis. Hence the stabilizer of F is a Lie subgroup of O(n,R) which may be identified with the product O(p,R) x O(n - p , R) ((16.9.8) and (16.8.8(i))). It follows (16.10.12) that there exists on the set G,,, a unique structure of differential manifold for which O(n, R) acts differentiably on G,,, ,. The set G,,,,endowed with this structure is called the (real) Grassmannian with indices n, p . When p = 1, the Grassmannian G,,,, is also denoted by P,-,(R) or simply P,,-] , and is called (real) projective space of dimension n - 1. The differential manifold G,,,,(R) is

,

72

XVI

DIFFERENTIAL MANIFOLDS

diffeomorphic to the homogeneous space O(n,R)/(O(p, R) x O(n - p, R)). If n - 1, it is also diffeomorphic to SO@,R)/H,, where H, is the subgroup of O(p,R) x O(n - p , R) consisting of pairs (t, t ' ) such that det(t) = det(t'). We remark that G,,, may also be considered as the space of spheres with center 0 and dimension p - 1 contained in the sphere S,,- : these spheres correspond one-to-one with the vector subspaces of dimension p in R".

p

,

(16.11.9) The orthogonal group O(p,R) acts differentiably on the right on the Stiefel manifold S,,, ,(R) by matrix multiplication ( X , T)H X T. For each matrix X E S,,, can be written as S E, where S E O(n, R) and E is the matrix whose columns are the vectors el,..., e,; thus X consists of the first p columns of S. The columns of E * Tare the images of el,. . . , ep under the element of the orthogonal group O(p,R) whose matrix relative to (ek)l sksp is T; hence these columns form ap-frame, that is to say E * T ES,,, and therefore also X . T = S . E . T ES,,,,. This shows also that the set of orbits for the above action may be identified with G , , , p .If we endow G,,, with the structure of differential manifold defined in (1 6.11.8) and identify Sn, with O(n, R)/O(n - p , R) and G,,,,with O(n, R)/(O(p, R) x O(n - p , R)), then the canonical mapping x : S,,, --t G,,, is a submersion (16.10.4); consequently, for the action of O(p,R) on S,,, defined above, the orbit-manifold exists and can be identified with the Grassmannian G , , , (16.10.3). Moreover, the orbits are each diffeomorphic to O(p,R). It follows from (16.11.6) and (16.11.7) that G,,,,(R) is compact and connected if n 2 1 and 1 5 p 5 n. There are analogous definitions and results for the complex and quaternionic Grassmannians G,,,,(C) and G,,,,(H), and in particular for the complex and quaternionic projective spaces P,,-l(C) and P,,-l(H). The dimensions of the differential manifolds G,,,,(R), G,,,,(C),and G,,,,(H) are therefore, respectively,

-

,

,

,

,

,

jn(n

- 1)

-jp(p

,

- 1) - t ( n - p)(n - p - 1) = p(n - p ) , nz - p2 - ( n - p)Z

n(2n

= 2p(n

-p),

+ 1) - p ( 2 p f 1) - ( n - p)(2n - 2p + 1) = 4p(n - p ) .

(16.11.10) There is another, equivalent, definition of the structure of differential manifold on the Grassmannian G,,,,(R).For the group GL(n, R) also acts transitively on G,,,,(R), and the stabilizer of the subspace F considered in (16.11.8) is the subgroup H of GL(n, R) consisting of matrices of the form

(0"

:),

where A is a square matrix of p rows and p columns. Since H is

clearly a Lie subgroup of GL(n, R) (it is diffeomorphic to the product GL(p, R) x GL(n - p , R) x RP("-,)) we obtain (16.10.12) a structure of dif-

11 EXAMPLES

73

ferential manifold on G,,,p . Since the action of O(n, R) on G,,, is obtained by restriction of the action of GL(n, R), the structure so obtained is the same as that defined in (16.1 1.8). Now let L,,, denote the subset of RnPconsisting of the matrices X of rank p , that is to say matrices X whose p columns xk (1 g k g p ) are linearly independent. This set Ln,p is open in RnP. More precisely, for each subset J consisting of p elements il < iz < . < ip of the set I = { 1,2, . . . ,n}, let Tj be the set of matrices X such that the matrix X, formed by the i,th, . .., ipth rows of Xis invertible; then it is clear that Tj is open in RnPand is canonically

-

diffeomorphic to GL(p, R) x Rp(n-p),and L n , pis the union of the

(3

sets

Tj . Note that GL(p, R) acts differentiably on the right on L,,, p , by matrix multiplication ( X , T )H X T. We assert that G,,, can be identified with the orbit-manifold of this action. Firstly, the orbit-manifold GL(p, R)\L,,, exists : for if R is the set of pairs ( X , Y ) of elements of L,,,p belonging to the same orbit, then the intersection R n (Tj x Ln,pJis the graph of the C"-mapping ( X , T ) H ( X X ~ - ~ T of ) ~Tj - ~x GL(p, R) into Rp(n-p),and the existence of the orbit-manifold now follows from (1 6.8.1 3) and (16.10.3). It is clear that there is a canonical bijection w of GL(p, R)\L,,, onto G,,, such that o(n(S . X)) = S n ' ( X ) for S E GL(n, R) and X E L n , P ,where n and n' are the canonical mappings of Ln,p onto GL(p, R)\L,,,, and G , , p , respectively. It follows (16.10.12) that o is a diffeomorphism, and our assertion is proved. From these considerations we can construct a convenient atlas for G,,,p . Let V, be the subset of Tj consisting of the matrices X E Tj such that XJ = Zp (the unit matrix). Then it is immediately seen that the restriction of n to V, is a bijection of V, onto U, = n(Tj). If 'pJis the inverse of this bijection, we have q,(n(X)) = XXJ-', from which we conclude (16.10.4) that 'pJ is of class C"; since nlVJ is also of class C", 'pJ is a dzfleomorphism of UJ onto VJ . AS VJ may be identified with Rp(n-p),we have an atlas of GL(p, R)\L,,, consisting of the charts (U, , 'p,, p(n - p ) ) . There are of course analogous results for complex and quaternionic Grassmannians. The real and quaternionic Grassmannians are real-analytic manifolds, and the complex Grassmannians are complex-analytic manifolds. (16.11.11) By virtue of (16.11.9), the projective space P,,(R) (resp. P,,(C), resp. P,,(H)) may be identified with the orbit manifold of the group consisting of the identity and the symmetry XH - x acting on S,, (resp. the group of rotations X H X ~ with 151 = 1 acting on SZn+lc C"", resp. the group of rotations x H x q , where q is a quaternion of norm 1, acting on S4n+3c H""). (16.11.12) The diflerential manifolds P,(R), P1(C), and Pl(H) are dzgeomorphic to S1, S2, and S4, respectively.

74

XVI

DIFFERENTIAL MANIFOLDS

We shall prove the assertion for Pl(H). For each pair of quaternions x = xo

+

+ ix, + j x , + kx, ,

y =yo

+ iyl + j y 2 + ky3

1 y I = 1 (where I x I is the Euclidean norm of x in R4),let such that 1 x I z =f(x, y ) = (zo,z l , z 2 , z 3 , z,) be the point of R5 (identified with H x R) defined by 2xJ = zo Since Z%

+ iz, + j z , + k z , ,

+ Z : + Z $ + Z:

24

= 1x1’

- IyI’.

= 41xJI2= 4(xIZly1’,

it is clear that z E S,. Moreover, z, = 2 1x1 - 1 can take all values in the interval [ - 1, 11, and for a given value of z4 # & 1, the quaternion 2xJ can take all values on the sphere of radius 1 - 1 z4 I ’. Hence f is a surjective C”mapping of S7 onto S , . Next, the relation f ( x , y ) =f(x’, y’) implies firstly that lx’l = 1x1, so that we may write x’ = xq, where q is a quaternion of norm 1;also it implies that x’y’ = x J , which gives y’ = y4-I = yq. Hence the mapping f factorizes as’follows :

+

s7

+

Pl(W

9 -+

s4

9

where g is bijective. It remains to show thatfis a submersion. Since the rotations x w q ’ x , y - q ” y (where q’,q” are quaternions of norm 1) are diffeomorphisms of S, which fix z , and transform X J into q’xJij”,it is enough to check thatfis a submersion at the points (x,y ) E S7 such that the quaternions x, y are scalars xo ,y o . It is then immediately verified (since xo , y o are not both simultaneously zero) that the Jacobian matrix off(extended to H2 = RE by the same definition) is of rank 5. This fact, together with the relation f ( t x , ty) = t2f(x, y) for all scalars t, proves that g is a diffeomorphism of P,(H) onto S4 (16.8.8). The proofs for P,(R) and Pl(C) are analogous but simpler: In the case of P1(C), we map the point (x, y ) E S, c R4 = Cz to the point f ( X ? Y > = (2XK 1 X l 2 -

blZ)

on S, c C x R = R3,and the mappingffactorizes as

where g is a diffeomorphism. We may therefore tfansport to S, by means of g the structure of complex-analytic manifold of P1(C).The sphere S, , endowed with this structure, is called the Riemann sphere.

11

EXAMPLES

75

PROBLEMS

(a) For any two points x , y

E S2,-,

c C", put

m(x, y) = arc cos(W(x Iy))

which is a real number between 0 and n-. If s, t are any two elements of the unitary group U(4, Put d(s, t ) =

sup

a ( s . x , t . x).

XES2n-1

Show that d is a bi-invariant distance on U(n). (b) For s E U(n), let eiei (I 5 j 5 m) be the distinct eigenvalues of s, so that C" is the Hilbert sum of the eigenspaces V, of s (1 5 j 5 m), the restriction of s to Vj being the homothety with ratio eieJ;we may assume that - n < 0 , s n for each j. Show that if 0(s) = sup I0,I, then d(e, s) = @s). (Minorize W ( xI s . x ) by using the decomposition 1SjSm

of x as a sum of vectors x j E V, .) (c) Let s, t be two elements of U(n) such that s and the commutator (s, t ) = s t s r ' t - ' commute, or equivalently such that s and u = tst-' commute. Show that if B ( t ) < &r, then s and t commute. (With the notation of (b), observe that if Vj is the orthogonal supplement of Vj ,and if W, = t(V,), then Wj is the direct sum of W, n V, and W, n Vl , and deduce from the hypothesis on t that W, n Vj = {O}.) When n = 2, the two charts q ~ , ,p), on S, defined in (16.2.3) are such that and p2 define on S , the structure of complex analytic manifold defined in (1 6.11.12). Let U be an open neighborhood of 0 in R". In the real-analytic manifold U x Pm-,(R), let U' be the subset consisting of points (x, z) such that for some system (z', . . ., z") of homogeneous coordinates for z we have xlzk - x*z' = 0 for all pairs of indices j , k (in which case these relations are satisfied for all systems of homogeneous coordinates for z). Show that U' is a closed analytic submanifold of dimension n in U x P,,-,(R) (consider the atlas of P.-,(R) defined in (16.11.10)). The restriction nu of the projection pr, to U' is a surjection of U' onto U and is proper; n, '(0) is a submanifold of U' isomorphic to Pn-,(R), and the restriction of n-u to U' - nU'(0) is an isomorphism of this open set onto U - {O}. Let r be the inverse isomorphism of U - {0}onto U' - ncl(0), and let f be any C'-function defined on an open neighborhood I of 0 in R and with values in U, such thatf(0) = 0 and T,(f) # 0. Then the function t Hr ( f ( t ) ) ,defined on 1 - {O} and with values in U', extends by continuity to a mappingf' : 1 + U', such that f'(0) is the canonical image in P,.. ,(R) of the vector T,(f) . I . Furthermore, iff is of class C', thenf' is of class C-'; and iff, g are two C'-functions defined on I, such that f(0) = g(0) = 0, which have contact of order >k 2 1 at the point 0, thenf' and g' have contact of order z k - 1 at the point 0 (Section 16.5, Problem 9). If V is another open neighborhood of 0 in R" and if u : U + V is an isomorphism of analytic manifolds (resp. a diffeomorphism), then if V' and nv are defined as above, there exists a unique isomorphism u' : U' + V' of analytic manifolds (resp. a unique diffeomorphism) such that nv u' = u 0 nu. Deduce that if X is a pure differential (resp. analytic) manifold of dimension n, and x a point of X, there exists a differential (resp. analytic) manifold X' of dimension n 0

76

XVI DIFFERENTIAL MANIFOLDS

and C m- (resp. analytic) surjection 7rx : X' + X with the following properties: (a) the restriction of 7rx to X - T?,'(x) is an isomorphism of X' - T?'(x) onto X - {x}; (b) there exists a chart (W, v, n) of X at the point x such that d x ) = 0 and @Y) =U is an open neighborhood of 0 in R", and a diffeomorphism (resp. an isomorphism of analytic manifolds) u of 7rx'(W) onto U' (with the notation introduced above) such that v(vX(x')) = T&(x')) for all x' E n H(by e,a(b, t>>, where 8,, is a C"-mapping. Moreover, it follows directly from this definition that if u, B, y are any three indices and if we denote by $ J a , I&, and $fa the restrictions of $, t,bys, and t,bya to (Van U,n U,) x Fa, (U,n U , n U,) x F,,

and (Uan U , n U,) x Fa,

respectively, then we have (1 6.13.1 .I)

= J/;,

0

Now consider two fibrations A = (X, B, n) and I' = (X', B, n') with the same base, and a B-morphism (resp. a B-isomorphism) g of X into X' (resp. onto X'). Then there exists an open covering (U,) of B such that for (16.13.2)

88

XVI

DIFFERENTIAL MANIFOLDS

each a the fibrations induced on U, by both 1 and 1' are trivializable, so that we have diffeomorphisms cp, : U, x F A + n'-'(U,)

cp, : U, x F, + z-'(U,),

satisfying (LT). The composite mapping

is then of the form

(b, f)H(b,o,(b, t)), where o, is a mapping of class C" (resp. a mapping of class C" such that o,(b, * ) is a diffeomorphism of F, onto FA for each b E U,). The mapping g, is called the local expression of g corresponding to cp, and cp; . Also, with the notation of (16.13.1) and analogous notation for the fibration A', if we put g,, = g. I ((U.n U,) x F,), the diagram

(u, n u,)

x F , ~ ( u ,n u,) x F:

is commutative for each pair of indices (a, fi). (16.1 3.3) Conversely, consider a differential manifold B and an open covering (U,) of B; suppose that for each index a we are given a differential manifold F, , and for each pair of indices (a, fi) a mapping $pa:

(Uun

Up) x

Fa

+

(Uu X Up) x Fp

of the form

(b, 0- (b, 0,,(b9 t)), where 0,, is of class C". Suppose also that: (1) for each b E U, n U,, the mapping 0,,(b, * ) : Fa + F, is a diffeomorphism (which implies (1 6.12.2.1) that $, is a diffeomorphism); (2) the "patching condition" (16.13.1 .I)(with the notation used there) is satisfied for each triple of indices (a, fi, 7).

This latter condition, together with the facts that the are homeomorphisms and (U, n U,) x F, is open in U, x F,, allows us to define first of all a topologicalspaceX bypatching together the topological spaces U, x F,

13 DEFINITION OF FIBRATIONS BY MEANS OF CHARTS

89

along the open sets ( V a n U,) x F, by means of the homeomorphisms $,a (12.2). Hence (loc. cit.) we have homeomorphisms cp, : U, x F, + X u , where the X, are open subsets of X which cover X, such that if q,, is the restriction of cp, to (U,n U,) x F a , we have (P,a((Ua

n U,) x F a ) = X a n X,,

$pa

=C P;' 0 Vpa *

Let us first show that X is metrizable, separable, and locally compact. There exists (12.6.1) a denumerable open covering (A,,) of B which is finer than the covering (U,); hence (12.6.2) a denumerable open covering (B,,) of B such that S,,c A,, for all n. For each n, let a(n) be an index such that A,, c U,,,,), and put Y,, = rp,(,,)(B,, x Fa(,,))c X,,,,,_ Since the interior q,,of Y,, in X contains cp,(,,)(B,, x Fa,,,), the open sets y,,cover X. By (12.4.7), it is enough to show that the sets Y,, are closed in X, and for this it is enough (12.2.2) to show that Y,, n X, is closed in X, ,for each index j?.This is evident if X,,,, n X, = 0, and if XUcn,meets X,, then Y, n X, is the image under cpac,,, of the set

,

(Bn

n Up) x F,

9

which is closed in U, x F,. Next we define a mapping n : X + B as follows. Each x E X belongs to some Xu, hence is of the form cp,(b,, t,) with (b,, t,) E U, x Fa; we define n(x) = b,, and from the hypotheses it is immediate that this definition is independent of the choice of the index a. Finally, we transport to X, by means of cp, the structure of (product) differential manifold on U, x Fa; the fact that the I), are diffeomorphisms ensures that the structures induced on X,n X, by those on X, and X, are the same. Hence we have defined a structure of diferentialmanifold on X (16.2.5). It is now clear that I = (X, B, n) is a fibration; it is said to be obtained by patching together the trivialfibrations (U, x Fa, U, ,pr,) by means of the $,. (16.13.4)

Keeping the hypotheses and notation of (16.13.3), consider another open covering (U;) of B which isfiner than the covering (U,). For each , put F; = F a ( y ) .For each index y let a(y) be an index such that U; c U a ( y )and pair of indices (y, d), let $;,:(U;nU;)xF;+(Uj,nU;)

xF;

denote the restriction of I).(a),a(y)to ( U b n U;) x Fb. It is clear that the $,; satisfy the same conditions as the $, and therefore define a fibration I' = (X', B, n') by patching together the trivial fibrations (Ui x F;, U;, prl) by means of the $ i y .This fibration A' is B-isomorphic to A. For if x' E X', then (with the obvious notation) we have x' = cp;(b, t ) with b E U; and t E F; for some index y ; to x' corresponds the point x = cp,(,,(b, t ) of X, and it is immediately seen that this point x does not depend on the choice of y, and that

90

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in this way we have defined a B-morphism g :X --* X. Conversely, for each point x E X we have x = cp,(b, t) with b E U, and t E F,, for some index a; there exists an index y such that b E U;, and to x corresponds the point x' = cp;(b, t). Once again, this point x' E X does not depend on the choices of a and y, and thus we have defined a B-morphism h : X + X'. Finally, it is straightforward to check that g 0 h and h o g are the identity mappings, and so our assertion is proved. (16.13.5) Still keeping the hypotheses and notation of (16.13.3), suppose that we are given, for each index a, a differential manifold FA, and for each pair of indices (u, /3), a mapping

+,;

: (U, n U,) x F: + (U, n Us) x F;

such that the conditions of (16.13.3) are satisfied by these mappings. Let 1' = (X', B, n') be the corresponding fibration. Suppose further that we are given, for each a,a mapping of class C" : c a : U ax F,+F:,

and that, if g, : U, x Fa + U, x F: is the mapping defined by g,(b, t ) = (b, c,(b, t)), the diagrams (1 6.13.2.1) are commutative. Then there exists a unique B-morphism g : X + X' such that g , = cp:-' 0 g cp, for each a.For if x is any point of X, there exists an index a such that x = cp,(b, t ) with b E U, and t E Fa; we put g(x) = cp:(g,(b, t)) and the commutativity of the diagrams (16.13.2.1) guarantees that this point does not depend on the choice of index a.The fact that g is a B-morphism is clear. In particular, if c,(b, * ) is a diffeomorphism for each a and each b E U,, then g is a B-isomorphism. Another particular case in which the preceding method may be applied is the definition of a Cm-sectionof the fibration 1:for such a section may be regarded as a B-morphism of the trivial fibration ( B , B, lB) into 1. 0

14. PRINCIPAL FIBER BUNDLES

We recall that a group G is said to act freely (or without fixed points) on a set E (cf. (12.10)) if for each x E E the stabilizer S, of x consists only of the idenfity element of G: in other words, if for each x E E the canonical mapping SHS * x of G into the orbit G . x is bijective. The group G then acts faithfully on E. (16.14.1) Let X be a diferential manifold and G a Lie group acting differentiably and freely on X; suppose that the orbit manifold X/G exists (16.10.3), and let n : X -P X/G be the canonical submersion. Then:

14 PRINCIPAL FIBER BUNDLES

91

(i) (X, X/G, a ) is a jibration. More precisely, each point of X/G has an open neighborhood U for which there exists a C"-mapping Q : U + X such that n(a(u)) = u for all u E U and such that the mapping (u, s ) w s a(u) is a d f e o morphism of U x G onto a-'(U). (ii) Let R c X x X be the set ofpairs (x,y ) such that xand y belong to the same orbit. For each (x, y ) E R,let ~ ( xy ), be the unique element of G such that y = z(x, y ) * x. Then z is a submersion of the submanifold R (16.10.3) into G.

-

(i) Since n is a submersion, it follows from (16.8.3) that every point of X/G admits an open neighborhood U for which there exists a mapping Q : U + X of class C" such that, for each u E U , we have n(a(u))= u and Tu(u)(~(U)) is a supplement of Tu(u)(a-l(u))in Tu(u)(X).Since by hypothesis the mapping cp : U x G + a - l ( U ) defined by cp(u, s) = s * Q(U) is bijective, it is enough to show that rp is a submersion (16.8.8(iv)). This is a consequence of the following more general result: (16.14.1.1) Let X be a diferential manifold and G a Lie group which acts differentiably on X such that the orbit manifold X/G exists. Let 71: X + X/G be the canonical submersion, and suppose that there exists a C"-mapping Q : X/G + X such that n 0 Q = IX,G. Then Q is an immersion, and the mapping cp : (X/G) x G + X defined by cp(u, s) = s * Q ( U ) is a surjective submersion. The fact that

Q

is an immersion follows from the relation

TU(Ud71) Tu(Q) = IT,(X,G). Next, we shall show that cp is a submersion at a point of the form (uo , e). Put xo = o(u0), so that n-'(u0) is the orbit G . x o . By virtue of (16.10.7), the canonical mapping G -P G . xo is a submersion of G onto the submanifold n-'(u0) of X, and we can apply (16.6.6) and (16.8.8). If now ( u o ,so) is any point of (X/G) x G, we remark that cp is the composition of the three mappings X H S O . x, (u, t)l+ t . a@), (u, S ) H ( U , SOIS), the first of which is a diffeomorphism of X onto itself (16.10), the third a diffeomorphismof (X/G) x G onto itself, and the second a submersion at the point ( u o , e). From this it follows that cp is a submersion at the point ( u o , so). O

(ii) Since the question is local with respect to B = X/G, we may assume that there exists a C"-section Q : B + X and that cp : (b, S ) H S * o(b) is a diffeomorphism of B x G onto X. Then the mapping p : xHpr2(cp-'(x)) is of class C", and hence the mapping 2, which is the restriction to R of ( x , Y)-P(Y)P(x)-'9

92

XVI

DIFFERENTIAL MANIFOLDS

is of class C". Moreover, for each x E X, the restriction of z to

{ x } x (G . X ) c R is a diffeomorphism of this submanifold onto G (16.10.7). Hence z is a submersion of R into G.

Examples Let G be a Lie group and H a Lie subgroup of G. It is clear that H acts freely on G (on the right) by the action (s, X ) H X S ; hence it follows from (16.10.6) and (16.14.1) that (G, G/H, n), where n : G + G/H is the canonical mapping, is a fibration. As another example, we have seen (16.1 1.I1) that the group G consisting of the identity mapping and the symmetry x H -x acts freely on S, and has the projective space P,(R) as orbit manifold; since S, is connected and G discrete, the fibration so defined is not triuializable. Again, with the notation of (1 6.11 .lo), the group GL(p, R) acts freely on the right on the space L,,p , and hence defines a fibration of L,, whose base is the Grassmannian Gn,p . It should be remarked that it can happen that a Lie group acts freely on a manifold X but that the orbit space X/G is not a manifold, even if G is discrete (cf. (16.10.3.4)). (16.14.2)

When the conditions of (16.14.1) are fulfilled, the manifold X endowed with the action of G is said to be a differentialprincipalfiberbundle (or simply a principal bundle) with structure group G ; the manifold B = X/G is the base of the bundle, and the fibers are the orbits of the points of x , and are diffeomorphic to G. Usually we shall regard the structure group of a principal bundle as acting on the right. The Riemann surface of the logarithm (16.12.4) is a principal bundle, with base C* and structure group Z. (16.14.3) Let X, X' be two principal bundles, B, B' their bases, n, n' their projections, and G, G their structure groups. A morphism of X into X' is by definition a pair (u, p), where u : X + X' is a C"-mapping and p : G -+ G' is a Lie group homomorphism, such that

u(x . s)

(1 6.14.3.1)

= u(x) * p(s)

for all s E G and x E X. The image under u of an orbit x * G is therefore contained in the orbit u(x) * G'; in other words, there exists a mapping u : B --* B' such that A' u = u n, and it follows from (16.10.4) that u is of class C". The mapping u is said to be associated with the morphism (u, p ) ; it is clear that (u, u ) is a morphism of fibrations (16.12). When p is an isomorphism of 0

0

93

14 PRINCIPAL FIBER BUNDLES

G onto G , it follows from (16.14.3) that the restriction of u to an orbit x . G is a diffeomorphismof x . G onto u(x) G'. If moreover v is a diffeomorphism of B onto B', then (u, u) is an isomorphism of fibrations (16.12.2). In these conditions, (u, p ) is said to be an isomorphism of the principal bundle X onto the principal bundle X . When G = G and p = lG,we shall say simply that u is a morphism of X into X'.

-

Example (16.14.4) Given a differential manifold B and a Lie group G, we define a right action of G on B x G by the rule

(b, t ) * s = (6, ts).

Since the orbits of this action are the sets pr;'(t) for t E G, and since pr, is a submersion, if follows that the orbit manifold exists and may be identified with B (16.10.3). Moreover, it is clear that G acts freely on B x G ; hence, with the above action, B x G is a principal bundle, called a trivial principal bundle. A principal bundle X with structure group G is said to be trivializable if it is isomorphic to a principal bundle of the form B x G. An isomorphism of X onto B x G is called a triuialization of X. (16.14.5) A (differentiable) principal bundle is trivializable if and only if it admits a COD-section.In particular, a principal bundle whose structure group is diiffeomorphic to RN is trivializable (1 6.12.1 1).

The condition is clearly necessary. Conversely, if a principal bundle X with structure group G and base B = X/G admits a Cm-section : B X, then it follows from (16.14.1.1) that the mapping (b, s ) H Q ( ~.)s is a bijective submersion, hence a diffeomorphism (16.8.8(iv)) and consequently an isomorphism of the principal bundle B x G onto X. (16.14.6) Let X be a principal bundle with structure group G, base B = X/G andprojection n :X + B. Let B' be a differential manifold and let f:B' + B be a C"-mapping. The group G acts differentiably and freely on the manifold X' = B' x X (16.12.8) by the rule (b', x ) * s = (b', x * s). With respect to this action, X' is a principal bundle with structure group G, and thejbration of X' may be identijied with the inverse image underfof the3bration 1 = ( X , B, n). Furthermore, if Y' is a principal bundle with structure group G and base B', and if u : Y' + X is a morphism for which f is the associated mapping, then there exists a unique B'-isomorphism w : Y' + B' x X such that u =f' w, where f ' :B' x X + X is the restriction of pr, . 0

94

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DIFFERENTIAL MANIFOLDS

The first assertion is obvious; with the notation of (16.12.8) the orbits of G in X are the fibers n'-'(b') of the fibration A' = (B' x X, B', x') =f*(3,). Since x' is a submersion, the orbit manifold X / G exists and the corresponding fibration of X may be identified with 2,by virtue of (16.10.3). The last assertion follows from (16.12.8). The principal bundle X' defined in (16.14.6) is called the inverse image of X by$ In particular, if B' is a submanifold of B and i f j : B' B is the canonical injection, then the inverse image of X by j may be identified with the submanifold x-'(B') of X, the action of G on this submanifold being the restriction of the action of G on X. This principal bundle is also called the bundle induced by X over B'. In this terminology, (16.14.1(i)) states that each point of B admits an open neighborhood U over which the induced principal bundle is trivializable (16.14.4). (16.14.7) Let X be a principal bundle with structure group G (acting on the right); let B = G\X be the base and n :X -,B the projection. Also let F be a diferential manifold on which G acts diferentiably (on the left). Then G acts diferentiably andfreely on the right on the product X x F by the rule

( x ,y ) * s = (x * s, s-l * y). For this action: (i) The orbit-manifold G\(X x F ) exists. We denote it by X x F , and the projection X x F + X x F by (x,y ) x * ~y. (ii) For each orbit z E G\(X x F), let xp(z) be the element of B which is equal to n(x)for all (x, y) e z. Then ( X x F , B, nF) is ajibration in which all the$bers are difeomorphic to F . More precisely, ifU is an open set in B such that n-'(U) is triviaIizable and ifa : U -+ n-'(U) is a Cm-section of n-'(U), then the mapping (b,y ) ~ a ( b* )y is a U-isomorphism of U x F onto n,'(U) (which is therefore trivializable). (i) Let R be the set of points ( x , x', y , y') E X x X x F x F (identified with the product manifold (X x F) x (X x F)) such that ( x ,y ) and (x', y') belong to the same orbit. With the notation of (16.14.1), R is identified with the set of points (r, y , o(r) * y ) E R x F x F, that is to say with the graph of the mapping (r,y ) H r ( r )* y of R x F into F. By (16.14.1(ii)) and (16.8.13), this is a closed submanifold of R x F x F, hence also of X x X x F x F. By virtue of (16.10.3), this establishes (i). (ii) It is sufficientto prove the second assertion, for which we may assume that U = B and that X is trivial. Then it follows from (16.10.4) that zFis a surjective mapping of class C". Next, for each x E X, put

s(x) = o(x, o(x(x))) E G

14 PRINCIPAL FIBER BUNDLES

95

(withthenotation of (16.14.1)), so that o(n(x)) = x * s(x). Iff: X x F + B x F is the mapping (x, y) H (n(x), s(x)-' * y), we havef(x . t, t-' y ) = f ( x , y ) for all t E G, because s(x * t ) = t - ' s ( x ) by definition. Since f is of class C" (16.14.1(ii)) there exists by virtue of (16.10.4) a mapping g : X x F -,B x F of class C" such that f ( x , y) = g(x .y ) , and it is immediately verified that g is the inverse of the mapping (b, y) Ho(b)* y . When X = B x G is trivial, so that we may take a(b) = b . e and identify X x F with B x F by means of g, we have (b, s) y = (b, s y ) . (16.14.7.1) With the same notation as above, every section cp of X x F over U may be uniquely expressed in the form cp : b H a ( b ) . $(b), where $ is a mapping of U into F. The section cp is of class C' (r an integer or + m) if and only if I) is of class C'. Since a is a diffeomorphism of U onto a submanifold a(U) of X, the inverse of c being the restriction of IC, we may also write $(b) = cP(o(b)),where 0 = $ 0 (n1 a(U)); the mapping @ is of class C' if and only if $ is of class C'. Moreover, by taking U to be sufficiently small, we may suppose that (D is defined on a neighborhood of a(U) in X, and is of class C' in this neighborhood if cp is of class C' (16.4.3). (16.14.7.2) For x E X, y E F, and t E G we have ( x * t) * y = x * (t .y). The relation x y = x * y' signifies that x = x * t and y' = t-' .y for some t E G ; hence y' = y , so that y Hx .y is a dzfeomorphism of F onto the fiber IC;'(K(X)). It should be noted carefully that the group G does not act canonically on a fiber ICE'@) of X x F: we can make G act on this fiber by choosing a point xo in z-'(b) and putting t (xo y) = xo (t . y); but this action depends in general on the choice of x,; for if xh = xo . t o , we have x; .y = xo * (to * y), so that on replacing xo by x; the new action of G on n;'(b) is 1

(4 xo * Y ) H X o

*

((to

- A,

which is not the same as the previous action unless the commutator subgroup of G acts trivially on F. This condition will be satisfied in particular if G is commutative. As in (16.10), putting m(x, y ) = x * y , we denote the tangent linear mappings T , b ( . r)) and T,(m(x, . )) by ?

(16.14.7.3)

h,Hh;y,

kYt-+x* k,,

respectively. Then we have (16.14.7.4)

h,

*

( t * y ) = (h,

*

t) * y ,

x .( t

. k,)

= (X

. t)

*

k,

96

XVI

DIFFERENTIAL MANIFOLDS

-

for all t E G , and the mapping k y H x k, is bijective. It follows (16.6.6) that

which implies that m is a submersion. The space X x F is called the bundle of fiber-type F associated with X and the action of G on F. This notion will be especially useful in Chapter XX. At this point, we shall make use of it to prove the following proposition: (16.14.8) Let X be a principal bundle with structure group G, and let H be a Lie subgroup of G. Then H acts on X (on the right) by restricting the action of G. The orbit-manifold H\X exists, so that X is a principal bundle with base H\X and group H (16.14.1). Also if x : H\X + G\X is the mapping which associates with each H-orbit the unique G-orbit containing it, then (H\X, G\X, n ) is a fibration whose fibers are difeomorphic to the homogeneous space G/H.

If R c X x X (resp. R c X x X)is the set of pairs ( x , y ) which belong to the same G-orbit (resp. the same H-orbit), then in the notation of (16.14.1) we have R = z-'(H), which shows that R is a closed submanifold, because T is a submersion ((16.14.1) and (16.8.12)). Next we remark that G acts differentiably on the left on G/H, so that we can define the associated bundle X x (G/H) over G\X. Let no :

xx

(G/H) -+ G\X

be the projection. We shall define a diffeomorphism

# : xx~(G/H)+H\x such that the diagram

(1 6.14.8.1)

G\X is commutative; this will prove the proposition. Let cp : G + G/H and p : X + H\X be the canonical projections. Let f : X x G + H\X be the composite mapping (x,s ) ~ p (.xs). For each t E H, we have&, st) = f ( x , s), so that we may writef ( x , s) = g(x, cp(s)), where g : X x (G/H) --+ H\X is a mapping of class C" (16.10.4). Further, for s' E G, we have

g(x . s', s'-' . cp(s))

=f ( x . s', s'-'s)

=f ( x , s) = g(x, cp(s))

14 PRINCIPAL FIBER BUNDLES

97

so that we may write g(x, q(s)) = u(x * q(s)), where u :X x (G/H) + H\X is a C"-mapping (16.10.4). Next, for each x E X, put f'(x) = x q(e), which defines a Cm-mappingf' : X --* X x (G/H). For each t E H, we have

-

f'(x

*

- -

t ) = ( x t ) q(e) = x (t p(e)) = x * q(e) =f'(x)

because t * q(e) = p ( t ) = p(e) since t E H. Hence we may writef'(x) = u'(p(x)) with u' a mapping of class C" (16.10.4). It remains to verify that u and u' are inverses of each other and that the diagram (16.14.8.1) is commutative, which is straightforward. In particular, and changing the notation : (1 6.14.9) Let G be a Lie group and H, K be two Lie subgroups of G such that K c H. Let n : G/K + G/H be the mapping which associates with each left coset of K the left coset of H which contains it. Then (G/K, G/H, n) is a Jibration withjbers diffeomorphic to the homogeneous space H/K. If K is a normal subgroup of H, then G/K is a principal bundle over G/H with structure group H/K.

The last assertion follows from the fact that H/K acts freely on G/K on the right, because xKt = xtK for all x E G and t E H. Examples (16.14.10) It follows in particular from (16.14.9) and from (16.11.4) and (16.11.6) that for p = 2 , . . . , n the Stiefel manifold S,,,(R) (resp. S,,,(C), resp. S,,,(H)) isjiberedover S,,,p-l(R) (resp. S,,,p-l(C), resp. S,,,,-,(H)) with fibers diffeomorphic to the sphere Sn-, (resp. S 2 ( n - p ) + lresp. , S4(n-p)+3). Again, by virtue of (1 6.11.9), S,,,,(R) (resp. S,,,,(C), resp. S,,,,(H)) is a principal bundle over the Grassmannian G,,,,(R) (resp. G,,,,(C), resp. G,,,,(H)) with structure group O(p,R) (resp. U(p, C), resp. U ( p , H)). In particular, the sphere S,, (resp. S2n+l,resp. S4n+3)is a principal bundle over the projective space P,,(R) (resp. P,,(C), resp. P,,(H)) with structure group {- 1, l} (resp. U(1, C), which is isomorphic to the multiplicative group U of complex numbers of absolute value I, hence also isomorphic to T, resp. U(1, H), which is

+

isomorphic to the multiplicative group of quaternions of norm 1).

More particularly, if we take n = 1 (having regard to (16.11.12)) we obtain a fibration of S1over S, with fiber-type { - 1, l}; a fibration of S, over S, with fiber-type S,; and a fibration of S7 over S, with fiber-type S, . Since S, is connected, it is clear that the first of these three fibrations is not trivializable, and it can be shown that the same is true of the other two (" Hopf fibrations").

98

XVI

DIFFERENTIAL MANIFOLDS

+

If F is any differential manifold on which the group G = { - 1, 1) acts, we obtain from the first of these three principal bundles an associated bundle with fiber-type F and base U = S1. Taking F = R, the action of - 1 on R being fH -t, we obtain the orbit-manifold (U x R)/G, where - 1 acts by (z, t ) ~ ( - z ,- t ) . This manifold is called the Mobius strip. Taking F = U, with - 1 acting by ZH -2, we obtain the orbit-manifold (U x U)/G, where - I acts by (z, z')H(-z, -2'). This manifold is called the twisted torus. Finally, taking F = U again, with - 1 now acting by complex conjugation ZH Z, we obtain the orbit-manifold (U x U)/G, with - 1 acting by (2,2') H (- 2, 2').

This manifold is called the Klein bottle.

PROBLEMS

1. Let X be a principal bundle with structure group G , base B = X/G, and projection n, and let (U.) be an open covering of B such that for each index a there exists a section a . of X over U. for which ya : (b, s) Hurn@) . s is a diffeomorphism of U. x G onto n-'(UJ. For each pair of indices (a, let cp,. denote the restrictionof cpa to U. n u@ , and put !IJ,= = c p ~ l c p ~ which ~, is a transition diffeomorphism of the form

m,

0

where e p , : U. n u@ G is a Cm-mapping. Further, for each triple of indices (a,8 and each point b E U, n Up n U, we have the " cocycle condition " --f

,~)

Conversely, let B be a differential manifold and (U.) an open covering of B; and suppose that we are given a C"-mapping : U, n UB+G for each pair of indices a,p, these mappings satisfying the cocycle condition (2). Show that there exists a principal bundle X with structure group G and base B, and for each index a a section a, of X over U. such that the transition diffeomorphisms are of the form (1). If for the same covering (U,) of B we are given another family of mappings & :U, n u@ + G satisfying the condition (2), and hence defining a principal bundle X' over B with structure group G, show that for X and X to be isomorphic it is necessary and sufficient that there should exist for each index a a C"-mapping pa : U., + G such that, for each pair of indices (a,B>

e,,

for all b E U. n u@

14 PRINCIPAL FIBER BUNDLES

99

Describe the relations between two families (8~8) and (8$) defining the same principal bundle, corresponding, respectively, to an open covering (U,) and a finer open covering (Ul). When G is commutative, given two principal bundles X, X over the same base B and with G as structure group, we can define (up to isomorphism) their composition X . X' as follows: for a given open covering (U.) of B, suppose that X (resp. X ) is satisfying the cocycle condition (2); then the defined by the family (88.d (resp. family (8,.8;,) also satisfies (2) and hence defines a principal bundle with structure group G, denoted by X . X'. Verify that up to isomorphism this bundle is independent of the choice of families (OD,) and (&) defining X and X . In this way the set of isomorphism classes of principal bundles with base B and structure group G is endowed with a commutative group structure. 2. Let X be a connected complex-analyticmanifold. Show that the ring O(X) of (complexvalued) holomorphic functions on Xis an integral domain (Section 16.3, Problem 3(a)). Let Ro(X) denote the field of fractions of O(X). If u,v E O w ) and v # 0, the function x Hu(x)/v(x) is defined and holomorphic on a dense open subset of X. If u/v = u l / q in the field Ro(X) (i.e., if the holomorphic function uvl - ulv is identically zero), then

the functions x Hu(x)/v(x) and x H ul(x)/vl(x) are defined and equal on a dense open subset of X. Hence, for eachfE Ro(X), there is a largest dense open set 6 ( f ) in X with the property that for each point xo E 6 ( f ) there exist two elements u, v in O(X) such that u/v = f and v(xo) # 0, so that u(x)/v(x) is defined and holomorphic in a neighborhood of xo . Putf(xo) = u(xo)/v(xo);then the complex numberf(xo) depends only on f (and xo). Hence we have defined a holomorphic mappingf: 6 ( f ) + C, and the mappingf-fis bijective. Usually therefore we shall identify fandA and say that f i s an elementary meromorphic function on X (by abuse of language), whose domain of definition is 6 ( f ) . I f f an d g are two elements of Ro(X), thenf+ g,fg, and l/f(iff# 0) are defined as elements of Ro(X); but all that can be said about their domains of definition is that S ( f + g) and &fg) contain 6 ( f ) n 6 ( g ) , and in general neither of the sets 6 0 , 6(l/f) is contained in the other. A meromorphic function on X is by definition a function f which is defined and holomorphic on a dense open subset U of X, such that for each x E X there exists a connected open neighborhood V, of x and an elementary meromorphic function fx E ROWs)such that 6(f,) = U n V, and such that fx agrees with f on U n V,; then there exists no holomorphic function on an open set U strictly containing U, which extends f. Show that the set R(X) of meromorphic functions on X Can be endowed with a field structure which induces the field structure of each Ro(V,). If X is compact and connected, then O(X) = Ro(X) = C (Section 16.3, Problem 3(b)). If X = P,(C) (which is compact and connected), show that, for each pair P, Q of nonzero homogeneous polynomials of the same degree on Cn*', there exists a meromorphic function f on X such that, if m : C"+'- {0}+ P.(C) is the canonical mapping, we have P(z)/Q(z) =f(m(z)) at all points z # 0 in C"+'such that m(z) E 6 0 . When n = 1, we obtain all meromorphic functions on P,(C) in this way (use Liouville's theorem (9.11 .I)).

3. Let X be. a connected complex-analytic manifold. Apredivisor on X is a pair consisting of a covering (U,) of X by connected open sets and a family (f.), where f . is meromorphic on U. (Problem 2) and not identically zero, such that for each pair of indices a,p, there exists a holomorphic function go. :U. n U b + C* such that

fa(x)= g8a(x)h(x)

100

XVI DIFFERENTIAL MANIFOLDS

at all points x E UQn U, at which f, and f p are defined. Two predivisors (0, (f.)) and ((Ui), (fi)) are said to be. equivalent if, for each x f X, there exists an open neighborhood V, of x contained in some U. and in some U; ,and a holomorphic function hz :V, +C*, such that f . ( y ) = h,(y)f;(y) at all points y E V, at which f, and f i are defined. A divisor on X is an equivalence class of predivisors. If D, D' are two divisors, then there exist two predivisors belonging to D and D , respectively, and corresponding to the same open covering (UJ. If ((Uo), cfo)) and ((Urn), (f:)) are two such predivisors, then we denote by D D' the divisor containing Show that D D' does not depend on the choice of the predivisor ((Ud, predivisors in D and D'. The mapping (D, D') HD D' defines a commutative group structure on the set Div(X) of divisors on X. The neutral element of this group (denoted by 0) is the divisor containing the predivisor consisting of X and the constant function 1. A principal divisor is a divisor containing a predivisor of the form (X, f), where f is a meromorphic function on X, not identically zero. The divisor containing this predivisor is called the divisor off and is denoted by Div(f). Two meromorphic functionsJ g. neither of which is identically zero,have the same divisor if and only if there exists a holomorphic function u on X which does not vanish at anypoint of X, such that f(x) = u(x)g(x) at all x E X, wherefand g are both defined. The principal divisors form a subgroup &c(X) of Diva, isomorphic to R*(X)/O*(X), where R*(X) is the multiplicative group of the field R(X) and O*(X) is the group of invertible elements of the ring 0 0 .

uQfL)).

+

+

+

4. We retain the hypotheses and notation of Problem 3. If (WE),(f.))is a predivisor on X, the functions gp. define (Problem 1) a principal bundle over X with structure group C*, and equivalent predivisors give rise in this way to isomorphic principal bundles, so that to each divisor D on X there corresponds, up to isomorphism, a principal bundle P@). The bundle P(D) is trivializable (as a holomorphic fiber bundle) if and only if D is principal. Two bundles P(D) and P(D') are X-isomorphic if and only if D D = Div(f), wherefis a meromorphic function on X. In this way we obtain an isomorphism of the quotient group Div(X)/Princ(X) onto a subgroup of the multiplicative group of isomorphism classes of principal bundles over X with structure group C* (Problem 1).

-

5.

L e t r b e thecanonicalmappingCn+l-{O}-+P.(C),andforj=O, 1, ...,nle tU ,be the image in P.(C) of the open set consisting of the points z = (zo, zl, .. ,z") f C"+' such that z' # 0. The U, are connected open sets which cover P.(C). For each j , let f i be the holomorphic function on U, whose value at r ( z ) E U, is zo/z'. Then ((U,), V;)) is a predivisor, and the corresponding divisor D1or D,(C) is called the fundamental divisor on P,(C). Show that when n 2 1, this is not a principal divisor (Section 16.3, Problem 3(b)) and that its class in Div(X)/hinc(X) generates a subgroup isomorphic to Z. It follows that there are infinitely many nonisomorphic holomorphic principal bundles over P.(C) with structure group C*. Show that the principal bundle P(D,(C)) is that defined by the action of the multiplicative group C* = GL(1, C) on the space L.+', '(C) = C"+' - {O}, (16.11.10). Consider the analogs for real-analytic manifolds of the definitions and results of hoblems 1-5. In analogous notation, show that the fundamental divisor D,@)

.

on P.(R) is such that 2D,@) is principal. (Consider the function R"+' - {Oh)

14

PRINCIPAL FIBER BUNDLES

101

6. Let B be a differential manifold, G an at most denumerable discrete group, and U, V two open subsets of B. Describe all the isomorphism classes of principal bundles over B with structure group G, such that the induced principal bundles over U and V are trivializable (cf. Problem 1). In particular, if U n V has exactly two connected components, then the isomorphism classes in question are in bijective correspondence with the conjugacy classes in G. 7. Define the product of two principal bundles X, X' with structure groups G, G' and bases B, B', respectively; also define the fiber product of two principal bundles X, X over the same base, with structure groups G, G', respectively. Show that if X is a principal bundle over B whose structure group G is the product of two subgroups G', G", then X is canonically isomorphic to the fiber product over B of two principal bundles X',X" over B with G',G" as respective structure groups; and that X is trivializable if and only if X' and X" are trivializable. 8.

Let X be a principal bundle with base B, structure group G, and projection T, and let E = X X F be a fiber bundle with fiber-type F and projection wF, associated with X. For each C"-sectionfof E over B, there exists a unique mapping rp, : X + F such that x . p,(x) = f ( ~ ( x ) )this ; mapping is of class C" and satisfies the relation

p,(x . s) = s-' . p,(x) for all x E X and all s E G. Show thatf Hp, is a bijection of the set of C"-sections of E over B onto the set of C"-mappings p : X + F such that p(x . s) = s-I . p(x) for all x E X and s E G. In particular, if there exists yo E F such that s . yo = yo for all s E G, then the bundle E admits a section over B; if G acts trivially on F, the bundle E is trivializable. 9. Let X, X' be two principal bundles, with bases B, B , structure groups G, G', and projections w , T' respectively; let E = X x F, E = X' x c' F' be the fiber bundles with fiber-types F, F associated with X, X', respectively, and let T F , wk, be their projections. Let (u,p) be a morphism of X into X (16.14.3) and let u : B + B' be the mapping associated with u. Show that for each mapping f: F + F such that f(s . y ) = p(s) .f(y) for all y E F and s E G, there exists a unique mapping w, :E + E such that the pair (u, w,) is a morphism of the fibration (E, B, wF)into (E, B', T;.), and that w,(x . y ) = U ( X ) . f ( y ) for all x E X and y E F. Such a morphism of fibrations is called a (u,p)-morphism. If G = G', X = X , and if u = lx ,p = l o , then w, is said to be an X-morphism.

10. Let X be a principal bundle with base B and structure group G , and let E = X x F be a fiber bundle associated with X, with fiber-type F. Let u : B' + B be a Cm-mapping, and let X = v*(X) = B x X be the inverse image of X by u. Show that there exists a unique mapping f:E = X' x F + u*(E) = B' x E such that f((b', x ) . y ) = (b', x * y ) forb' E B', x E X, and y E F, and that /is a B'-isomorphism of fiber bundles. What factorization property analogous to (16.1 2.8(iii)) can be stated in this context? = (E, B, T ) be a fibration in which B is a connected differential manifold. In order that should be B-isomorphic to a fibration (X X F, B, wF) associated with a principal bundle X, it is necessary and sufficient that there should exist: (1) a Lie group G acting differentiably and faithfully on a differential manifold F diffeomorphic to

11. Let

102

XVI

DIFFERENTIAL MANIFOLDS

the fibers of E; (2) an open covering (U,) of B and for each a a diffeomorphism vm:U,, x F 'IT-~(U=) such that P ( ~ ~ (yX) ) ,= x for all (x, y) E U, x F;(3) for each pair of indices (a, a C"-mapping g,= : U. n UB--f G such that if rpaais the restriction of 'p. to (U. n U,) x F, then the transition diffeomorphism a/~,* = cpG' vPD is of the form (x, y) H(x, g,.(x) . y) (cf. Problem 1). 0

12. Let X, Y be pure differential manifolds, of dimensions p and q, respectively. Let R'(X) (resp. R:(X)) denote the set of invertible jets of order r from X to Rp (resp. invertible jets of order r from X to RP,with source x ) (Section 16.9, Problem 1). We can define on R'(X) a structure of differential manifold as in Section 16.12, Problem 6; with respect to this structure, the group G'(p) (Section 16.9, Problem 1) acts differen-

tiably on the right on R'(X) and defines on R'(X) a structure of principal bundle over X. Show that the fibration (J'(X, Y), X x Y, (n,d))(Section 16.12, Problem 6) is isomorphic to a fibration associated with the principal bundle R'(X) x R'(Y), with fiber-type L;, q ; likewise that the fibration (J'(X, Y ) ,X, 'IT) (resp. (J'(X, Y). Y, 'IT')) is isomorphic to a fibration associated with the principal bundle Rr(X) (resp. R'(Y)), with fiber-type Ji)(Rp, Y) (resp. J'(X, Rq)o).

13. Show that the twisted torus (16.14.10) considered as a fiber bundle with base S , and fiber-type S, , i s trivializable.

14. With the hypotheses of (16.14.8), suppose in addition that H is a normal subgroup of G. Show that the quotient group G/H acts differentiably and freely on the right on the manifold H\X, and that the orbit-manifold may be canonically identified with G\X, so that H\X is a principal bundle with base G\X and group G/H.

15. Let (X, B, n) be a principal bundle with structure group G, and let H be a Lie group acting differentiably on the right on X. Suppose that (x .s ) . t = (x . t ) . s for all x E X, s E G, and t E H;this implies in particular that for each b E B the set w-*(b). t is a fiber n-'(b') for some b' E B, so that if we write b' = b . t, then H acts differentiably on B, and equivariantly (16.10.10) on X and B. Suppose further that H actsfreely on B and that the orbit-manifold H\B exists, so that B is a principal bundle over H\B with structure group H (16.14.1).

(a) Show that H acts freely on X, that the orbit manifold H\X exists, and that if 'IT' : H\X H\B is the unique mapping which makes the diagram --f

X

& H\X

commutative, then (H\X, H\B, n') is a principal bundle with structure group G. (Reduce to the case where (B, H\B, q) is trivial.) (b) Let F be a differential manifold on which G acts differentiably on the left. Show that there exists a unique differentiable right action of H on X x G F such that (x . y) . t = (x . t ) . y for t E H, x E X, and y E F. Furthermore, show that the orbit manifold H\(X x F) exists and is canonically diffeomorphic to (H\X) x F. (c) For each pair (x, x') of elements of X such that ~ ( x=) 'IT(x'),let T ( X . x') denote

14 PRINCIPAL FIBER BUNDLES

103

the element of G such that x = x’ . T(X, x’). Suppose that there exists a Cm-section u of X over B and a Cm-mapping p : H + G such that u(b . t) . t-’ = u(b) . p ( t ) for all b E B and all t E H. (a) Show that /?is a homomorphism of H into G. (#?) For each x E X, put f ( x ) (n(x), T(u(T(x)), x)), so that f is a Cm-mappingof X into B x G. Show that the unique mapping g which makes the diagram

X

-

H\X

commutative (where H acts on the left on G by the rule (I, s) Hs/?(t -I)) is an isomorphism of principal bundles with base H\B and structure group G. 16. Let (X, B, v ) and (X’, B , n‘) be two principal bundles with structure groups G, G’, re-

spectively. Suppose that G acts differentiably on the left on X and that s.(x’.t’)=(s.x’)*t‘

(which we shall denote by s . x’ . t‘) for all s E G, x’ E X’, and t‘ E G’; then G also acts differentiably on B , and equivariantly on X and B’. Show that the unique mapping p :X x X X x B’, which makes the diagram --f

XXX

-

XXGX

commutative, is such that (X x X , X x B , p ) is a principal bundle with structure group G’, the action of G’ being such that ( x . x’) . f’ = x . (x’ . f ’ ) for all x E X, x’ E X , t’ E G’. (Reduce to the case where X is trivial.) Furthermore, the composite mapping p’:

1 X X d

X x X - X x B P x X ~ B

is such that (X x X , X x B , p’) is a principal bundle with structure group G x G’ (the action of G x G’ on X x X being defined by ( x , x’) . (s, t’) = (x . s, s-’ . x’ . t’)). Let F be a differential manifold on which G’ acts differentiably on the left; then G acts differentiably on the left on the fiber bundle X x G’ F associated with X with fiber-type F’, by the rule s . (x‘ . z’) = (s . x’) . z’. Show that there exists a unique diffeomorphism (called canonical) : (X x X‘) x G’ F’+X x

(X’ x O’ F’)

for which the diagram

XXXXF’

(X

XG

X)

XG’

F‘ 7 x X G (X’

XG’

F)

104

XVI DIFFERENTIAL MANIFOLDS

is commutative, where f and g are the canonical mappings. Hence on the space Y = (X x X') x G' F' we have two structures of fiber bundle. one with base X x B' and fiber-type F', and the other with base B and fiber-type X' x G' F'. 17. Let (X, B, a)be a principal bundle with structure group G, and let p : G +G' be a homomorphism of the Lie group G into a Lie group G . Then G acts differentiably on the left on G by the rule (s, t') H p ( s ) t ' , and we may consider the fiber bundle X x G associated with X by this action. Show that G' acts on the right on X x G' by the rule ( x . s') * t' = x . (s't') and that with respect to this action (X x G', B, a') (where a' is the canonical mapping) is a principal bundle with base B and group G'; this bundle is called the p-extension of X. If X is a principal bundle with base B and structure group G', and if u : X + X is such that (u, p) is a morphism (16.14.3) of X into X , show that there exists a unique isomorphism u : X x G + X such that u = u 0 q, where :X + X X G is the canonical mapping x H X * e' (e' being the

neutral element of G).

18. Let (ek)l S k S Z n be the canonical basis of C2n,and identify R2" with the real vector subspace of C""spanned by the ek; then C2"= R2"@ iR2", that is to say every vector z E Cz" can be written uniquely in the form z = x iy with x, y in RZn.Put x = Wz. Let B(z, w) be the symmetric bilinear form on C2" for which B(e,,, eJ = ,a (Kronecker delta). Let V be a totally isotropic subspace of C2"of maximum (complex) dimension n, relative to the form B. Then the mapping z HWz is an R-linear bijection of V onto R"". Let fv be the inverse bijection, and put j v ( x ) = W(ifv(x)) for x E RZn, so that jvis an R-linear bijection of R"" onto itself.

+

(a) Show that j(b).When B = B' andf= l,, we say that g is a (linear) B-morphism of vector bundles. The set of linear B-morphisms of E into E' is then denoted by Mor(E, E') (cf. (1 6.1 5.8)). An isomorphism (f,g) of E onto E' is a morphism of vector bundles which is an isomorphism for the corresponding fibrations (16.12). For (f,g) to be an isomorphism it is sufficient thatfshould be a diffeomorphism and gb bijective for each point b E B (16.12.2). The local expression of a morphism of E into E' relative to the fibered charts corresponding to charts (U, $, rn) and (U', $', m') on B and B', respectively (16.15.1), is therefore of the form

where F is a C"-mapping of +(U) into $'(U') and X H A ( X )is a C"-mapping of $(U) into the set Hom(R", R"')(resp. Hom(C", C"')),(identified,if we prefer, with the set of matrices of type (n', n) over R (resp. C)). Example (16.15.3) Let B be a differential manifold and F a real (resp. complex) vector space of dimension n, and consider the trivial fibration (B x F, B, pr,) (16.12.3). By transporting the vector space structure of F onto the set

{b} x F = pr;'(b)

108

XVI DIFFERENTIAL MANIFOLDS

by means of the mapping t w ( b , t), we obtain on B x F a structure of a real (resp. complex) vector bundle of rank n corresponding to the trivial fibration, because the condition (VB) is satisfied by taking U = B and cp to be the identity mapping. Such a vector bundle is said to be trivial. A vector bundle E over B is said to be trivializable if there exists a B-isomorphism of E onto a trivial vector bundle; such an isomorphism is called a trivialization of E, and it is precisely the inverse of a framing over B (16.15.1). (16.15.4)

The tangent bundle of a digerential manifold.

Let M be a differential manifold and let T(M) be the union of the (pairwise disjoint) tangent spaces T,(M) as x runs through Mt. Let oM :T(M) -+ M be the mapping which associates with each tangent vector h, E T,(M) the point x E M (the " origin " of h,). We shall show that there exists a unique structure of diyerential manifold on T(M) such that z = (T(M), M, oM) is a jibration (the tangent spaces T,(M) being the fibers) and such that the following condition is satisfied:

(TB) For each chart c = (U, cp, n) on M, the mapping (cf. (16.5.3))

of U x R" onto oi'(U) is a digeomorphism. An equivalent condition is that, for each h E R",the mapping x~(d,cp)-' * h is a Cm-section of T(M) over U. For if this is so, and if for each vector e, of

the canonical basis of R" we put Xi(x) = (d, q)-' the mapping (x, t',

.,., 5") H c n

i= 1

*

ei,and h =

c tiei,then n

i= 1

('Xi@)is a diffeomorphism, by virtue of

(I 6. I2.2), hence (TB) is satisfied. The existence of these sections shows at the same time that the vector space structures of the T,(M) and the fibration z define on T(M) a structure of a real vector bundle; this vector bundle is called the tangent vector bundle, or more briefly the tangent bundle, of the differential manifold M (cf. Section 16.12, Problem 6).

t By definition (16.5), an element of TJM) is an equivalence class of mappings of R into M, hence a subset of the set S(R,M) of all mappings of R into M; consequently T,(M) M)), and the sets {x} x T,(M) are pairwise disjoint in the set is a subset of %3(9(R, M x @(F(R, M)). As a set, T(M) is defined to be their union.

15 VECTOR BUNDLES

109

To establish the existence of the differential manifold T(M), consider an open covering (U,) of M such that for each a there exists a chart (U, , cp, ,nu) on M. Let a, f? be two indices such that U,n U, # @, which implies that nu = n, = n say; let .cp, and cpa, be the restrictions of cp, and cps ,respectively, to U,n U,, and let fs, = qsU cpGi be the transition diffeomorphism. Then (16.5.7) we have 0

Dfs.((P,(x)) = (4cp,3 O (dx),cp. for x E U, n U, , and the mapping $su:

-

(U,n U,) x Rn+(U, n U,) x Rn

defined by

$&,

h) = (X, D f s u ( ' P p ( 4 )

*

h)

is evidently of class C", hence a diffeomorphism by (16.12.2.1). Further, if for each triple of indices (a, f?, y) we denote by f;,, f;,, andf: the restrictions of the mappings f,,,f y B, and j;,to (P,(Uu n up n U J , cp,(uu n u, n UJ,

and

cp,(U, n u,

UY),

respectively, then we have&{ =f,ys o f i , and by (8.2.1) the patching condition (16.1 3.1 .I) is satisfied. This establishesthe existence ofthe fibration z (16.13.3); moreover, it follows from the construction in (16.1 3.3) that this fibration satisfies (TB) and is the unique fibration with this property; for the condition (TB) implies that each mapping (16.15.4.1) is aframing of T(M) over U. This framing 1,9= is said to be associated with the chart c. (16.15.4.2) A section of T(M) over a subset A of M (16.12.9) is called a tangent vector JieId (or simply a vector field) over A. With the preceding (16.5.1) for notation, the mappings XHX,(X) = (dXq)-'* e, = O;?Je,) 1 i 5 n are C"-vector fields on U which form a frame of T(M) over U. These vector fields are called the vector fields associated with the chart c, and the frame they form is the frame ussociuted with c. Every vector field on U is uniquely expressible in the form (16.15.4.3)

X H X(x)

c ai(x) n

=

*

Xi(X),

i= 1

where the u' are n scalar-valued functions on U. For X to be of class C', it is necessary and sufficient that the ai should be of class C'. (16.15.4.4)

With the notation of (16.15.4.1), we see that

(oii(W, (cp x 1m) 0 $,

',2n)

110

XVI DIFFERENTIAL MANIFOLDS

is aJibered chart on T(M) (16.15.1), called the fibered chart associated with the chart c = (U, cp, n) on M. If (U, cp', n) is another chart on M with the same domain of definition, and if u = cp cp'-' : cp'(U) + cp(U) is the transition diffeomorphism, then the transition diffeomorphismfor the associated fibered charts on T(M) is the mapping 0

of cp'(U) x R" onto q(U) x R". If a vector field X on U is given by (16.1 5.4.3), its local expression relative to the fibered chart associated with (U, cp, n) (16.15.1.3) is the mapping (16.15.4.6)

i= 1

of cp(U) into cp(U) x R". (16.15.5) When M is an open subset of R",the inverse of the framing associated with the chart (M, 1, ,n) is a trivialization of T(M), called the cunonical trivialization and given by h x H( x , zx(hx)) (16.5.2). Usually we shall identify T(M) with M x R" by means of this trivialization. A vector field on M is then of the form X H ( x , f(x)), where f is a mapping of M into R". (16.15.6) Let M, N be two differential manifolds and let f:M -+ N be a Then the mapping mapping of class C' (where r is an integer 2 1, or + a). (16.15.6.1)

(which we shall write in the more legible form

is a mapping of class C'-' (with the convention that r - 1 = 00 if r = co) of T(M) into T(N). For if (U, cp, m) and (V, $, n) are charts on M and N , respectively, such thatf(U) c v, and if F is the local expression offrelative to these charts (16.3), then the local expression of T(f) relative to the associated fibered charts (16.15.4) is the mapping (16.15.6.3)

-

(x, h)H(F(x), F'(x) h)

of cp(U) x R" into $(V) x R".Iffis of class C", then (f, T(f)) is a morphism of oector bundles (16.15.2). It is clear that

15 VECTOR BUNDLES

Ill

Hence iffis a subimmersion (resp. an immersion, resp. a submersion), so also is T(f). Also it is clear that T(1M) = lT(M), and that if g : N + P is another C'mapping, we have T(g of) = T(g) T(f). Iff is a diffeomorphism, then so is TCf), and T(f-') = T(f)-'. If M and N are two differential manifolds and pr, ,pr, are the projections of M x N onto M and N, respectively, then the mapping (T(pr,), T(pr,)) is a canonical isomorphism of T(M x N) onto T(M) x T(N) (16.6.2); usually we shall identify these two vector bundles over M x N. As an application of these results, we remark that if a Lie group G acts on a differential manifold X, then the mapping (16.10.1.2) 0

(hs, k,)Hs- k,+ h;x

of T(G) x T(X) into T(X) is of class C". Likewise, if (X, B, n) is a principal bundle with structure group G and if E = X x F is the associated fiber bundle with fiber-type F, then the mapping (16.14.7.2)

of T(X) x T(F) into T(E) is of class C". (16.15.7)

The tangent bundle of a vector bundle.

If 1 = (X, B, n) is a fibration, then (T(X), T(B), T(n))is also a Jibrution. For if U is an open subset of B such that there exists a diffeomorphism rp : U x F + II-'(U) with II o cp = pr, , then it follows that T(q) is a diffeomorphism of T(U) x T(F) onto T(n-'(U)) such that T(n) T(cp) = pr, . We recall (16.12.1) that the tangent vectors h, E T,(X) such that T(a) * h, = 0 in Tn(.JB) are said to be trertical, or along thefiber. Now let E be a vector bundle with base B and projection II.Then T(E) is also a vector bundle with base T(B) and projection T(II). Since the question is local with respect to B, we may assume that E is trivializable, and hence we reduce to the case where E = B x F, with F a vector space of dimension n and B an open set in R".Then (16.15.6) T(E) may be identified with T(B) x T(F), and the canonical trivializations of T(B) and T(F) (16.15.5) therefore finally identify T(E) with (B x R") x (F x R").Since F x R" is canonically endowed with the structure of a vector space of dimension 2n, it follows that T(E) is endowed with a vector bundle structure over T(B); but it has to be shown that this structiire is independent of the trivialization of E from which we started. 0

112

XVI

DIFFERENTIAL MANIFOLDS

Now a transition diffeomorphism from one trivialization to another is of the form

-

$ : (b, Y) I--r (b, &) Y), where A : B + GL(F) c End(F) is a C"-mapping; and then T($) is the diffeomorphism (16.15.7.1)

-

((b, h), (Y, k))H((b, h), (A(W * y, 44 k + (A'(b) * h) * y))

((8.1.3) and (8.9.1)), and our assertion follows from the fact that the mapping

+

(y, k)HA(b) * k (A'@) * h) . y is linear. Hence we have two vector bundle structures on T(E): one with base E and projection oE,the other with base T(B) and projection T(n); moreover we have oE T(n) = n oE. Furthermore, if w = oB T(n) = n 0 oE, then (T(E), B, m) is ajbration over B; but although, for a given trivialization of E, each fiber of this fibration can be endowed with the structure of a vector space of dimension 2n m, yet it is not possible to define in this way a vector bundle structure on the fibration. For the preceding calculation shows that the righthand side of (16.15.7.1) is not linear in (h, y, k), although it is linear in (y, k) and in (h, k), which correspond to the two vector bundle structures on T(E) previously described. In the particular case where E = T(M), the tangent bundle of a differential manifold M, the two vector bundle structures on T(T(M)) both have the same base T(M), but are quite distinct from each other. 0

0

0

+

(16.15.8) Let E be a real (resp. complex) vector bundle with base B and projection n, and let &(B; R) (resp. B(B; C)) be the set of C"-mappings of B into R (resp. into C), which is an R-algebra (resp. a C-algebra). Consider also the set Mor(E, E') of B-morphisms of E into another real (resp. complex) vector bundle E' over B. If u', u" E Mor(E, E'), we define u' U" to be the mapping of E into E' such that (u' + u")b = u; + u; for all b E B, and it follows from (VB) that u' + u" is a B-morphism. Again, for each function f E B(B; R) (resp.fE B(B; C)) and each u E Mor(E, E'), we define a mappingf. u : E + E' by the rule (f. u)b =f(b)ub, and f * u is again a B-morphism. Hence we have defined on the set Mor(E, E') a structure of 8(B; R)-module (resp. &(B; C)module). In particular, the set T(B, E) or T(E) of all Cm-sectionsof E may be considered as the set of B-morphisms of the trivial bundle (B x {0}, B, pr,) into E, and therefore T(E) is an B(B; R)-module (resp. an &(B; C)-module). If there exists a frame over B (in other words, if E is trivializable), then this module is free, and every frame over B is a basis of it.

+

15 VECTOR BUNDLES

113

By applying these remarks t o the vector bundle induced on n-'(U), where

U is open in B, we define a structure of &(U; R)-module (resp. $(U; C)module) on the set T(U, E) of Cm-sectionsofE over U. For each u E Mor(E, E'), the mapping S H U o s of T(U; E) into T(U; E') is &(U;R)-linear (resp. &(U;C)-linear). Remark (16.15.9) In exactly the same way we can define the notion of a real (resp. complex) vector bundle over a real-analytic manijold B; such a bundle is called a real- (resp. complex-) analytic vector bundle over B. Everything goes through as before, with the exception of (16.15.1.2). When the base B is a complexanalytic manifold, thecorresponding notion of vector bundle is of interest only when the fibers are complex vector spaces; such vector bundles are called holomorphic vector bundles. When M is a real- (resp. complex-) analytic manifold, the tangent bundle T(M) is a real-analytic (resp. holornorphic) vector bundle over M. All the developments of Sections 16. I6 and 16. I 7 (with the exception of (I 6.17.3)) extend immediately to real-analytic and holomorphic vector bundles.

PROBLEMS

1. A differential manifold M is said to be paralfelizabfe if the tangent bundle T(M) is trivializable. Show that the differential manifold underlying a Lie group is parallelizable. In particular, the spheres S1 and Snare parallelizable.

2.

Let m be a bilinear mapping of R' x

R" into R" such that

Ilm(r, x) II = l l II ~. Ilx II

(the norms being Euclidean norms).

', we have (m(y, x) I m(y', x)) = (y 1 y') IIx ]I2, and for all x, x' in R" (a) For all y, y' in R we have (m(y, x) I m(y, x')) = Ilyllz(xI x'). For each y f 0, the mapping x H m(y, x) therefore belongs to O(n,R). If (ef)lslsk is the canonical basis of Rk and if we put v(x) = m(ek,x), then the mapping (x, y) H mo(y, x) = m(y, v-'x) has the same property as m, and we have mo(ek,x) = x. In order that there should exist such a bilinear mapping m, ,it is necessary and sufficient that there should exist k - 1 elements u l ~ O ( n , R( )I s i i k - 1 ) such that u 2 = - 1 , u i u , t u , u , = O f o r 1si,j6 n and i # j (in the algebra EndCR")). For each x E S.-l, the k - 1 vectors (x, uf(x))E Tx(Sn-i) (where Tx(Sn-l)is identified with a subspace of T,(R")) are then linearly independent, and so we have k - 1 vector fields on S.- I which are linearly independent a t each point of

114

XVI DIFFERENTIAL MANIFOLDS

(b) We shall assume the existence of an (associative) algebra C, over R, of dimension 2", generated by the unit element 1 and m elements c, (1 i 6 m) such that c f = - 1, CICJ cj CI= 0 for i # j , so that the element 1 and the products el ct2 * * * ctpfor

+

1 6il < it

< ... < ipz m

form a basis of C, . The algebra C, is called the Cliffordalgebra of index m. In order that there should exist a mapping m with the property considered in (a) above, it is necessary and sufficient that there should exist a homomorphism of Cr-, into End(R"). (Observe that if r is the (finite) group generated by the imam u1of the c1 in GL(n, R), then the bilinear form (s . x 1s * y) is positive-definite on R"and invariant under I?.) aar

(c) It can be shown that for 0 5 m following table: m=O

1 2

3

4

5 7 the Clifford algebras C,,, are given by 5

6

the

7

R C H H x H M 2 0 M*(C) Me.@) Ma@)xMa@) and that Cm+8is isomorphic to C, &M16(R). Deduce that if each integer n 2 1 is expressed in the form n' * 2c(n)16d(") with 0 = 1 ;the formula (16.20.8.4) then shows that the inverse image under f of a form of class Cs is of class inf(r - 1, s). (1 6.20.15 )

Vector-valueddcyerentialforms.

Instead of considering in (16.20.1) the vector bundle T(M*) = Hom(T(M), M x R) or AT(M)*

= Hom(hT(M),

M x R),

we may more generally consider the vector bundle Hom(T(M), M x F) or Hom(A T(M), M x F), where F is a real vector space of finite dimension. A section of Horn(AT(M), M x F) over M is called a uector-ualueddlfferential p-form on M, with values in F. If (a,), is a basis of F, such a p-form is uniquely expressible as a =

r

i= 1

cli a,,where

the a, are differential p-forms in

the sense defined earlier, or (as we shall sometimes call them) scalar-valued differential p-forms. For each X E M, a ( x ) takes its values in the space of P

linear mappings of AT,(M) into F (identified with (M x F)*); if h, , . . . , h, are p tangent vectors belonging to TJM), we denote by a(x) . (h, A . A hp) the value of a(x) at the p-vector h, A h, A ..* A h,, so that we have (16.20.15.1) a ( x ) . ( h ,

A *-. A

h,) =

r

C (cli(x), h, A ... A hp)a,.

i= 1

If X , , . . . , X,, are p vector fields on M, we denote by

146

XVI DIFFERENTIAL MANIFOLDS

(1 6.20.15.2)

~.(X,AX,A*.-AX,)

the function on M, with values in F, x H a ( x ) * (Xl(x)

A

X,(x)

A

A

X,(x)).

In particular, iff = C f iai is a C1-mappingof M into F, then df = C (&)ai i

i

is a differential 1-form with values in F, called the diflerential off; it is precisely the mapping XI--, d, f (16.5.7). For example, if E is a finite-dimensional real vector space, the mapping XHZ, (16.5.2) is the differential d(l,) of the identity mapping. Iff: M’ + M is a C”-mapping, we define the inverse image ‘f(a) of a under f by the same condition as in (16.20.8): for each x’ E M’ and each p-vector h; A h; A . * * A h;, where the hi belong to T,,(M’), we put (16.20.15.3)

‘f(a)(x’)

*

(hi A . * * A hi)

= a(f(x’)) . (T,.(f)

*

hi A * *

A T,.(f)

*

hb).

Equivalently, we have (16.20.1 5.4)

~ ( a =)

f

i= 1

‘f(ai)ai

which brings us back to scalar-valued p-forms. There is no “exterior calculus ” for vector-valued differential p-forms analogous to the exterior algebra of scalar-valued differential forms, but we shall be led to consider in Chapter XX operations analogous to the exterior product in certain particular cases. Consider three finite-dimensional real vector spaces F , F”, F, and let B : F’ x F”+ F be a bilinear mapping. If we are given vector-valued differential I-forms a’,a“on M, with values in F and F , respectively, we can associate with them a differential 2-form with values in F, in the following manner: the mapping (16.20.15.5)

(h, , k,)W B ( ~ ’ ( X*) h, , o”(x) * k,) - B(~’(X). k, , o”(x) * h,)

of T,(M) x T,(M) into F is bilinear and alternating, and hence there exists a 2

unique linear mapping of AT,(M) into F, for which the right-hand side of (16.20.15.5) is the value at the bivector h, A k,. We denote this mapping by (a’A a”)(x), and thus we have defined a 2-form a’A a”with values in F. (When there is no risk of confusion we shall write a‘A a“,but it should be remarked that W” A a’in general has no meaning.) If (a& is, is a basis of a basis of F”, and if we put F and (a;)1

20

DIFFERENTIAL FORMS

147

(16.20.15.6)

where the wl and w; are scalar-valued 1-forms, then we have (1 6.20.15.7)

o’A

(ofA w;)B(ai

0’’=

, a:).

i,j

In the particular case in which F’ = F” = E, o’= o”= o,and the bilinear mapping B is alternating, we do not use the notation o A a.The mapping

is already bilinear and alternating, and we denote by B(o, o), the corre2

sponding linear mapping of A T,(M) into F. If a’,o”are 1-forms with values in E. we have

B(o’ + a”,w‘

(16.20.15.9)

and if we put o = (16.20.15.10)

+ a”)= B(o‘, a‘)+ a’A

o”f B(o”, a”),

c o i a i ,where (ai)is a basis of E, then 1 B(o, o) - c (mi oj)B(ai, aj) 2 i

=

i,j

A

(summed over all pairs of indices ( i , j ) ) ,which can also be written in the form (1 6.20.15.11) (1 6.20.16)

B(o, o)=

1(miA oj)B(ai, aj).

i< j

Diflerential forms on complex manifolds.

All the definitions in this section can be transposed to the context of a complex manifold M, by replacing C“-mappings throughout by holomorphic mappings, and real vector bundles by complex vector bundles. The cotangent bundle T(M)* is a holomorphic bundle (16.15.9) of (complex) rank n, if M is a pure manifold of (complex) dimension n. If we denote by MI, the differential manifold of dimension 2n underlying M, then the cotangent bundle T(M,,)* is a real vector bundle of rank 2n over MI,, and its complexification (T(MIR)*)cc) is a complex vector bundle of (complex) rank 2n over MI,. The results of (16.5.13) show that there exists a C-automorphism ‘J of this fiber bundle such that ‘J2 = - I ; the images of the two morphisms p’ = -)(I - i ‘J), p ” = *(I + i‘J) are, respectively, the cotangent bundle T(M)* and its complex conjugate T(M)*, so that (T(M,R)*)(C) = T(W* 8 T(M)*.

148

XVI DIFFERENTIAL MANIFOLDS

To any complex functionfof class C' on MI, we can associate two sections d ' f = p ' 0 df, d"f=p" df of T(M)* and T(M)*, respectively, so that the complex differential form df is equal to d ' f + d"$ The function f is holomorphic if and only if d"f = 0. If (U, cp, n) is a chart on the complex manifold M, where cp = (cp'), s j 6 n , then the dcp' form a frame of T(M)* over U, and the @ a frame of T(M)* over U. 0

PROBLEMS 1. Let A be one of the rings &(M), B(')(M) (where r 2 1) on a differential manifold M. For each x E M, let m, denote the ideal of functions belonging to A which vanish at x.

Consider the mapping f~ d, f of m, into T,(M)*. Show that, if A = &M), the kernel n, of this mapping is equal to m: , but that if A = b(')(M) with r finite, then m', is of infinite codimension in n, . (Observe that in the latter case the product of two elements of m, has a local expression which admits derivatives of order r 1 at the point of the chart corresponding to x.)

+

2.

Let M be a differential manifold. (a) For each real-valued Cm-functionf'on M, let $denote the C"-function on T(M) defined by h,H = 1, the sphere S, is a two+ P,(R) is the canonical projection, then sheeted covering of P,(R). If K :s,,, for each z E P,(R) the two points of n-'(z) are antipodal on S, . If m = 2n - 1 is odd, we shall show that there exists on PznT1(R)a (2n - 1)-form a' such

21 ORIENTABLE MANIFOLDS A N D ORIENTATIONS

157

that ' a ( d )= a, in the notation of (16.21.10). Each point z E P,n-l(R) has a connected open neighborhood U on which are defined two Cm-sections, u, : U + a-'(U) and u, : U + n-'(U), which are diffeomorphisms of U onto two disjoint open subsets U1,U, of a-'(U), whose union is a-'(U); also we have u,(z) = s(ul(z)) for all z E U. It follows immediatelyfrom (1 6.21.10.2) that 'ui(a I U,) = 'uZ(a I U,); if 06 denotes this (2n - 1)-form on U, then it is clear that for each open subset V of PZn-,(R)over which the covering n-'(v) is trivial, the restrictions of a; and a; to U n V are the same. Hence the existence of the (2n - 1)-form d,which is clearly #O at each point. (1 6.21.12)

The projective spaces P,,(R) are not orientable (n 2 1).

With the same notation as in (16.21.11), suppose that there does exist a continuous differential 2n-form p on P,,(R) which is nonzero at every point. Then the same is true of 'a(p) on S2,,and therefore ' ~ ( p= ) f * a, wherefis a continuous real-valued function on S,, which is never zero. However, by definition we must have 's('n(p)) = 'a@), because a = a s; and since by (16.21.10.2) we have 's(o)(x) = -a(x), it follows thatf(-x) = -f(x) for all x E S,, . Since S,, is connected, this contradicts the fact thatf(x) # 0 for all x E s,, . 0

Let X be a pure complex-analytic manifold. Then the dzTerentia1 manifold Xo underlying X is orientable.

(16.21.13)

Let 2I be an atlas of X, and consider two charts (U, cp, n) and (U', cp', n) belonging to 2I, such that U n U' # 0.Let $' = cp'I(un U'),8 = $-I; $ = cpl(U n U'), 8 is a holomorphic mapping of an open set in C" onto an open set in C". For is a C-linear bijective mapping of each z E cp(U n U'), it follows that C" onto C". The proposition will therefore result from the following lemma: (16.21.13.1) r f u : C" + C" is a C-linear mapping and q u o : R2"+ Rz"is the same mapping u considered as an R-linear mapping, then

(1 6.21.I3.2)

det(uo) = 1 det(u) 1 '.

To see this, take a basis (bj)l of u is upper triangular (A. 6.10):

of C"with respect to which the matrix

158

XVI

DIFFERENTIAL MANIFOLDS

If r j = sj + i t j , where sj and ti are real, then the matrix of uo relative to the basis of Rz" formed by the bj and the ibj (1 S j 5 n) is of the form

R1 P12 ..*

P1n

a triangular array of blocks of order 2, in which

The formula (1 6.21 .I3.2) follows directly by calculating the determinant of this matrix (A. 7.4). We remark that there is a canonical orientation on Xo, with the property that for each chart (U, cp, n) of the complex manifold X the corresponding chart (U, cp, 2n) of X, preserves the orientation, where R2"is endowed with the canonical orientation (16.21.4) and C" is identified with R2"via the mapping ([',t2,. . . , H (W[',Y[', . . . , a(?', Yr). It is easily verified that the forms which belong to the canonical orientation of Xo are those which, for each chart (U, cp, n) of X , have a restriction to U which can be written as

r)

f * dcp'

A

d@' A dcpz A d?

A

-

*.

A

dcp" A d p ,

where f(x) > 0 for all x E U. (16.21.14)

The manifold underlying a Lie group G is orientable.

Suppose that dim G = n, and let z : be a nonzero n-covector at the identity element e of G. Then XHY(X)Z: is a C" differential n-form on G (16.20.13) which clearly is everywhere # 0. We remark that a homogeneous space of a Lie group is not necessarily orientable; for example, we have just seen that P,,(R) is not orientable ((16.11.8) and (16.21.12)). (16.21.15) We have seen in (16.21.12) that the nonorientable manifold P,,(R) admits an orientable two-sheeted covering. This is a general fact: (16.21 .I 6) Every pure manifold X of dimension n admits a canonical orientable

two-sheeted covering.

21 ORIENTABLE MANIFOLDS AND ORIENTATIONS

159

n

In the bundle A T(X)* consider the open set Z which is the complement of the zero section. The multiplicative group R* acts differentiably and freely n

on Z, because AT(X)* is a line bundle, and by taking a fibered chart of n

A T(X)* it is immediately seen that Z is a principal bundle over X with struc-

ture group R*.Now apply (16.14.8), taking H to be the subgroup RT of R consisting of the positive real numbers; since R*/R*, is the group of two elements, it follows that X' = Z/RT is a two-sheeted covering of X. To show that X is orientable, we shall construct an atlas of X' satisfying condition (b) of (16.21.1). To do this, we start with an atlas CU of X such that, for each chart (U, cp, n) belonging to CU, the open set U is connected and the inverse image of U in X' is the disjoint union of two open sets U', U" such that the canonical projections p' : U' + U and p" : U"+ U are diffeomorphisms. If n : Z +X, n' : Z +X are the canonical projections, then by hypothesis there is a canonical morphism of fibrations (16.15.4) $ : cp(U) x R* +z-'(U), and z'-'(U') and n'-l(U") are each equal to one of the images under $ of q(U) x R*, and q(U) x (-RT). Let s be the reflection of Rn with respect to the hyperplane t' = 0. If n'-'(U') = $(cp(U) x R:), we take as chart of U' the triplet (U', cp op', n); otherwise we take (U', s 0 cp op', n ) ; and similarly for U". We have now to show that the condition (b) of (16.21.1) is satisfied by the atlas of X so defined. For this we may limit ourselves to considering two charts corresponding to charts (U, cp, n), (U, cp', n) of X having the same domain of definition. Let $' : cp'(U) x R* + n-'(U) be the canonical morphism corresponding to the second chart; if F : cp'(U) 4cp(U) is the transition diffeomorphism, then the composite morphism $' $ - I is given (16.20.9.4) by 0

(A t

) ( F ~( 4 , J(x)-'t),

where J(x) is the Jacobian of F at the point x. Suppose for example that n'-'(U') = +(cp(U) x R:). If J(x) > 0 in cp'(U), then we have also n'-'(U') = $(cp'(U) x RT), and the transition diffeomorphism for the charts (U', cp "P',n), W', cp' "PI,n>

is then F. If on the other hand J(x) < 0 in cp'(U), then we have z'-'(U') = $'(cp'(U) x (-RT)), and the transition diffeomorphism for the charts (U', cp op', n) and (U', s 0 cp' op', n) is then F 0 s. In both cases, the transition diffeomorphism has Jacobian >O. The argument is similar when n'-'(U) = @(P(U) x (-RT)), and the proof is complete. We remark that if X is orientable, then the covering X' is trivializable, because Z admits a section over X and is therefore trivializable.

XVI

160

DIFFERENTIAL MANIFOLDS

PROBLEMS

1. Let G be a connected Lie group, H a closed subgroup of G. Suppose that, at the point xo E G/H which is the image of e, the endomorphisms h x 0 wt * hXoof T+,(G/H), where t E H, have determinant 1. Show that G/H is orientable. (Using the hypothesis, show that there is a differential form on G/H of highest degree which is invariant under the action of G.) Hence give another proof of the orientability of spheres (16.11.5). Generalize to Stiefel manifolds. Show in the same way that the homogeneous spaces

Sob, R)I(SO(P,R) x Sob - P, R)) = G,p(R) are orientable. G.4.p(R)is in one-one correspondence with the set of orientedp-dimensional subspaces of R". Show that GL,,(R) is a two-sheeted covering of the Grassmannian Gm, p(R). 2.

Show that the Mobius strip and the Klein bottle (16.14.10) are not orientable (same method as for projective spaces). Generalize to the situation of a principal bundle over S , , obtained by making an arbitrary finite subgroup of S1 = U act by translations.

3. Let X be a pure differential manifold of dimension n. Define a canonical differential

+ 1)-form on the manifold A TO()* which does not vanish at any point. I

(n 4.

If M is any pure differential manifold, show that the tangent bundle T(M) is orientable (use the chart construction of T(M) (16.15.4)).

22. CHANGE O F VARIABLES IN MULTIPLE INTEGRALS. LEBESGUE MEASURES

(16.22.1) Let U, U' be two open subsets of R",and let u be a homeomorphism of U onto U' such that both u and u-' are of class C'. For each x E U , let J(x) be the Jacobian of u at x (8.10). Let I , and I,, be the measures induced on U and U' by Lebesgue measure I on R".Then the image under u (13.1.6) of the measure I JI * I , is equal to I,. .

This means that iff is any function in S ( R 7 with support contained in U', then (16.22.1 .I)

f(x) w 4=

I

f(u(x)>I J(x)I d4-4

("formula for change of variables in a multiple integral").

22 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS

161

The proof is in several steps.

(1) Let (U,) be an open covering of U, so that the U: = u(U,) form an open covering of U', and let u, : U, -+ U: be the restriction of u to U, . Sup-

pose that the theorem is true for each u,. Then it is true for u. For if &,= is the restriction of 1 to U, and if pu, is the image under u of I J I * A", then its restriction to U: is the image under u, of (I J I 1 U,) * AUI , hence is the restriction to U: of A", , by hypothesis; hence we conclude from (13.1.9) that pus= A,. . (2) Let u' be a homeomorphism of U' onto an open subset U" of R",such that u' and u'-' are of class C', and put u" = u' u, which is a homeomorphism of U onto U". If the theorem is true for u and u', then it is true for u". For if J'(x) is the Jacobian of u' at the point x , then for any functionfe S ( R " ) with support contained in U" we have 0

kx)

d m ) = Sf(uYx)) I JW I d4x) = S f ( u W ) ) ) I J'(u(x)) I

*

I J(x)I a x )

and the Jacobian of u" at x is J'(u(x))J(x) (8.10.1). (3) The theorem is true when u is the restriction to U of an affine mapping X H U + w(x). For then we have Du = w (8.1.3); whence J(x) = det w for all x E R",and the formula (16.22.1.1) follows from (14.3.9). (4) The theorem is true for n = 1. We have then J(x) = Du(x), and every point of U is the center of a bounded open interval in which Du is bounded and keeps the same sign. By (1) above we may therefore assume that U = ]a, b[ is a bounded interval in R,u being the restriction to U of a continuous function, differentiable and monotonic in [a, b]. Then we have U' =]u(a), u(b)[ if D(x) > 0 for x E U, and U' = ]u(b),#(a)[ otherwise, and the formula (16.22.1 .I)reduces to (8.7.4). ( 5 ) The theorem is true for n arbitrary and u of the form

where 8 is of class C' and J(x) = D,B(x) # 0 for all x E U. For each point x' = (t2, . ., tn)E Rn-' such that the section U(x') # the mapping tl H O(t, , t2,. . ., 5") is a homeomorphism of the open set U(x') c R onto an open set in R,and both it and its inverse are of class C'. Hence, by (4),

.

a,

162

XVI

DIFFERENTIAL MANIFOLDS

and therefore, by virtue of the definition of the product measure on (1 3.21.2),

R"

(6) The results established so far show immediately that in the general case the theorem will result from the following lemma: (16.22.1.3) Under the hypotheses of (16.22.1), for each x E U there exists un open neighborhood V of x such that the homeomorphism of V onto uw), obtained by restricting u, is of the form up 0 0 * 0 u l , where each uj is a homeomorphism of an open subset of R"onto an open subset of R", of one of the types considered in (3) and ( 5 ) above. 9

By replacing u by t u t'-', where t and t' are translations, we may assume that x = u(x) = 0. Replacing u by (Du(O))-' u, we may assume moreover that Du(0) = lRm. Hence we may write u(x) = (ul(x),. . . , un(x)), where, for 1 4 j 5 n, ujisa C'-mapping of Uinto R sucb that Di ~ ~ (= 0 6, ) (Kronecker delta). Put 0

0

0

(where x = (tl, ...,&)); it foliows from the implicit function theorem (10.2.5) that there exists an open neighborhood V of 0 in U such that, for each j, wj = vj I V is a homeomorphism of V onto an open neighborhood of 0. We may therefore write

22 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS

163

where u is a permutation of {1,2, . . . ,n}, or a mapping of the form

so

0

(Wj

0

w,:-l1) s, , 0

where the permutations u and z are chosen so that this mapping is of the type (16.22.1.2). Q.E.D.

For an important example of the appljcation of (16.22.1) (“change to polar coordinates” in R”),see (16.24.9). (16.22.2) Let X be apure differential manijdd of dimension n. Then there exists apositive measure p on X with the following property: for each chart ( U , q, n) of X , the image under cp of the induced measurep, is of the form f 0 (A,,,,), where A is Lebesgue measure on R”and f is a function of class C“ which is # 0

at every point of cp(U). Moreover, any two measures p, p’ on X with this property are equivalent and each has a density of class C“ with respect to the other.

There exists a sequence of charts (uk,qk,n) of X such that the u k are relatively compact and form a locally finite open covering of x. Let v k be the image under the homeomorphism cp;‘ of the measure induced by A on Vk(uk). Also let (fk) be a C“-partition of unity subordinate to the covering (U,) (16.4.1). Then there is a measure pk on x which coincides w i t h f , * vk on u, and with the zero measure on the complement of Supp(f,) (13.1.9), and this measure pk is clearly bounded. Moreover, since each compact subset of X meets only finitely many of the uk, the sum p = pk is defined and is a k

positive measure on X. We have to show that it has the property stated above. Let g be a function in .X(X) with support contained in U . By definition, we have j g dp =

!/

dpk =

J:dk)

=

dvk

(9 pF’)(fk pL1) dA,

‘?k(unuk)

the summation being over the finite set of indices k such that Uk intersects U. If 8, : cp(U n U k )-+ qk(Un U,) is the transition diffeomorphism and J(6,) its Jacobian, then by (16.22.1.1) we obtain J g dP = J,(uk7

where h

= k

(A

0

c p - 9 dA,

q-’)I J(8,)l is a function which does not depend on g but

only on the charts. Since J(8,) is nonzero at each point of cp(U n U,) and

164

fk

XVI

DIFFERENTIAL MANIFOLDS

9-l vanishes in a neighborhood of each point of q ( U ) - q(U n Uk),it follows that h is of class C" and f O at each point z E cp(U), because at least one of the functionsf, q - l is #Oat this point. The last assertion of (16.22.2) is evident, because the inverse of a nonvanishing C"-function is of class C". 0

0

Measures p satisfying the condition of (16.22.2) are called Lebesgue measures on X . The set of negligible functions (resp. measurable functions) is the same for all these measures. 'So also is the set of locally integrable functions, because the density of one Lebesgue measure relative to another is continuous and locally bounded. Whenever we speak of negligible, measurable, or locally integrable functions on X without specifying the measure, it is always a Lebesgue measure that is meant. We remark that submanifolds of X of dimension < n are negligible, because vector subspaces of dimension < n in R" are negligible for Lebesgue measure ((14.3.6) and (13.21.12)). If E is a fiber bundle over X, the notion of a measurable section of E over an open subset of X is well defined, independently of the choice of Lebesgue measure on X. So also is the notion of a locally integrable section if E is a vector bundle; this is clear when the bundle is trivial, and since the notion is local with respect to X, it is enough to verify that on an open set over which E is trivializable, the notion is independent of the trivialization chosen, and this follows immediately from the definitions (1 6.1 5.3).

PROBLEMS

1. Let u be a C'-mapping of an open subset U of R" into R",and let J(x) be its Jacobian atxEU. (a) Let K be a compact subset of U such that J(x) = 0 for all x E K. Show that there exists a real number c > 0 and, for each sufficiently small E > 0, a real number &(&) > 0 with the following property: for each x E K and each cube C with center x and side length 26 < a&), contained in U, we have A(u(C)) 5 cd(C). (By using the uniform continuity of J on K,show that if the image of R" under Du(x) has dimension p < n, and if (bJ1 6,8p is an orthonormal basis of this space, and (c& S k 5 n - p an orthonormal basis of its orthogonal supplement, then u(C) is contained in the parallelotope with center u(x), constructed from the p vectors ZSnMb,, where M is the least upper bound of JIDu(y)IIin K,and the n - p vectors e8n''*ck .) (b) Deduce from (a) that, for each A-measurable subset A of U, we have h*(u(A)) 5 JA*

I J(x) Idx(x).

(Reduce to the case where A is relatively compact. Let K be the set of points x E A at which J(x) = 0, and let V be a relatively compact open neighborhood of K such that

22 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS

h(V) < h(K)

165

+ E . Cover K by small closed cubes with pairwise disjoint interiors, and

use (a) to show that if W is the union of the concentric open cubes of twice the side

length, then h*(u(W)) can be made arbitrarily small. Next, show that there exists a partition of A n CW into a finite number of integrable sets G,, each of which is contained in an open set U,, such that the restriction u, of u to U, is a homeomorphism of U, onto an open set u(U,) such that both u, and u;' are of class C'. Finally, use (16.22.1) in each U, .) (c) Deduce from (b) that the image under u of each set N c U of measure zero is of measure zero. If also N is closed in U, then u(N) is a meager set, a denumerable union of nowhere dense compact sets of measure zero. (d) If E is the closed subset of points x E U such that J(x) = 0, show that uQ is a meager set, a denumerable union of nowhere dense compact sets of measure zero. Deduce that if M is any meager subset of U, then u(M) is meager in R".(Show that if B is any compact and nowhere dense subset of U, then u(B) is nowhere dense. For this purpose, consider a decreasing sequence (V.) of open neighborhoods of E, whose intersection is E, and consider as in (b) above a suitable partition of B n CV,,into integrable sets.) 2.

With the notation of (16.22.1) show that if u is a homeomorphism of U onto U', of class C' (but whose inverse is not necessarily of class C'), the formula (16.22.1.1) remains valid. (Use Problem 1.) Furthermore, the set E = {x E U :J(x) = 0) is nowhere dense, but not necessarily of measure zero. (To prove this last point, use Problem 4 of Section 13.8.)

3. (a) Let F be a closed subset of R".Show that there exists a real-valued function g of class C" on R",such that g(x) = 0 for all x E F and g(x) > 0 for all x $ F. (Use Problem 4 of Section 16.4.) (b) Let fl ,f2, . . .,f. be n real-valued functions of class C' on an open set A c R". Show that if the Jacobian of thefJ vanishes on A, then for each compact B c A there exists a Cm-function g on R" such that the set g-'(0) is nowhere dense in R"and such that g(f,(x), ...,fn(x)) = 0 for all x E B. (Use (a) and Problem l(d).) 4.

(a) Let f be a holomorphic function in an annulus S : r < l z l < R in C, and let

f(z) =

+m

C

11E-W

a. Z be its Laurent expansion (9.14.2). Show thatf(S) is open in C and that

h*(f(S)) 5

+m

C n l ~ . 1 ~ ( R-~v'") "

-m

(the right-hand side being interpreted as +a, if the series does not converge). Iffis injective on S, the two sides are equal. (b) Let fbe a holomorphic function for IzI > 1, and suppose that its Laurent series is of the form

+ c b.z-". m

f ( z )= 2

"=O

Iff is injective, show that

(Use (a).) Under what conditions is Ibl 1

= 1?

166

XVI

DIFFERENTIAL MANIFOLDS

(c) With the same assumptions on f as in (b), suppose moreover that f(z) # 0 for 1 z 1 > 1. Show that 1 b0 I 5 2. (Remark that there exists a function g, holomorphic for (z(> 1, such thatf(.z2) = (g(z))', by using Section 10.2, Problem 8.) (d) With the same assumptions o n f a s in (b), show that

for IzI > 1. (a) Let fbe a function holomorphic in the disk D : ( z I < I , with Taylor series of the f o r m f ( z ) = z + a z z 2 + ~ ~ ~ + a . ~ + . . . . S h o w t h a t i f ~ i s i n j e c t i v e i then n D , la2/5 2 . For which functions is 1 a2 1 = 2? (Bieberbach's theorem: consider the function g(z) = f(z-')-l for (z(> 1 and use Problem 4.) (b) With the same hypotheses, show that the open setf(D) contains the open disk with center 0 and radius &. (If c .$f@), consider the functionf(z)/(l - c-'f(z)).) Let B be a positive definite symmetric bilinear form on R", and let A > 0 be its discriminant relative to the canonical basis of R".Show that

1,.

exp(-B(x, x)) dh(x) = n"/zA-l'z.

(Consider an automorphism u of the vector space R" such that the matrix of the transformed form B(u(x), u(y)) is diagonal.) Let P. denote the open subset of RnZ= M.(R) consisting of the positive definite symmetric matrices. (a) Let

is a row matrix and Xl is a positive definite matrix of order where 'z = (xlzxla... xln) n - 1 . Show that (det X1)-'(det X)= x ,

- 'z . X;'

*

Z.

(Reduce to the case where X , is a diagonal matrix by means of an orthogonal transformation in R"-'.) (b) If Y EP, and s > )(n l), show that

+

(Reduce to the case Y = Z by means of an orthogonal transformation. Integrate first with respect to xll,by taking as new variable of integration u=(detXl)(xll-f~~Xi'~~) and then with respect to x1z , . . . ,x l nby using Problem 6, and finally with respect to XI; hence obtain a reduction formula for the integral.)

23 SARD'S THEOREM

167

23. SARD'S THEOREM

Let X,Y be two differential manifolds,f : X + Y a C"-mapping. Generalizing the definition of (16.5.11), a point x E X is said to be critical for f if f is not a submersion at x (16.7.1), in other words if

r k ( f 1 < dim,&I.

If E is the set of critical points o f f , then Y -f(E) is called the set of regular values off. For each y E Y -f(E), the fiberf-'(y) is therefore either empty or a closed submanifold of X (16.8.8). (16.23.1) (Sard's theorem) Let f : X Y be a C"-mapping, E the set of criticalpoints ofJ Then f(E) is negligible in Y , and Y -f(E) is dense in Y.

The latter assertion follows from the former and the fact that the support of any Lebesgue measure on Y is the whole of Y. To prove the first assertion of the theorem, observe that if (U, , qk,n,) is a sequence of charts of X such that the U, cover X and such that eachf(U,) is contained in a chart of Y, it is enough to show that f(E n U,) is negligible for each k (13.6.2). We may therefore assume that Y = Rp and that X is an open subset of R". The proof will be by induction on n ; the case n = 0 is trivial. Put f = (fi , . . . ,f,)where each4 is a real-valued function of class C" on X. Put E, = E, and for m 2 1, let Emc E denote the set of points x E X such that all the derivatives of order S m of all t h e 4 vanish at x. The theorem will be established if we prove the following two statements: (i) For each m, f(Em- Em+,)is negligible. (ii) For m ZnIp, f(Em)is negligible. (16.23.1.1) Proof of (i). Since the topology of R"has a denumerable basis, it is enough to show that, for each xo E Em- E m + 1there , exists an open neighborhood V of xoin X such that f(Em n V) is negligible. By hypothesis, there exists an index j and a derivative D'fi of order [ a ( = m + 1 which is f O at x, . By permuting the coordinates, we may assume t h a t j = 1 and that DYl = D1w, where w is a C"-function. Consider the mapping h : X +R" defined by h(x) = (

~ ( ~ 15 23 , ..

*

9

5n)T

where x = (5, , Tz, .. . ,5.) E X. Since Dlw(x,) # 0, it follows immediately from (16.5.6) that there exists an open neighborhood of x, in X such that h 1 V is a diffeomorphismof V onto an open subset W of R".Let g = ( g1 , . . . , gp) denote the restriction o f f h-' to W, which is therefore a Cm-mappingof W into RP. 0

168

XVI DIFFERENTIAL MANIFOLDS

We distinguish two cases: (a) m

= 0 and

(b) m 2 1.

(a) m = 0. By definition, we may assume that w =fi,sothat g,(x) = t1 for all x E W. Next, the set E of critical points of g is equal to h(E n V), hence f(E n V) = g(E') and it is enough to prove that g(E') is negligible. Identify R" with R x R"-', and for each x = (c, z) E W, put g(x) = gc(z)). Then the Jacobian matrix of g at the point x is of the form

(c,

hence in order that x E E' it is necessary and sufficient that z E E;, where E; is the set of critical points of gr. Consequently, for each C E R,

((51

x R P - 9 (7 g(E') = {el x g m .

Now the inductive hypothesis implies that g,.(E;)is negligible in RP-'; on the other hand, E' is closed in W, hence is a denumerable union of compact sets, and consequently g(E') is a denumerable union of compact sets, hence is Lebesgue-measurable in RP (1 3.9.3). It follows now from (13.21 .lo) that g(E') is negligible. (b) m 2 1. By definition, we may assume that w(x) = 0 for x E Em, hence h(E, n V) c (0)x R-"I. For each point (0,z) E W n ((0)x R"-'), put g(0, z) = go(z). Since all the first derivatives of g vanish at each point of h(E, n V), all these points are critical points of go.The inductive hypothesis therefore implies that go(h(E, n V)) is negligible in RP,and this set is precisely f(E, n V). (16.23.1.2)

Proof of (ii). Let IJxI(denote the norm sup i

I til on R" and on

RP. For each real number a > O and each k = (k,,..., k,)ER", let I(k,a) denote the cube in R" defined by the inequalities k j 5 ti5 ki+ a (1 5 i 5 n). Then it is evidently enough to show that f(E, n I(k, a)) is negligible for some a > 0 such that I(k, a) c X. Let M be the least upper bound of l(f('"+')(x)lIi n I(k, a); it follows from Taylor's formula (8.14.3) that if x E E, n I(k, a) and x + t E I(k, a), then

Now observe that, for each integer N > 1, the cube I(k, a) is the union of the N" cubes I(s, a/N), where 5 = (k1

+ ( a , / N ) , . . . > kn +(asnlN))

and the integers sirange independently from 0 to N - 1. The set f(E, n l(k, a)) is therefore contained in the union of the N" sets f(E, n I(s, a/N)); but for

23 SARD'S THEOREM

169

each s such that Emn I(s, a/N) is nonempty, if xo is a point of this set, it follows from (1 6.23.1.3) that for every other point x E Emn I(s, a/N) we have [If(x)- f(x,)(I 5 M(a/N)"+', and therefore

I(f(E, n I(s, a/N))) S MP(~/N)P(m+l), where 1is Lebesgue measure on Rp. Hence I(f(E,,, n I(k, a))) S Mpap(m+l)Nn-p(m+l). Since by hypothesis m 2 n/p, the right-hand side of this inequality tends to zero with l/N, and the proof is complete. Sard's theorem implies, in particular: Let X, Y be pure differential manifolds of dimensions n, p , respectively, such that n < p , and let f:X + Y be a C"-mapping. Then Y -f(X)is dense in Y . (1 6.23.2)

I n other words, for C"-mappings there do not exist phenomena of the type of the "Peano curve" (Section 4.2, Problem 5, or Section 9.12, Problem 5).

PROBLEMS 1. (a) Let m, n, p , r be integers >O. Let f be a C"-mapping of a n open set U C R" into R",and let g be a Cm-mapping of a n open set V 3 f(U) in R" into RP. Put h = g f. Show that for each x E U we have 0

D'h(x) =

c c. . I

q = o (11..

o,(il, . . . , iq)Dqg(f(x)) (Dilf(x), . . . 0

* i,)

where in the inner sum, (il ,. . ., i,,) runs through all sequences of q integers 21 such that il . . . iq = r . Furthermore, in this formula, the constants o,(il , . . . , i,) are rational numbers which depend only on the ij, q, m,n,p , and r, and nor on the functions f and g. (b) Let xo E U. Assume now only that, for some s < v, f is a mapping of class C'-" of U into R",and that g is a mapping of class C' of V 3 f(U) into Rp. Suppose also that Dkg(f(xo))= 0 for k $ s. For each x E U and each integer k E [0, r], let

+ +

c c.... k

hk(x)=

q=s+1

(ii.

uk(il, . . . , i,)Dqg(f(x))

iq)

0

(D"f(x),

.. . ,D',f(x))

E

-Yk(Rm;Rp),

the second sum being over the same sequences ( i , ,. . . ,i,,)as in (a), so that ij 5 k - s for all j , and hence the function h, is well-defined on U. Ry applying Taylor's formula, show that hk(x) can be written in the form r-k

hdx) =

C

j=O

a,

*

(x

- x0)(j)

+ Rdx),

170

XVI

DIFFERENTIAL MANIFOLDS

where the ajk are constant elements of 9,(Rm; ..Yk(Rm;Rp)) = 9k+,(Rm;Rp), and IIRk(x)111JIx- xoIr-’ tends to 0 as x +xo . Further, the aJkdepend only on the values of the derivatives o f f at xo and of g at f(xo). Deduce that j!=jk = hk+J(xO)

f o r j $ r - k . (Use(a).) (c) More generally, let A be a closed subset of U, B a closed subset of V containing f(A), and suppose that Wg(y) = 0 for all y E B and 0 5 k 5 s. Defining the hkas in (b) above, show that for xo E A, h&’) =

J=oJ.

hk+j(X) *

(X’

- X)(’)

+ Rk(X’,

X),

where IIRr(x’, x)ll/Ilx’- x IY-k tends to 0 as x, x’ tend to xo whilst remaining in A. (d) Under the hypotheses of (c), deduce that there exists a mapping H : U+RP of class C‘, such that H(x) = g(f(x)) for all x E A and DkH(x) = 0 for all x E A and k 5 s (Kneser-GIueser rheorem). (Use Whitney’s extension theorem (Section 16.4, Problem 6).) Let U be an open subset of R”, f a mapping of U into Rp, of class C‘. Show that if r 2 max(1, n - p l), the image f(E) of the set of critical points off is of measure zero in Rp.(Proceed as in the proof of (16.23.1), using the result of Problem 1 to deal with the case m 2 1 in (i).)

+

The notation is that of Section 13.21, Problem 2. The simple arc K =g([O, 71) is therefore a subset of R2 of measure >O. Let KObe the union of {go(0)},{g0(7)} and the three sets Kol = g0([l, 21), Koz = g0([3,4]), and KO3 = go([5, 61). Let FObe the function on KOwhich takes the value 0 at g0(0), 4 on KOI, 4 on KO2,2 on K03, and 1 at g0(7). Define sets K. inductively as follows: K. is the union of K.-1 and the sets K,, where I is any sequence of n terms (il , .. ., in),each term of which is equal to 0,2,4, or 6; for any such s, K, is the union of Dethe three sets Ka, I = Bn(Ua([l, 21)), Ka. 2 = ~dua([3,41)),and K.. 3 = 8n(01([5,61)). fine functions F. on K. inductively as follows: F,(K.-, = Fn-l, and F. is constant on each K,, (1 5j S 3): putting a = g.(u,(O)), p = gm(us(7)), then

F. = fFjn-l(ar) F. = W n - h ) F n = bFn-i(a)

+ &Fn-lCs> + +F.-I@) + PFn-lGB)

on L 1, on IL.2 , on Ka, 3 .

(a) Let G be the function on the union of the K. which is equal to F. on K. for each n. Show that G extends by continuity to a real-valued function F on K. (b) Show that there exists a constant c > 0 such that, for any two points x, y E K, we - ~ 1 1 for ~ ’ a ~suitable choice of the sequence (a”).(Consider have 1 F(x) - F(y)l =< c * JJx the least integer II such that the two points g;l(x), g;’(y) do not belong to the same interval o,([O, I),for some sequence s of n terms.) (c) Deduce from (b) and Whitney’s extension theorem (Section 16.4, Problem 6) that there exists a function f:RZ-+R of class C1 for which all the points of K are critical points, and yet such that f(K) is the interval [0,1].

24 INTEGRAL OF A DIFFERENTIAL n-FORM

171

=< inf(p, n), let M, be the subset of the space Rp" of p x n matrices consisting of the matrices of rank k . Show that Mk is a differential submanifold of RpD,of dimension

4. For k

k(p

+ n - k). (Observe that an p x n matrix of the form

(z

:),where

A is a k x k

invertible matrix, is of rank k if and only if D = CA-'B; for this purpose, multiply the matrix under consideration on the left by a suitably chosen invertible matrix so that the product is of the form

(; :/).

5. Let U be an open set in R" and let f : U RP be a C" mapping. Suppose that p 2 2n. Show that for each E > 0 there exists a p x n matrix A = (a,,) with I u,, I 6 E for all i, j , such that the mapping x ~ g ( x = ) f(x) A * x of U into RP is an immersion.(For each k < n, consider the C" mapping Fk : Mt x U -+ Rp" (notation of Problem 4) defined by F&Z, x) = Z - Df(x), and remark (16.23.1) that the image of Fk is of measure zero in Rp", by using Problem 4; the complement of the union of the images of the FI for 0 =< k 5 n - I is therefore dense in RP".) -+

+

24. INTEGRAL O F A DIFFERENTIAL *FORM PURE M A N I F O L D O F DIMENSION n

OVER A N ORIENTED

(1 6.24.1) Let X be an oriented pure manifold of dimension n 2 0. Then there exists a C" differential n-form uo on X, belonging to the orientation of X. We shall define on X a positive Lebesgue measure (16.22) pvo depending on uo . To do this it is sufficient to define a Lebesgue measure on U, for each chart (U, cp, n) of X with U connected, and then to show that if (U',cp', n) is another chart with U' connected, and if p, p' are the measures defined on U and U', then the restrictions of p and p' to U n U' are equal (13.1.9). By hypothesis, if x = (t', ... , E cp(U), we may write

r)

uo(cp-'(x))

=f(C', . . . , 5")

dt'

A

dt2 A

A

dt",

and since U is connected, f is of the same sign throughout cp(U). More precisely, if R" is endowed with the canonical orientation (1 6.21.4), thenf(x) > 0 (resp. f ( x ) < 0) in cp(U) if cp preserves (resp. reverses) the orientation. By definition, the measure p on U is the image under 'p-' of the measure Ef. I, where 1 is the measure on q(U) induced by Lebesgue measure on R",and E is + 1 or - 1 according as cp preserves or reverses the orientation. Likewise, if x E cp'(U'), we have uo(cp'-'(x)) = f ' ( x ) dC;' A dtz A A d y , and we define the measure p' on U' as the image under q'-' of ~ ' *fA', ' where I' is the measure on cp'(U') induced by Lebesgue measure, and E' is + I or - 1 according as cp' preserves or reverses the orientation. Now let -

0 : cp(U n U')

+ cp'(U n U')

0

172

XVI

DIFFERENTIAL MANIFOLDS

be the transition diffeomorphism, and let J(0) be its Jacobian, which is > O in cp(U n U') if E' = E , and c0 if E' = - E . For each x E cp(U n U') we have

Ax) =f'(e(x))J(e)(x)* To prove that the measures induced on U n U' by p and p' are equal, we may assume that U = U'. Then, by (16.22.1), the image under 0 of the measure cf * 1= ( c ( f ' 0)) * (J(0) .1)is equal to cf' * At if e' = E , and to - ~ f l A' if E' = - E . Hence in both cases it is equal to ~lf'* I, which proves our assertion. The fact that the measure puois a Lebesgue measure follows from the fact that uo(x) # 0 for all x E X; moreover it is clear that this measure is positive. If u1 is another C" differential n-form on X, belonging to the orientation of X, then u1 = hu,, where h is a C"-function which is >O at all points of X, and it follows immediately that p,, = h * puo (13.14.5). 0

1

(16.24.2) With the same hypotheses and notation as above, consider now an arbitrary differential n-form u on X. We have u = go,, where g is a realvalued function on X. The n-form u is said to be integrable (or integrable over X) if g is p,,-integrable, and the number g dp,, is called the integral of u (or

s

/

Ix

the integral of u over X) and is denoted by u, or u, or Jx u(x). We must show that this definition is independent of the C" n-form u, chosen in the orientation of X. Now, if u1 is another n-form of class C" in the orientation of X, then u1 = hu, , where h is a C"-function on X, everywhere >O. Hence u = gh-'ul; but also p,, = h * p u o , and g is puo-integrableif and only if gh-' is h * p,,-integrable (13.14.3). Moreover we have P

P

(1 3.14.3), which proves the assertion.

The integrable differential n-forms on X clearly form a real vector space, and UH u is a linear form on this space, taking positive values on forms u 2 0 (in the sense defined in (16.21.2)). If we fix a C" n-form u in the orientation of X, it is clear that the linear form f- fu on 2 ( X ) is a positive Lebesgue measure, and conversely all positive Lebesgue measures on X are of this type for a suitable choice of u. These measures are also called volumes on the oriented manifold X, and the corresponding forms u are called volume-forms.

s s

(16.24.3) A differential n-form u = gu, on X is said to be measurable (resp. locally integrable, resp. negligible) if the function g is measurable (resp. locally

24 INTEGRAL OF A DIFFERENTIAL n-FORM

173

integrable, resp. negligible) with respect to the measure p v 0 . It is immediately verified that these notions are independent of the choice of the n-form u, in the orientation of X (13.15.6). If u is locally integrable and of class C", then fu is integrable for each measurable functionf which is bounded and of compact support. Let u be a locally integrable differential n-form on X, let (uk) be a locally finite covering of by relatively compact open sets, and let (uk) be a continuous partition of unity subordinate to (uk). Then u is integrable if and only if

x

the series

zj1gukl dpuois convergent; in which case the series juku

k s

k

converges and has u as its sum. If each U, is the domain of definition of a chart ( U k ,q k ,n), where (Pk is orientation-preserving, and if

(16.24.4) Let X, be an orientable pure manifold of dimension n ; let X be the oriented manifold obtained by endowing X, with an orientation, and let -X be the oppositely oriented manifold. If u is an integrable differential n-form on X, then u is also integrable over - X , and

(1 6.24.5) Let f:X' + X be a diffeomorphism of a connected oriented manifold X' onto an oriented manifold X. Then, for each integrable n-form u on X, the form 'f(u) is integrable over X', and we have n

n

(16.24.5.1)

where the sign is + or - according asf'preserves or reverses the orientation. For by the method of calculation indicated above, we reduce immediately to the situation where X and X' are open subsets of R", where the result follows immediately from the definitions and the formula (16.22.1.1) for change of variables.

174

XVI DIFFERENTIAL MANIFOLDS

(16.24.6) Let Y be an oriented manifold of dimension m, X a manifold of dimension n >= m, and f : Y + X a mapping of class C ' (r 2 1). We have seen (16.20.9) how to define the inverse image 'f(u) of an m-form (i on X. If this inverse image is integrable, we may consider its integral 'f((i).When Y is an oriented submanifold of X, andf is the canonical injection, we shall often write

sy

.Jy (i in

place of .Jy ~((i).

Example (16.24.7) Integration on a sphere. Let n > 0. Consider the closed parallelotope P in R" defined by the inequalities

-nienjn (the @ being the coordinates of a point of P). The interior P of P is defined by -tnrej6$n

(isjsn-i),

the same inequalities with all the signs 5 replaced by 0 (16.21.2) almost everywhere on Y. Then for almost ally E Y the (n - m)-$orm o/[(y) (16.21.7) is defined and integrable over f - ' ( y ) (endowed with the orientation induced by f from the orientations of X and Y ) ;the form y~ [ ( y ) /I- ,(y) u / [ ( y ) is integrable over Y, and n

( u k

n

n

There exists a denumerable open covering ( u k ) of X and charts , ( P k , n), ( f ( u k ) , $ k , m ) of X and Y, respectively, such that (Pk(Uk)

= $k(f(uk)

I"-",

where I = ] - I , I[, and f l u k = Fk ( P k , where Fk is the restriction to q , ( u k ) of the canonical projection of R" onto R". Using the integrability 0

0

176

XVI

DIFFERENTIAL MANIFOLDS

criterion of (16.24.3), we reduce immediately to the situation in which Y is an open set in R", X = Y x I"-" and f = pr, . Then we have u(x) = u(x) d t l

A

dt2 A

A

dy,

[ ( y ) = w(y) dt'

A

* *.

A

dt",

where u is integrable with respect to Lebesgue measure R on X, and w is locally integrable and # O almost everywhere with respect to Lebesgue measure A' on Y. The form u/[(y), defined for almost all y = (t', ..., t"), may be written as

. . c)

Now the function (t', . , H u(t', . .. , 5")/w(t1,... , 5") is measurable with respect to the measure w * R on R",and this measure may be thought of as the product measure (w * A') 63 A", where 1" is Lebesgue measure on R"-" (13.21.1 6). The proposition is therefore a consequence of the LebesgueFubini theorem (13.21.7) and the definition of the orientation on the fibers f +(Y). (1 6.24.9) Application : calculation of integrals in polar coordinates. Let n 2 2. We shall apply (16.24.8) by taking X = R" - {0},Y = R*,= 10, co [, f(x)= llxll = ((5')' * * * ((?)2)"2, so that f -'(u), for u > 0, is the sphere u Sn-' homothetic to S,,-,. Also take [(t)= t-' d t , and take u to be an n-form g * u o , where uo is the canonical n-form dt' A * - * A d y , and g is Lebesgue-integrable over X. The (n - 1)-form uo/[(u) on u * Sn-, may be calculated as follows: Let 0' be an (n - 1)-form on a neighborhood of u * S,,-l in X, such that uo = 'f([) A a'. Let h, : x w u x be the homothety of ratio u on X. Then we have

+

+ +

9

-

'h,(oo) = t~u(Y(o) A 'W') in a neighborhood of S,,-'; but it is immediately seen that 'h,(uo) = 24" * uo

9

'h,(Y(CN

=

m.

In view of the uniqueness of uo/[(u),it follows that 'hu(uo/[(u))= U" . S,-l, whence by (16.24.5.1)

1

u.s,-,

s(uo/~(u)) = u" I s n - g ( u * z)a("-')(z).

Hence the formula (1 6.24.8.1) gives

on

24 INTEGRAL OF A DIFFERENTIAL n-FORM

177

Using the formula (16.24.7.4), we obtain finally the formula for calculating an integral over R" - {O} in polar coordinates:

Of course, we should arrive at the same formula by calculating the Jacobian of the diffeomorphism of P x R; onto an open set in R" - (0) with complement of measure zero. In the notation of (16.24.7) (with n replaced by n - 1) this diffeomorphism is u * b,t. Applying (1 6.24.9.1) to the particular case where g is the characteristic function of the unit ball B,, : llxll 5 1 in R", we obtain

=Is

1 V,=-C&, n

where Q,, a("-"iscalled the solidangle in R" or the superjicialmeasure n- 1 of Sn-l. From the preceding calculations and the formula (14.3.11.3), its value is (1 6.24.9.3)

nn=

n71"J2

T($n

+ 1)'

or equivalently,

(1 6.24.9.4)

2nKn - I

1 3*5

0

.

.

(2n - 3)'

We remark that, with the definition of do)given in (16.21.10), the formula (16.24.9.1) remains valid for n = 1 , for it then takes the form

178

XVI

DIFFERENTIAL MANIFOLDS

(16.24.10) The interesting thing about the preceding method is that it applies without modification when f is any positive and positively homogeneous function of degree 1 on R" - {0}(i.e., such that f(u * x) = uf(x) for all u > 0) of class C" and whose differential df is nonzero at allpoints of R" - (0).If El is the submanifold given by the equation f ( x ) = 1 in R",and of is the form uo/y([(l)) on E l , then by the same reasoning as before we shall obtain the following generalization of (1 6.24.9.1) :

From a practical point of view, if the set U in El of points where aflat' # 0 for some index i has a complement of measure zero, we may determine of by the method of (16-21.9.1): We consider, in a neighborhood of U in R" - (0) the following (n - 1)-form: (16.24.10.2)

of = (- l)i-'

d5'

A

A

0

dt'

A

A

dc

amti

and we take for of the form induced by

0;

on U.

(16.24.11)

(Stokes' formula, elementary version) Let V be an open subset of R"-', U an open subset of R", F a function of class C" on U such that D,F = dF/a[' # 0 in U and such that the mapping

T")H(F(tl,* . . , T"), t2,. . ., 5") is a diffeomorphism of U onto I x V, where I is an open interval in R. For each u E I, let E, be the closed submanifold of U defined by the equation F(C1, . . ., = U, whose image under $ is { u } x V , and let o, be the ( n - l)-form on E, equal to $ : (5'

9

* *

3

r)

uo/'F(e,*),where e,* is the unit covector in T,(R)*. Let [a,b] be a closed interval contained in I. Then, for every C'-jiunction f on U,we have (1 6.24.11.I)

1.., 1.

D l f ( t l , . .. ,5") dt' d t 2 * - . d r

=

f(z)DIF(z)ob(z)

-

6.

f(z)Dl

F(z)ae(z),

where Ua,bis the set of x E U such that a 5 F(x) b, and the orientations on E,, and Eb are induced by F from the canonical orientations of R"-' and R (16.21.9.2).

24 INTEGRAL OF A DIFFERENTIAL n-FORM

Put ui,b= ]a, b[ x V, and let g Fubini theorem, we have

s

D,g(u, t2,...,Y) du

U'a.b

= Jv d5'

A

A

*

dr"

A

J:

= f o

I)-' on I x V. By the Lebesgue-

dt2 A . .*

A

dc

D,g(u, t', . . ., r") du

t', .. .,5") - g(a, t2,... ,5")) d5'

= s?(b,

179

A

* * *

A

dy.

5', .. .,Y), the projection of z on R"-' is

Observe that if z = $ - I @ ,

The calculation of a b indicated in (16.21.9.1) shows that this (n - 1)-form is induced on E, by the (n - 1)-form x ~ d 5 'A * . . A d 1, Wo,(C) = A,@), and for 1 5 i 5 n,

+

where pu is the orthogonal projection on the hyperplane (x I u) = 0. Show that each of the functions WI, is increasing and continuous on @", and that W,.(aC) = a"-'Wl.(C) for all a > 0. If A , B are convex bodies belonging to L, , such that A u B is convex and A n B has a nonempty interior, then W&

u B)

+ W d A n €3)

= Wln(A)

+ W,.(B).

(Observe that if u E A and b E B, there exists a point of A n B in the segment with a,b as endpoints. Consequently, if H is a supporting hyperplane of A n B, then H is a supporting hyperplane of either A or B. Deduce that P,(A n B) =pU(A) np,(B).)

In particular, nW,,.(C) 6.

= .dn-l(C)

(Problem 4).

If C is a compact convex body in R" containing the origin, recall that the function of support of C is the function H(z) = sup(x1z) (Section 16.5, Problem 7). For each

+

XEC

u E Sm-], let b(C, u) = H(u) H(- u) (the "width" of C in the direction u, cf. Section 14.3, Problem 9(a)). Show that, in the notation of Problem 5,

182

XVI

DIFFERENTIAL MANIFOLDS

(Proof by induction on n. For each u E S.- ,let E(u) denote the hyperplane (XI U) = 0 in R",and let u("-*) be the differential (n - 2)-form on n E(u) which is the image of a("-') by a rotation of R" transforming Sn-' into Sn-l n E(u). Show that the integral n- 1

a - l nE(u)

b(P"(C),v). U:-')(v)

can be written in the form

where P is the submanifold of S"-lx S.-l consisting of pairs (u,v) such that (u Iv) = 0, and w is a (2n - ])-form on P obtained by the procedure of (16.21.7); use (16.24.8).)

7. (a) Show that if P is a compact convex polyhedron of dimension n in R",then the area S ~ " - ~ ( Pis) equal to the Minkowski area of P (Section 14.3, Problem lqc)). Deduce that, for each p > 0, in the notation of Section 3.6,

(b) Show that, for each compact wnvex body C in R",we have

(Steiner-Minkowski formula). (Prove the formula first in the case that C = P is a compact convex polyhedron of dimension n, by using (a) and induction on n. Then pass to the limit in L. .) (c) Deduce the formula

sn

and r 2 0. (Observe that V,+,(C)= V,(V,(C)) and use the Steinerfor 0 5 i Minkowski formula.) (d) Deduce from (b) and (c) that formula (1) is valid for any compact convex body C . In particular, .dn-l(C)is equal to the Minkowski area of C (Section 14.3, Problem 1O(C)). 8.

Let LA be the set of all nonempty compact convex sets in R",endowed with the Hausdorff distance (Section 3.1 6, P.roblem 3). 9; is a compact space in which L,,is dense. Show that the functions W , . defined on 9. extend by continuity to S;(induction on n). If A is a compact convex set of dimension t n in R",then

25 EMBEDDING AND APPROXIMATION THEOREMS

183

9. Let A c RP, B C Rq be compact convex sets. Show that

(Apply the Steiner-Minkowski formula to A x B, and use the Lebesgue-Fubini theorem.) Hence calculate the values of Wl.(C) when C is a cube in R". 10. Let C C Rn be a compact convex body of dimension n. Show that the set CV,(CC) is convex (express it as an intersection of translates of C). For each r > 0, we have

V,(CV,(CC)) c C. By using the continuity of the functions W,.(C) with respect to C, deduce that the function rH/\.(CV,(CC)) has a derivative on the right at the origin, equal to - .dn-l(C).

11. Let X, Y be two oriented pure differential manifolds, of respective dimensions n and m ;let f:X .+ Y be a submersion, x a point of X, y =f ( x ) . We shall use the notation of (16.21.7). Let 5 be a C"m-form on Y belonging to the orientation of Y.For each k

rn-k

k 6 m, 5 defines a canonical isomorphism z Y w@ r ( y ) ( ~ y ) of A T(Y) onto A T(Y)*, such that @r(y)(z,) = zy _I ( ( y ) . Let u be a C" (n - m k)-form on X with compact k

A

+

support. To each k-vector zy E T,(Y) there corresponds an (n - m)-form f - ' ( y ) such that, for x E f - ' ( y ) , we have

&, on

in the notation of (16.21.7). This form is independent of the choice of 5 in the orientation of Y. We give each fiberf - I ( y ) the orientation induced by f from the orientations of X and Y (16.21.9.1). Show that there exists a unique C" k-form y on Y such that, for all y

EY

and all z,

k

E

A T,(Y),

This form is denoted by ub and is called the integral of u along the fibers ofJ (Reduce to the case (16.7.4).) If 3!,' is a C" k'-form on Y,then cd A j?= (a A *f(j?))b.

25. EMBEDDING A N D A P P R O X I M A T I O N THEOREMS. T U B U L A R NEIGHBORHOODS

(16.25.1) Let X be a dijierential manifold, U a relatively compact open subset of X. Then there exists an integer N and an embedding (16.8.4) of U in RN. There exist a finite number of charts (uk, (Pk,nk) of X (1 6 k 5 m) such that the U, cover the compact set 0. Next, there exists a family (Vk)lSksrn of open subsets of X which cover 0 and are such that ?k c U, for each k

184

XVI

DIFFERENTIAL MANIFOLDS

x,

(12.6.2). Finally, there exists a family of C"-functions (fk)lsksm on with values in [O, 11, such that ~upp(f,)c uk andfk(x) = 1 for all x E 0,. We shall show that the mapping : XH((fk(X))i~k~m? (fk(X)(Pk(X))ljk4rn)

of 0 into RN = R" x

n m

R"kgives the desired embedding by restriction to

k= 1

U. It is clear that g is of class C" (16.6.4). Next, g is injective. For if x, x' are two distinct points of X such that fk(x) =fk(x') for I r k r m , then since x E v k for some k, it follows that &(X) = 1 and thereforefk(x') = 1, so that x' E uk; but then, since x # x', we have fk(x)qk(x) = (Pk(X)# (Pk(X')=f,(x')qk(x') so that g(x) # g(x'). Since 0 is compact, it follows that g is a homeomorphism of 0 onto g ( 0 ) (3.17.12), hence of U onto g(U). Hence it remains to show that g is an immersion (16.8.4) at each point x of U. Let k be such that x E vk , and let p be the projection of RN onto the factor Rnk,so that p o g = (Pk on v k . Since T,(q,) = Tecx,(p)0 T,(g) is of rank nk, it follows that T,(g) is of rank nk . This shows that g is an immersion at x (1 6.7.1) and finishes the proof. One can in fact show that there exists an embedding of the whole of X into an RN,and that if X is pure of dimension n , one can take N = 2n + 1 (Problems 2 and 13(c)). The above embedding theorem will enable us to extend to manifolds the Weierstrass approximation theorem (7.4.1), polynomials being replaced by C"-functions. We shall begin by establishing two extremely useful auxiliary results on " tubular neighborhoods " of a submanifold of RN. (16.25.2) Let X be a pure submanifold of RN,of dimension n. Let j : X -+ RN be the canonical injection and U a relatively compact open subset of X . For each x E X let M, be the n-dimensional subspace of RN which is the image of T,(X) under the mapping r , o T,(j) (16.5.2). Let d be the distance function on RN derived from the scalar product (x I y) = t j q j , and let N, be the orthogonal i

supplement of M, in RN (6.3.1). Suppose that we have de$ned, on an open neighborhood V of 0 in X , N - n mappings uj (1 5 j 5 N - n ) of V into RN, of class C", guch that for each x E V the uj(x) form a basis of N,. Then there exists an open neighborhood T of U in RN and a dixeomorphism Y H (n(Y), B(Y))

of T onto U x RN-"such that, for each y whose distance from y is equal to d(y, X).

E T, ~

( yis) the unique point of X

25 EMBEDDING AND APPROXIMATION THEOREMS

185

Consider the mapping g : V x RN-"+ RN defined by g ( x ; t1 7 . .

9

tN-,,)

=x

+

N-n j = 1

tj

uj(x).

Clearlyg is of class C". Moreover, for each point a E V, the tangent mapping T(,, ,&)is bijective, hence (16.5.6) there exists an open neighborhood W, c V of a in X and an open ball B, with center at the origin in RN-"such that the restriction of g to W, x B, is a diffeomorphism of this open set onto an open neighborhood T, of a in RN. Let y ~ ( n , ( y ) O,(y)) , be the inverse diffeomorphism. We shall show that there exists a ball S, c T, with center a such that, for each y E S, , n,(y) is the only point x E X such that d(y, x ) = d(y, X). We shall use the following lemma: (1 6.25.2.1) For each a E X there exists a ball S: in RN with center a such that, for each y E S: , there exists at least one point x E X for which d(x, y ) = d(X,y). For such a point x, the vector y - x is orthogonal to M, .

Since X is locally closed in RN,there exists a closed ball S L with center a and radius r, in RN such that X n Sg is closed in Se, and therefore compact. Let S: be the closed ball with center a and radius Sr, . For each y E S: ,we have d ( y , a) S +r, , and for each z E X such that z 4 S: , we have d(y, z) 2 %r,, so that d ( y , X) = d ( y , X n Se). Hence X n contains a point x such that d ( y , x) = d(y, X) (3.17.10). Moreover, the function z-h(z) = (d(y,2)' = n

j= I

(q' - [ j ) 2 is of class C" on X and admits a minimum at the point x ; hence

d,h = 0 (16.5.10). However, relation d,h = 0 may be written as N

1 (qj -

j= 1

cj)

dcj = 0 ;

SJs

also, for each vector t E M,, the numbers (t, d") ( I N) are the components o f t with respect to the canonical basis of RN.Hence the vector y - x is orthogonal to M,. Having established the lemma, we shall now argue by contradiction. Suppose that there exists a sequence (y,) of points of T,, tending to a, and a sequence (x,) of points of X such that x , # n,(y,) and d ( y , , x,) = d ( y , , X). Since d ( y , , x,) 5 d ( y , , n,(y,)), it follows that d(y,, x,) +0 and hence that x, + a ; so we may suppose that x , E W,. By virtue of the lemma (16.25.2.1), we can write y , - x ,

N--n

= j = 1

tj,uj(x,);and since x , # x,(yY), the point

186

XVI

DIFFERENTIAL MANIFOLDS

ctit,

..

N--n

does not belong to the ball B,. If we put r,' =

therefore bounded. Since

N-n

j= 1

i= 1

the sequence (r;') is

(r;1tjv)2= 1, we may, by passing to a sub-

sequence of the sequence (yv),assume that each of the sequences (r;'tjV) has a limit ti (1 S j S N - n); we have then t j Z = 1; but on the other hand,

1

rv-'(yv - x,) --f 0 as v + 03, and hence

i

i

t j ui(a) = 0. This is impossible, since

the tJ are not all zero and the uj(a) are linearly independent. If b is another point of V such that S, n Sb # @, then it follows from above that n, and q,agree on S, n s b . Hence there is a unique function n, defined on the union S of the open sets S, (a E V), which extends each of the functions n, . For each a E V, let W: c W, be an open neighborhood of a in X , and B: c B, an open ball with center 0 such that g(W: x B:) c S,. Cover 0 by a finite number of open neighborhoods W:, (1 5 i s r), and let B" be an open ball with center 0 in RN-" (hence diffeomorphic to RN-") contained in the intersection of the B:, . Then g(U x B") is an open subset of RN contained in S and containing U, and for each (x, t) E U x B" we have n(g(x, t)) = x

and

t = g(x, t) - x .

This proves that g l ( U x B") is a bijection of U x B" onto g(U x B"), and hence a diffeomorphism of U x B" onto an open neighborhood of U in RN (16.8.8(iv)). (16.25.3)

There does not always exist a system (ui) of mappings of V into

RN having the properties of the statement of (16.25.2). For example, if X is a

nonorientable compact manifold embedded in RN (16.25.1), the existence of an open neighborhood of X in RN diffeomorphic to X x RN-" would contradict (16.21.9.1), since every open subset of RN is an orientable manifold. However, there is the following weaker result:

(16.25.4) Let X be apure submanifold of RN of dimension n. Then there exists an open neighborhood T of X in RN and a C" submersion n of T onto X with thefollowing property:for each y E T, n(y) is the onlypoint of X whose distance from y is equal to d(y, X), andfor each x E X, the3ber A - ' ( x ) is the intersection of T and the linear manifoldx + N, (the space N, being defined as in (16.25.2)). It is enough to prove that, for each a E X, there exists an open neighborhood T, with center a in RN such that X n T, is closed in T,, and a surjective submersion n, :T, + X n T, such that, for each y E T,, n,(y) is the unique point of X whose distance from y is equal to d(y, X), and such that, for each

25

EMBEDDING AND APPROXIMATION THEOREMS

187

+

x E X n T,,, the fiber a;'(x) is the intersection of T,, with x N,; the union T of the T,, will then have the required properties. By virtue of (16.25.2), it is enough to show that there exists a relatively compact open neighborhood V of a in X on which N - n functions uj can be defined so as to satisfy the conditions of (16.25.2). By means of a displacement we may assume that a = 0 and that Ma is the space R" spanned by the first n vectors of the canonical basis ( e i ) 1 5 i sof N RN.Consequently (16.8.3.2) there exists a relatively compact open neighborhood U of 0 in R" such that a neighborhood V of 0 in X is formed by the points of U x RN-"which satisfy the equations ( * ' j = jj(t', . .., 5") (1 j 5 N - n), where the fi are C"-functions defined on U which vanish, together with their first derivatives, at the origin. For each point x = (5'

9

**.,

t",fi(t',. . * , t"),

. * * , f N - n ( t l , * *51) *,

of V, the space M, is defined by the N - n linear equations ["+j

-

n

i= 1

Difi(t1,... , 5") - [' = 0

(1

sj 5 N - n)

and consequently the functions uj(x) =

-

n

i= 1

DiJ;.(tl,. . . , Y)ei

(1

5 j S N - n)

satisfy the required conditions. We can now state the approximation theorem: (16.25.5) Let X, Y be two dzTerentia1 mangolds, K a compact subset of X, andf : K + Y a continuous map. Let d be any distance which defines the topology of Y, and let E > 0. Then there exists an open neighborhood U of K in X and a C"-mapping g : U + Y such that

4 f 0 ,d-9)5 E for all x E K.

Let U, be a relatively compact open neighborhood of K in X, and Vo a relatively compact open neighborhood of f ( K ) in Y (3.18.2). We may assume that Uo is embedded in R" and V, in R" (16.25.1). Let T be an open neighborhood of V, in R"having the properties of (16.25.4), and let 6 < +E be a positive real number such that every point of R" whose distance from f ( K ) is j S lies in T (3.17.11). By the Weierstrass approximation theorem, there exist n polynomials hi (1 51 n) in m variables such that if h = (h, , , . . ,h,,),we have

188

XVI

DIFFERENTIAL MANIFOLDS

d(f(x),h(x)) =< 6 for all x E K (7.4.1). Since h(K) c T, there exists an open neighborhood U c Uo of K in X such that h(U) c T. For each x E U, put g(x) = n(h(x))E Vo . Sincef(x) E Vo , we have by definition (16.25.4) and therefore dcf(x), g(x)) 6 E for all x E K.

PROBLEMS

1. Let M be a pure submanifold of R", of dimension n, and let A be a compact subset of

M. Let Rl be the set of linear mappings u :R" +RZ"+'such that U I M is of rank n at every point of A, and let R2 be the set of linear mappings u :Rm-+R2"+' such that u I A is injective. Show that, if m 2 2n 1, R l and R2are dense open sets in the vector space 9 ( R m ;R2"+l).(Show first that the complements Ql ,Q2 of a,, R2 are closed, by using the compactness of A; then show that DZ are meager subsets of 9(R"; Rz' l), by covering A with a finite number of charts, and using Sard's theorem.)

+

+

2.

Let M be a pure manifold of dimension n, and let (u&1 be a locally finite denumerable open covering of M such that each uk is relatively compact and is the domain of a chart (uk,pk, n) of M, where p k = (pi, ...,pi) is such that pk(uk) is the cube in R" defined by 18' 1 I < 4 for 1 s j 5 n. For each k, let v k be an open set such that v k C u k and the v k cover M. Let gk be a Cm-mappingof M into [0,1], with support contained in uk, and equal to 1 on v k .

-

m

(a) Show that the function uo =

k=l

kgk is of class Cmand is a proper mapping (Sec-

tion 12.7, Problem 2) of M into R+ . (k 2 1, 1 6 i $ n ) ar(b) Let (fdh)hZl denote the sequence of functions gk and ranged in some order. For each x E M, let u(x) denote the point of E = R") whose coordinates (all but a finite number of which are zero) are the uh(x)for h >= 0. Show that u is injective. (C) k t Ak (resp. B,) denote the union of the uh (resp. t h ) for h 5 k. Then U(&) 3 U(&) is contained in a vector subspace Ek of E, of finite dimension which we may assume to be 2 2 n 1. Show that the restriction of u to Ak is an embedding of Ak in & . (d) Let E'+ denote the topological product RT, and let F+ = E Y + l . An element v E F+ is therefore of the form v = (vl, .. ., V Z . + ~where ), vj = (tj,,),,~~, the sequence (.fJ;h)htOof numbers 2 0 being arbitrary. We may identify u with the linear mapping of E into RZ"+'which maps the point ( q & O of E (in which all but a finite number of

+

F+ consisting of the v E F, such that the restriction of u to u(Ak)is an immersion at each point of u(Bk),and such that the restriction of v to u(Bk) is injective. Use Problem 1 to show that Rkis a dense open subset of F+ . Deduce that the intersection of the 0, is dense in F+ . In particular, there exists w = ( w l , . . .,w ~ ~in +this ~ intersection ) such that the.coordinate of index 0 in w1 is fO. Show that the mappingf= w u is aproper embedding of M in R2"+'(Whitney's embedding theorem). 0

25 EMBEDDING A N D APPROXIMATION THEOREMS

189

3. Let M be a submanifold of dimension n in R". Show that for each point a E M there exists a number E > 0 with the following property: for each b E B n M, where B is the open ball in R" with center a and radius E , the orthogonal projection of B n M on the linear manifold Lb tangent to M at b (16.8.6) is a bijection of B n M onto a convex open subset of Lb . (Reduceto the case where a = 0 and T,(M) is the subspace Rnof R" generated by the first n vectors of the canonical basis (el)'6 1 s m , so that in a neighborhood of the origin M is defined by n - m equations ...,& (1 5 n - m).k t u l (1 5 i 5 n) be the vectors in Tb(Tb(M))which project orthogonally onto the el (1 5 i 5 n). Orthonormalize the sequence (ul, .. .,u., en+,, . ..,em), thus obtaining an orthonormal basis (vJl of R" whose elements are C"-functions of b, and the first n vectors of which form a basis of Tb(Tb(M)). Using the impkit function theorem, show that if E is sufficiently small, B n M is identical with the set

@+'=fJ(t',

sj

of points

m

'=1

71vL,where the point y = (T'),

S16m

runs through an open set Hb in

R", and the q"+*= Fl(y,b) for 1 5 i 5 m - n are C"-functions of (y, b) in a neighborhood of the origin in R" x M. Finally show that if E is sufficiently small, the funcn

tion G(y)= C (7'- at)' 1=1

+I=C (Fj(y,b) - F,(a, b))' m--n

1

is convex in a neighborhood of

0 in R",where a = (a')E R" is sufficiently small.) 4.

5.

Let M be a pure manifold of dimension n. Show that there exists a locally finite denumerable open covering (Ar) of M such that every nonempty intersection of a finite number of the sets Ar is diffeomorphic to R".(Use Problems 2 and 3.) Let X be a pure submanifold of R", and let T : X

-+

B be a surjective submersion. Let

Y be the graph of T,which is a submanifold of X x B diffeomorphic to X, and consider Y as a submanifold of R" x B. Show that there exists an open neighborhood T of Y in Rm x B and a submersion p of T onto Y with the following property: for each 6 E B and y E T n pr;'(b), p ( y ) is the only point of Y n pr;'(b) whose distance from y in R" x {b} is equal to d ( y , Y n pr;'(b)) (dbeing the Euclidean distance on R" x {b}). (Use (16.7.4).)

6.

Let M be a pure differential manifold of dimension n and f a continuous mapping of M into R". Let F be a closed subset of M such that the restriction o f f t o F is of class C' (where r is an integer >0, or +a)in the following sense: at each point x E F there , such thatfo 9-l is of class C' at q ~ ( x )in , the sense of Section 16.4, is a chart (V, q ~ n) Problem 6. Let S be a neighborhood of the graph off in M x R". Show that there exists a mapping g : M +R" of class C, which coincides withfon F and is such that ( x , g(x)) E S for all x E M. (We may assume that M c R'"+' and extendfto R'"+' by the Tietze-Urysohn theorem. Let S' be a neighborhood of the graph offin R2"+'x R" such that S' n ( M x R") c S, and for each x E R'"+' let r ( x ) be the distance from (x,f(x)) to CS'. Show first, using Weierstrass' theorem and a partition of unity, that there exists a mapping h : R'"+' 4 R m of class C" such that Ilf(x) - h(x)ll< ar(x). Using Whitney's extension theorem (Section 16.4, Problem 6) show that there exists a mapping u : Rz"+'+R" of class C' which is equal to f- h on F and is such that Ilu(x)ll< t r ( x ) for all x.)

7. Let (X, R, T)be a fibration, F a closed subset of B, s a continuous section of X over B. Suppose that s is of class C' (r an integer >0, or f a ) in F,in the following sense: for

190

XVI

DIFFERENTIAL MANIFOLDS

each b E F there exist charts (U, p, n) of B and (V, $, m) of X such that s(U) c V and such that $ s 0 q~-' is of class C' at p(b) in the sense of Section 16.4, Problem 6. Show that for each neighborhood S of the graph of s in B x X, there exists a section s1 of X over B, of class C' in B, such that (s(b), sl(b)) E S for all b E F.(Embed X in R" (for some m) and apply the results of Problems 5 and 6.) 0

8. A Co-fibration h = (X, B, T)is defined by replacing, in the definition (16.12.1), dif-

ferential manifolds by topological spaces, and diffeomorphisms by homeomorphisms. X i s also said to be a Co-fiber bundle. If (X, B, T)is a fibration in the sense of (16.12.1), it defines a Co-fibration (called the underIying Co-fibration) by regarding X and B as topological spaces. Generalize to Co-fiber bundles the definitions and results of Section 16.12. Define likewise the notion of a Co-principd bundIe and a Co-vectorbundle, and generalize the definitions and results of Sections 16.14-16.19. Show that if a vector bundle E (in the sense of (16.1 5)) is such that the underlying Co-vector bundle is Co-trivializable, then E is trivializable. (Use Problem 7.) Show likewise that if two vector bundles (in the sense of (16.15)) E, F over B are such that the underlying Co-bundles are isomorphic, then E, F are isomorphic. (Consider the bundle Hom(E, F) and use (16.16.4)J

9.

Let (X, B, n-) be a Co-fibration. It is said to have the section extensionproperty if every continuous section of X over a closed set A which is the restriction of a continuous section of X over an open neighborhood of A is also the restriction of a continuous global section of X (which need not agree with the preceding one on the open neighborhood of A). Suppose that B is separable, metrizable, and locally compact. Let (U.) be an open ) the section covering of B such that for each a the fibration induced on T - ~ ( U =has extension property. Then (X,B, T) has the section extension property. (Let A be a closed subset of B, and let Vo be an open neighborhood of A which is equal to the set of points a t which a continuous mapping go : B [O, I ] is strictly positive, the function go being equal to 1 at all points of A. Let (Vn)nzl be a locally finite open covering of B, and let (g.).21 be a continuous partition of unity such that V. = { x E B :9.h) > 0) for all n 2 1. Then the functions ho = g o and R, = ( 1 - go)gm (n 2 1 ) form a continuous partition of unity. Let s be a continuous section of X over A which extends to a continuous section of X over VO. Proceed by induction on n by introducing the functions fn = ho hI * h., and reduce the problem to the following one: Let u,v be two continuous functions on B with values in [0, I], and let U, V be, respectively, the open sets on which u(x) > 0, u(x) > 0; suppose that T - ' ( V ) has the section extension property. At the points x E U n A (resp. V n A) we have u(x) = I (resp. u(x) = I ) and there is a continuous section s over A such that s((U n A) extends to a continuous section s' over U. Show that sI ((U U V) n A) extends to a continuous section s" over U u V. For this purpose, consider the function w on U u V which is equal to 1 at points x such that u(x) u(x), and equal to u(x)/v(x) otherwise; observe that the set V n w-'(I) is closed in V, and extend the section which is equal to s' in V n w - I ( I ) and to s in V n A, to a continuous section over V. --f

+ + + - 7

10.

Let M be a pure differential manifold, and (U.) an open covering of M.Show that there exists an integer N, depending only on the dimension of M,and a denumerable locally finite refinement (V.) of the covering (Urn), consisting of relatively compact open sets V., such that no point of M belongs to more than N of the sets V.. (Embed M in some

25 EMBEDDING AND APPROXIMATION THEOREMS

191

Rm (Problem 2) and consider the U, as the intersections of M with open sets UL in Rm;this reduces us to considering only the case M = R". Let K,be the closed cube in Rmdefined by lz'l 6 n for 1 6 j 5 m ; decompose each set K. - K,-l into equal closed cubes, sufficiently small that each one is contained in some U: (3.16.6); then enlarge each of the cubes slightly to a concentric open cube.)

11. Let (gn).ro be a continuous partition of unity in a separable, metrizable, locally compact space B. Let V,, be the set of b E B at which g.(b) > 0. For each finite subset J of N let W(J) be the set of all b E B such that gi(b) > gi(b) for all i E J and j $ J. If J and J' are two distinct subsets of N having the same number of elements, then W(J) n W(J') = 0.For each 6 E B, let J(b) be the set of integers n such that gn(b)> 0, so that J(b) is finite. Then W(J(b)) C V. for all n E J(b). For each integer m > 0, if W, is the union of all the sets W(J(b)) such that J(b) has m elements, the W, form an open covering of B. Using these results and Problem 10, show that if B is a pure differential manifold and X a fiber bundle over B, then there exists afinite open covering (WJlsiSm of B such that X is trivialiable over each W r. Generalize Problem 5 of Section 16.19 by replacing the relatively compact open subset U of B by B itself. 12.

(a) Let M be a differential manifold of dimension n, let f : M +RP be a Cm-mapping, and let N be a compact subset of M such that f is of rank n at each point of N. Also let v be a chart of M with domain W, let V be a relatively compact open set such that 0 c W, let U be a relatively compact open set such that 0 c V, and let u : R"-+ [0,11 be a C"-mapping such that u(z) = 1 for z E @) and u(z) = 0 for z $ @). Suppose that p 2 2n. Show that for each E > 0, there exists a p x n matrix A such that: (1) IIA .I(( 5 E for all z E v(V); (2) the function ZH~(V-'(Z)) + A .z is of rank n in y(V); (3) the function ZH~(P)-~(Z)) u(z)(A .z) is of rank n in v(N nv). (Use Problem 5 of Section 16.23.) Deduce that the function g : M +RP defined by g(x) = f(x) for x $9,g(x) = f(x) u(v(x))(A . cp(x)) for x E W, is of class C", is of rank n at each - f(x)/lg E for all x E M. point of N u 0, and is such that (b) Let M be a differential manifold of dimension n, let f : M + Rpbe a C"-mapping. and let N be a compact subset of M such that f is of rank n at each point of N. Let 6 be a continuous real-valued function on M, everywhere >O. If p 2 2n, show that there exists an immersion g : M + RP which agrees with f on N and is such that Ilg(x) - f(x) I[ 2 6(x) for all x E M. (There exists an open set T 3 N in which f is of rank n. Consider a denumerable locally finite open covering (W,) of M such that the W, are domains of definition of charts and are contained in either T or M - N. Construct g by induction, using (a).)

+

+

13. (a) With the same notation as in Problem 12(a), suppose that f is an immersion, and that the restriction off to an open set S is injective. Put u(x) = u(~+-)), and suppose that p 2 2n 1 . Show that, for each E > 0, there exists a vector a E Rp such that: (1) /lal/5 E ; (2) the function xt+f(x) o(x)a is an immersion of M in RP;(3) therela tion g(x) = g(y) implies v(x) = o(y) and f(x) = f(y). (To show that the third condition can be satisfied, consider the open set D in M x M consisting of pairs (x, y ) such that u(x) # u(y), and the image of the mapping (x, y ) - (f(x) ~ - f(y))/(o(x) - a(y)) of D into Rp.) (b) Let M be a pure differential manifold of dimension n, let f : M +RP be an immersion, let S be an open subset of M such that f 1 S is injective, let N be a closed subset

+

+

192

XVI

DIFFERENTIAL MANIFOLDS

of M contained in S, and let 6 be a continuous real-valued function on M, everywhere >O. If p 2 2n 1, show that there exists an embedding g : M + Rp which agrees with f on N and is such that Ilg(x) - f(x)ll5 8(x) for all x E M. (Consider a denumerable locally finite open covering (W,) of M, where the W, are domains of definition of charts, the restrictions f I Wk are injective, and each Wk is contained either in S or in M - N. Construct g by induction on k, using (a).) (c) Deduce from Problems 12@) and 13(b) a new proof of Whitney's embedding theorem. (Problem 2; first define a proper mapping of class C" of M into R2"+'.)

+

+

Let M be a manifold of dimension n. Show that there exist 2n 1 real-valued functionsf, of class C" on M (I zj 5 2n 1) such that every C"-function on M is of the form F(fl, .. ,fZn+J, where F is a C"-function on R2"+'. (Use Whitney's embedding theorem.) Show that the B(M)-module B,(M) of C" differential p-forms on M is generated bythep-forms df,, A ... A df;, (1 6 jl

+ (1 - tlg(x))E y ,

then it is clear that cp is a homotopy of g into f. For by virtue of (3.17.11) applied to cp-'(T), there exists a > 0 such that tf(x) (1 - t)g(x) belongs to T for -a < t < 1 tl and all x E K. Moreover, iff1 K and g I K are of class Cp, then the restriction of cp to K x ] -a, 1 a [ is of class Cp, from which the second assertion follows.

+

+

+

Remark (16.26.4.1) The above proof shows that if f ( x o ) = g(xo) for some xo E K, then the homotopy cp constructed above is such that cp(x, , t ) = f ( x o ) for all t E [O, I]. (16.26.5) Let X , Y be diferential manifolds, K a compact subset of X , and f a continuous mapping of K into Y . Then there exists a relatively compact open neighborhood U of K in X and a C"-mapping g of U into Y such that g I K is homotopic to j:

Let d be a distance defining the topology of Y , and let E > 0 be a real number for which (16.26.4) holds. By virtue of (16.25.5) there exists a relatively compact open neighborhood U of K in X and a C"-mapping g of U into Y such that dcf(x), g(x))5 E for all x E K. The result now follows from (1 6.26.4). (1 6.26.6) Let X , Y be diferential manifolds, K a compact subset of X , andf, g homotopic continuous mappings of K into Y . I f f I K and g I K are of class Cp,

then they are CP-homotopic.

By hypothesis, there exists a compact interval I = [a,81 in R and a continuous mapping cp : K x I + Y such that cp(x,a) =f ( x ) and cp(x,p) = g ( x ) for all x E K. Choose E > 0 for which (16.26.4) holds for X, Y , f , and K ; for X, Y , g , and K ; and for X x R, Y , cp, and K x I. Then there exists an open neighborhood U (resp. J) of K in X (resp. of I in R) and a C"-mapping $ : U x J + Y such that d(cp(x, t), $(x, t)) 5 E for all ( x , t ) E K x I. If we put f i ( x ) = $(x, a) and g,(x) = $(x, p) for x E U, then $ is a C"-homotopy of f i into 9 , . However, we have d(fi(x),f(x)) E and d(g(x),g,(x)) E ; hence b y (16.26.4) it follows that f 1 K and fi I K are CP-homotopic, and that g , I K and g1 K are CP-homotopic. Hence, by (16.26.3), f I K and g I k are Cphomotopic.

196

XVI DIFFERENTIAL MANIFOLDS

Iff, g are two diffeomorphism of a differential manifold X onto a differential manifold Y, a C"-homotopy cp o f f into g , defined on X x J, where J is an open interval in R,is said to be a C"-isotopy (or an isotopy of class C") if for each t E J the mapping x Hcp(x, t ) is a diffeomorphism of X onto Y. The same argument as in (16.26.3) shows that the relation "there exists a C"-isotopy off into g " is an equivalence relation on the set of diffeomorphisms of X onto Y. (1 6.26.7)

(16.26.8) Let X be a differential manifold, U a connected open subset of X , and a, b two points of U.Then there exists a C"-isotopy cp of the identity map l X into a direomorphism h of X onto X , such that h(a) = b and such that cp(x, t ) = x for all t and all x # U.

(I) We consider first the following particular case: X ball llxll
0, put gk(X) = k"g(kx),so that g k ( X ) dA(x) = 1. The sequence (gk) is said to be a regularizing sequence. Since the support of gk is k-'l", for fixed q the support of each of the functions vks= gk * us is compact and contained in U for all sufficiently large k ((3.18.2) and (14.5.4)). Let us show that the functions U k , with support contained in U form a dense set in 6(U) (resp. in each cV(U)). Since Ukq(x)= /gk(X - y)u,(y) d ~ ( y )it, follows from (13.8.6) that I)kq is of class C" and that, for each multi-index v ,

s

(17.1.2.1)

1 THE SPACES g(r)(U)

235

Fix a real number E > 0 and two integers m and r. We shall show that for each f~ B(U) it is possible to find a ukq such that I D’(f-ukq!(x) 1 5 e for all

s

x E K, and all v such that I v I r ; this will prove the assertion above. Let F be a C“-function which is equal to f on K,+, and has support contained in K,+, (16.4.3). By (13.8.6) the functionf, = F * gk is of class C“, and we have

It follows therefore from (14.11.1) that for sufficiently large k we have I D’(fk - f ) ( x ) I 5 j e for all x E K, and I v I 5 r. On the other hand, if k is sufficiently large, then for all x E K, we have, by (3.18.2) and (14.5.4)

I D‘(f, - Vkq)(x)I =

Is. +.

Dvgk(x - y)(F(y) - uq(Y)) dA(Y)

5 NI(D”Sk)

SUP

ysKm+i

I

I F(Y) - U,(Y) I.

Now, for a fixed and suitably large k, we can by hypothesis find a large q so that the right-hand side of this inequality is S+Efor 1 v I 5 r. This completes the proof of (i) for b ( U ) ; the proof for @)(U) is similar. (ii) If a sequence (f,)is bounded in b ( U ) , then each of the sequences ( D i f , )(1 6 i S n) is uniformly bounded in each K,, hence it follows from (7.5.1) and the mean value theorem that (f,)is equicontinuous. By definition of the bounded sequences in b(U), it follows that for each multi-index v the sequence (Dvfk)is equicontinuous. By applying Ascoli’s theorem (7.5.7) and the diagonal procedure (cf. 12.5.9) we can therefore find a subsequence (fk,) such that each of the sequences (DYfkk,) converges uniformly on each of the K, . In other words, the sequence (A,) converges in b(U),and this proves (ii). (17.1.3) For each multi-index v, the linear mapping f is continuous.

For pS,,(D”f) 6 ps+ (17.1.4)

H D’fof

b ( U ) into b ( U )

,,(f),from which the assertion follows (12.14.11).

For each function g

E

b ( U ) (resp. g E SCr’(U)),the linear mapping

f~ f g of’E(U) (resp. cV(U)) into itself is continuous.

For each pair of integers s, m (resp. s 5 r and m), let as,,, be the greatest of the least upper bounds of the I D’g I on K, for I v I s. By Leibniz’ formula

236

XVll

DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

(8.13.2), there exists a number cs,,,,independent off and g such that ps, ,,,ug)S cS,,,,aS,,,,ps,,,(f). Hence the result (12.14.11). (17.1.5) Let cp be a mapping of class C" (resp. Cr) of an open set V c R" into U. Then the linear mapping f wf 0 cp of B(U) into B(V) (resp. of &)(LJ) into s(')(V)) is continuous.

Put cp = (ql,cp2, ..., cp,), where the cpj are scalar-valued functions. Let (KL) be a fundamental sequence of compact subsets of V used to define seminorms pi, ,,,on &(V) as in (17.1.I). For each pair of integers s, m, let a:, ,,,be the greatest of the least upper bounds of the functions sup(1, I D"cpjI) on KL for I v I 5 s and 1 sj 5 n. Finally let q be an integer such that cp(Kk) c K,. By repeated application of the formula for the partial derivatives of a 'composite function (8.10.1) we obtain, for each f E B(U) ~ sm ,( f

where c:,

0

CP) 5 4,rn

~ s q ,( f L

,,, is a constant independent off and cp. This proves the proposition.

2. SPACES O F C"- (resp. C'-) SECTIONS O F VECTOR BUNDLES

Let X be a pure differential manifold of dimension n and let (E, X, n) be a complex vector bundle of rank N (16.15) over X. If U is an open subset of X, we recall (16.15) that the complex vector space of Cm-sectionsof E over U is denoted by T(U, E). Likewise we shall denote by T(')(U, E) the vector space of Cr-sections of E over U. We propose to generalise the results of '(17.1) to these spaces: b(U) is the space T(U, E) when X = R" and E = X x C is the trivial complex line bundle over X. Since in general we cannot attach a meaning to the notion of partial derivatives of a section of E, we shall reformulate the problem as follows: we have to show that the space T(U, E) (resp. rcr)(U,E)) can be endowed with the structure of a Hausdorff locally convex topological space, defined by a sequence of seminorms and having the following property: (**) A sequence (uk)of sections of T(U, E) (resp. T(')(U, E)) converges to 0 ifand only if, for each chart (V, cp, n) of X over which E is triuializable, each difeomorphism

zH(cp(n(z)),

vl(z),

vN(z))

of n-'(V) onto q(V) x C N, where the v j are linear on each fiber n-'(x), each compact K c cp(V) and each multi-index v (resp. such that I v I 5 r), the sequence

2 SPACES OF Cm- (resp.Cr-) SECTIONS OF VECTOR BUNDLES

((D'wjk)I K ) k t uj(uk((P-'(t)))for

converges uniformly to 0 for 1 S j S N, where

237

wjk(t)

=

(Pv).

Here again, by virtue of (3.1 3.14), the topology is necessarily unique. To establish existence, consider an at most denumerable family of charts (V,, cp, n) of U such that the V, form a locally finite open covering of U, and such that E is trivializable over each V, (12.6.1). For each u, let z H ( ( P a ( 4 Z ) ) , vla(z),

u,a(z))

* * *9

be a diffeomorphism of n-'(V,) onto q(V,) x CN,the via being linear on each fiber n-'(x). Let (KmO;)mnhl be a fundamental sequence of compact subsets of cp,(V,), and let p i , , , , , be the corresponding seminorms (17.1 .I) on &(cpa(Va)) (resp. &(r)(cpa(Va))). For each section u E T(U, E) (resp. u E T(")(U,E)) we define (17.2.1)

where u, is the restriction of u to V,. It is clear that the ps, are seminorms, and that if p o , ,,,,(u) = 0 for all m and all a, thenu(x) = 0 in eachV, and therefore u = 0. To show that the topology defined by these seminorms satisfies the condition (**), we have to show that if ( u k ) tends to 0 in the topology defined by the seminorms ps,,,,,, then the DYwjkconverge uniformly to 0 on K (in the notation of (**)). The compact set cp-'(K) meets only a finite number of the open sets V, ;by applying the Borel-Lebesgue axiom to the union of these V,, it follows that there exists an integer m and indices uh (1 h 5 q) such that the C~&~(K,,,,) cover q-'(K). If w h j k is the restriction of w j k to cp(V n V,,), then clearly it is enough to show that the restrictions of the D"whjkto K n cp(cp;'(K,,,)) converge uniformly to 0. Now, for each z E n-'(V n V,,), we can write N

=

vj(z)

1

I= 1

cIjh(.rr(z))ula,(z),

where the C l j h are C"-functions on V n V,, . Let $h : dv va,> qah(V be the transition diffeomorphism. Tf we put -+

Uklk

= Olah

then we have

('k

I vah)

vah)

q,',

N whjk(t>

=

CIjh((P-'(f))UhZk(~h(t)), I= 1

and the assertion is now a consequence of (17.1.4) and (17.1.5), because each sequence ( u , l k ) k z 0 converges to 0 in the space d(cp,,(V,,)) (resp. &(r)(qah(~ah))).

238

XVll

DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

Equivalently, we may say that a sequence (uk) converges to 0 in T(U, E) if and ody if, for each a, the sequence of restrictions ttk I V, converges to 0 in Tor,, E); and T(V,, E) is isomorphic to (8(cp,(Va)))N,so that we are brought back to the spaces &(U).

(17.2.1.1)

The spaces T(U, E) and l?)(U, E) are separable Frichet spaces. Moreprecisely, there exists a sequence of Cm-sectionsof E over U, with compact support contained in U, which is dense in each of the spaces T("(U, E) and T(U, E). Euery bounded subset of T(U, E) is relatively compact in T(U, E). (17.2.2)

Let (Uk) be a Cauchy sequence in T(U, E). By virtue of (17.1.2), there exists for each a a section f, E T(V,, E) which is the limit of the sequence (UkIV,), and for any two indices a, fi the restrictions of f, and fBto V , n V, coincide, so that the fa are restrictions of a section f E T(U, E). By the definition of the topology of T(U, E), the sequence (uk)converges to f. In the same way we may show that T(')(U, E) is complete. To prove the second assertion, we remark that it follows from (17.1.2) that for each a there exists in T(V,, E) a sequence (Uk,) of sections with compact support contained in V,, which is dense in T(V,, E) and in each r@)(V,,E). If (g,) is a (?-partition of unity subordinate to the covering (VJ, then the sections which are linear combinations of the gaUk, are dense in T(U, E) and in each T"'(U, E). For each compact K c U meets only a finite number of sets Supp(g,), say those with indices ah (1 5 h 4 4);for each section u E T(U, E), the restriction of u to K therefore coincides with the restriction to K of 4 h= 1

g,, u, and by virtue of (17.1.2) and (17.1.4), each section gahu may be

approximated arbitrarily closely by the g,, Ukah ; whence the assertion follows. Finally, if (Uk) is a bounded sequence in T(U, E), then it follows from (17.1.2) that for each u there exists a convergent subsequence of the sequence (ukI V,) in T(V,, E). By applying the diagonal procedure, we obtain a convergent subsequence of the sequence (uk). Remark (17.2.2.1) The last part of the proof may be used, more generally, to show that a vector subspace H of T(U, E) is dense in T(U, E) provided that, for each v E H, the sections gav also belong to H; it is eqough to verify that, for each a, the restrictions to V, of the sections belonging to H form a dense set in T(V,, E). (17.2.3) If F is another complex vector bundle over X, u,, an element of T(U, E) and vo an element of T(U, F), then the linear mappings ti Hu 63 vo

2 SPACES OF Cm-(resp.C'-) SECTIONS OF VECTOR BUNDLES

and

VHU~

239

@ v of T(U, E) and T(U, F), respectively, into T(U, E @F)

are continuous. Likewise, if wX

E

T(U,

P

A E*), then

u++i(u)w,* and w * ~ i ( u ~ ) of w *T(U, E) and T(U,

the linear mappings

A E*), respectively, into

r(U,'AE*) are continuous. The same is true of the mappings SHE tHsOA t of T(U,

A

A

A to and

E) and T(U, E) into T(U, T E ) . All these propositions follow froin (17.1.4) by virtue of the definitions of (16.18), by reducing to the case where E is trivial, as we may assume by use of the general principle stated in (17.2.1 .I).

PROBLEMS

Let M,N be two differential manifolds. For each integer r 2 I , let B(')(M; N) denote the set of all C'-mappings of M into N. (a) LetfE B(')(M; N). Then there exists a denumerable locally finite covering of M by compact sets K., such that each K. (resp. each f&)) is contained in the domain of definition of a chart (Ua, 'p., ma)of M (resp. a chart (Ve,&, n.) of N). For each E > 0 and each a,let W(f. K., QJ=, (CI., E ) denote the set of all CI-mappings g : M 4N such that d K J c V. and IIDY& 0 (flId)0 cp;')(x) - DV& (sl K.) 0 p:l)(x)ll 4E for all x E va(K,) and I Y I 5 r . Show that there exists on B(')(M; N) a unique topology Yc (called the coarse C'-topology) such that, for each f~ B(')(M; N), the finite intersections of the sets W(f,K., cp#, #,, 1 ln) (where the K., cpm, &, are fixed and n is variable) form a fundamental system of neighborhoods off(cf. Section 12.3, Problem 3). Show that the topology so defined is independent of the choice (for eachfe B(')(M; N)) of the families (&), (&, ($,) satisfying the stated conditions. (b) If N is a vector bundle over M, show that 37 induces on r(')(M; N) the topology defined in (17.2). (c) If N is a submanifold of a differential manifold P,then B(')(M; N) is a subset of 6(')(M; P). Show that the topology induced on LP(')(M; N) by the coarse Cr-topology of B(')(M; P) is the coarse C'-topology. (d) Let M, N, P be three differential manifolds. Show that the mapping (f,g ) ~ og f of 8(*)(M;N) x 8(')(N; P) into b(')(M; P) is continuous with respect to the coarse CI-topologies on these three spaces. 0

(a) With the notation of Problem 1, for each family (6.) of numbers 6. > 0, let WCf, (6.)) denote the intersection of the sets W(f, K., cpa, $*, 6.) for all a.Show that there exists on B(')(M; N)a unique topology Y f(called the fine CI-topology) such that, , are fixed and the 6, for each f e b(')(M; N), the sets W(f, (6.)) (where the K., g ~ =(6. variable) form a fundamental system of neighborhoods off. Show that the topology so defined is independent of the choice (for each f~ b ( ' ) ( M ; N)) of the families (Id), (v,,), (k)satisfying the stated conditions.

240

XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

(b) If N is a submanifold of a differential manifold P, then &(')(M; N) is a subset of B")(M; P). Show that the topology induced on B(')(M; N) by the fine C-topology on 8')(M; P) is the fine C'-topology. (c) Give an example of an element f~ B(')(M; N) for which there does not exist a denumerable fundamental system of neighborhoods of f for the fine C1-topology (take M = N = R). (d) Suppose that fo E b(')(M; N) is a homeomorphism of M ontofo(M) and thatf, is of rank equal to dim,(M) at each point x E M. If P is another differential manifold, show that for eachgo E B(')(N; P) the mapping ( f , g ) ~ofof g B(')(M; N) x &(')(N; P) into B(')(M; P) is continuous at the point (fo ,go) with respect to the fine C-topologies. (e) With the notation of (d), give an example in whichfo is of rank equal to dim,(M) at each point x of M and is injective, but is not a homeomorphism of M ontofo(M), and the mapping g H g o f o fails to be continuous at some point go with respect to the fine C'-topologies (cf. (I 6.8.5)). (f) Suppose that fo E @)(M ;N) is of rank equal to dim,(M) at each point x E M. Show that there exists a neighborhood Vo of fo in B(')(M; N) for the fine C-topology such that each g E Vo is also of rank equal to dim,(M) at each x E M. If moreover fo is injective, show that there exists a neighborhood V1 = W(fo ,(6.)) of fo contained in Vo such that the restriction of each g E V, to each K, is injective (argue by contradiction). Give an example in which fo is injective and such that there exist noninjective mappings of M into N in each neighborhood of fo for the fine C'-topology (cf. (1 6.8.5)). If, however, fo is a homeomorphism of M onto fo(M), then there exists a neighborhood V2 of fo contained in V, (for the fine C-topology) such that each g E V, is a homeomorphism of M onto g(M). (Consider first a covering of M by compact sets L. such that L a c K, for all F. If d is a distance defining the topology of N, show that d(fo(L.), fo(M - K.)) > 0, and deduce that for a suitable choice of the family (a:), the functions g E W(fo, (6;)) C V, are injective. Put L(f0) =fo(M) -fo(M). By considering sequences (x,) in M with no cluster values, show that if g E V, and if the sequence g(x.) converges, then its limit must belong to L(fo). Finally remark that the distance from fo(K,) to L(fo) is >0, and use this remark to construct a neighborhood Vz c W(fo ,(6;)) such that g(M) n L(fo) = 0 for all g E V2.) If moreoverfo(M) is closed, then so is g(M) for each g E V2 . Finally, if M and N are connected and iffo(M) = N, theng(M) = N for all g E V, . 3. (a) Let U be an open subset of R",A a compact subset of U, and V a neighborhood of A such that V C U. For each C-function f on U and each E > 0, show that there exists a C-mappingf, : U + R which coincides withfon U - V and is of class C" on a neighborhood of A, and such that I D'fi(x) - D'f(x) I 6 E for all x E U and all v such that 1 Y I 5 r. (Let g be a C'-function equal tofon a compact neighborhood of A contained in V, and zero outside this neighborhood; take a suitable regularization h of g and consider the function f, =f+ (h - g).) (b) Let M, N be two differential manifolds. Show that for each functionf, E Q(')(M; N) and each neighborhood W(fo, (6.)) of fo in the fine C-topology (Problem 2), there exist C"-mappings of M into N in this neighborhood (proceed by induction as in (16.1 2.1 I), using (a) above). 4.

Show that in B(')(M; N)the set of mappings which are transversal over a submanifold

Z of N is a dense open set in the fine C'-topology (cf. Section 16.25, Problem 16).

3 CURRENTS AND DISTRIBUTIONS

241

3. C U R R E N T S AND D I S T R I B U T I O N S

(17.3.1)

Let X be a pure differential manifold of dimension n, and consider

(A

T(X)*)(,, on X, whose global sections are the the complex vector bundles complex-valued dzfferential p-forms on X (1 6.20.1), for 0 5 p 5 n. For p = 0, these are by definition the complex-valued functions. For brevity we shall denote by bz)(X) (resp. b,(X)) the Frtchet space T'"(X,

(

( A T(X)*)(,,)

(resp. T(X, ),T(X)*)o,)) of complex differential p-forms of class C' (resp. Cm) on X. For each compact subset K of X we denote by 9g)(X; K) (resp. g P ( X ; K)) the vector subspace consisting of the complex differential p-forms of class C' (resp. C") with support contained in K; this is clearly a closed subspace of 8g)(X) (resp. b,(X)). We denote by g;)(X), (resp. .9,(X)) the union of the subspaces @)(X; K) (resp. 9JX; K)) as K runs through all the compact subsets of X, i.e., the space of all complex differential p-forms of class C' (resp. Cm) with compact support. When p = 0 we shall drop p from these notations. We can now proceed exactly as in the definition of a measure (1 3.1), except that the Banach spaces X(X; K) are now replaced by FrCchet spaces. A p-current (or a complex pcurrent) (or a current of dimension p ) on X is by definition a linear form T on 9JX) whose restriction to each FrCchet space 9JX; K) is continuous; in other words (3.13.14), in order to verify that a linear form T on g P ( X )is a p-current, it must be shown that for each sequence (ak) of C" differential p-forms, with supports contained in the same compact set K. and which converges to 0 in b,(X), the sequence (T(ak))tends to 0 in C. A 0-current on X is called a distribution. With the notation of (17.2.1), T is a p-current if and only if, for each compact subset K of X, there exist integers s, m and a finite number of indices a l , . .. , a,, together with a constant aK 2 0, such that, for each C" p-form o with support contained in K, we have (17.3.1 .I)

IT(o) I 5 aK

* I

ps,rn,

(17.3.2) Suppose that a p-current T is such that, for each compact subset K of X, the restriction of T to gP(X;K) is continuous with respect to the topology induced by that o f g ; ) ( X ; K). In that case T is said to be a p-current of order I-r . The order of a current is the smallest integer r with this property (when such integers exist). If there exists no integer r with this property, then T is said to be a current of infinite order. It is clear that if T is the restriction to

242

XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

gr)(X)

gP@)of a linear form T' on whose restriction to each 9r)(X;K) is continuous, then T is of order s r . Conversely, we shall show that if T is of order S r , then it is the restriction of such a linear form T', which moreover is unique. Let K be a compact subset of X and let K be a compact neighborhood of K in X. Then there exists a C"-function h which is equal to 1 on a compact neighborhood of K and which is 0 on X - K (16.4.2). For each pform b E S%g)(X;K), there exists a sequence ak in gP(X) which converges to /3 with respect to the topology of &$)(X)(17.2.2). The sequence (hak),which belongs to gP@; K'), therefore also converges to /I in @)(X; K') by (17.1.4). In other words, the closure of 9,(X; K ) in Sr)(X) contains 9$')(X; K) and is contained in 9r)(X;K). The existence and uniqueness of T' now follow immediately by applying (12.9.4). We shall often write T in place of T'. Examples of currents (17.3.3)

Let x be a point of X, and let z, be a tangent pvector at x (i.e., an

element of A T,(X)). Then the mapping ct H ( 2 , ,a(x)) is a continuous linear form on S$')(X), hence is a p-current of order 0; it is called the Diruc pcurrent defined by z,, and is sometimes denoted by E=*. When p = 0, we obtain the scalar multiples of the Dirac measure (13.1.3). P

(17.3.4) It follows from (17.3.2) that distributions of order 0 on X are continuous linear forms on each of the Banach spaces 9io)(X; K) = X(X; K), and are therefore precisely the (complex) measures on X. (17.3.5) Let T be a p-current and w a C" differential q-form (i.e., an element of S,(X)), with q j p. For each (p - q)-form B E BP-,(X; K), we have w A /I E O p ( X ;K), and it follows therefore from (17.2.3) that the linear form B -T(w A B) is a (p - q)-current, which is denoted by T A w. If T is of order s r , then so is T A U ,and in this case we can also define T A O when w is a differential q-form of class C'. When q = 0, so that w is a complex-valued function g, we write T * g or g T in place of T A w ;if T is a measure, this definition agrees with that of (13.1.5), because of the fact that the closure of 9(X; K')in VC(X) contains X ( X ; K) for each compact neighborhood K of K.

-

(17.3.6) Suppose that p > 0, and let Y be a C" vector field on X.For each pform a E 9,(X; K), we have iy a E 9 p - , ( X ; K) (16.18.4). For each (p - 1)current T, it therefore follows again from (17.2.3) that the linear form a t+T(iy * a) is a pcurrent, which is of order j r if T is of order j r, and which is denoted by 'iy * T.

-

3 CURRENTS AND DISTRIBUTIONS

243

(17.3.7) If X, Y are two locally compact metrizable topological spaces, a continuous mapping u : X -,Y is said to be proper if for each compact subset K of Y,the inverse image u-'(K) is a compact subset of X. It then follows that if F is any closed subset of X, its image u(F) is closed in Y. To see this, let (y,,)be a sequence of points in u(F) converging to a point y E Y. Then the set K consisting of the y,, and y is compact, and therefore F n u-'(K) is compact; choose for each n a point x,, E F n u-'(K) such that u(x,,) = y,,, then the sequence (x,,) has a subsequence (x,,,) converging to a point x E F. Since u is continuous, it follows that u(x) = y, that is to say y E u(F). Now let X, X be differential manifolds and u :X -,X' a mapping of class C', where r 2 1. If a' is any p-form on X' of class C"(s 2 0), then by the formula (16.20.9.3) the inverse image %(a') is defined and is a p-form on X of class C i n f ( r - 1 , s ) . ,moreover it is clear that Supp('u(a')) is contained in u-'(Supp(a')).

If we suppose that the mapping u is proper, it follows that for each compact subset K of X the mapping a'H'u(a') is a linear mapping of 9r-')(X'; K) into 9f-')(X; u-'(K)). Furthermore, this mapping is continuous; this follows immediately from (17.2), the local expression of 'u(a') (16.20.9.2), and (17.1.4). Hence, for each pcurrent T of order S r - 1 on X, the linear form a'HT('u(a')) on 9r-"(X) is a p-current of order S r - 1; it is denoted by u(T) and is called the image of T by u. If u is a proper mapping of class C' of X' into another differential manifold X", then u u is proper of class C', and we have (u 0 u)(T) = u(u(T)) for each p-current T of order 5 r - 1 on X. If u is a diffeomorphism, then u(T) is defined for every current T on X and has the same order as T. If T is a distribution on X, then u(T) is the distribution on X' such that u(T)(g) = T(g u) for each functiong E @'-')(X'). But this formula makes sense also for functions g E &')(X); hence u ( T ) is defined also for distributions of order r. When u is a homeomorphism and T is a measure, we recover the definition of (1 3.1.6). If G is a Lie group which acts differentiably on X on the left (resp. on the right), the image of a current T under the diffeomorphism XHS x (resp. XHX * s) will be denoted by y(s)T (resp. 6(s-')T. When X = G, so that G is acting on itself by left translation (resp. right translation), and T is a measure, then 7(s)T (resp. 6(s)T) coincides with the measure so denoted in (14.1.2). If X is a differential manifold, Y a closed submanifold of X, then the canonical immersion j : Y -,X is proper; hence, for each p-current T on Y, the imagej(T) is defined and is a p-current on X. By considering the local expressions it is immediately verified that j(T) has the same order as T. For measures, this notion agrees with that defined in (13.1.7). 0

0

(17.3.8) The set of all p-currents on X forms a vector space, which we denote by 93b(X). The subspace of currents of order r is denoted by g;(')(X). When p = 0 we suppress p from the notation, so that 93'(X) denotes the space of

244

XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

distributions on X, and 9'(')(X) the subspace of distributions of order r. If a is a differential p-form and T a pcurrent, we shall often write or in place of T(a). For example, for a Dirac p-current (17.3.3), we have

a = (z,

9

44).

If T is a distribution, we write (T,f) or (f, T ) or even

(1 7.3.8.1)

in place of T(f), for a functionfe 9(X), whenever there is no risk of ambiguity. 4. LOCAL DEFINITION O F A CURRENT. SUPPORT O F A CURRENT

(17.4.1) Let U be an open subset of a differential manifold X. For each compact subset K of U, it is clear that the mapping a Ha I U is an isomorphism of QP(X; K) onto QP(U;K) (resp. of 9:)(X; K) onto 9$')(U; K)); the inverse isomorphism sends a differential p-form jE gP(U;K) to the p-form j"which is equal to fi in U and is zero in X - U. (By abuse of notation we shall often write a in place of a I U when u is a differential p-form on X with support contained in U.) For each p-current T on X, the mapping jHT(fl') is therefore a p-current on U; it is said to be the p-current induced by T on U, or the restriction of T to U, and is denoted by T, . The order of T, is at most equal to that of T, but it may be strictly less than that of T. It should be noted that a current on U is not necessarily the restriction of a current on X (Section 17.5, Problem 2), and that when it is the restriction of a current on X, the latter need not be unique. However, there is the following result, which generalizes (1 3.1.9): (17.4.2) Let (U,),EL be an open covering of X. For each 1 E L let T, be a p-current on U,, such that for each pair of indices 1,p the restrictions of T, and T, to U, n U, are equal. Then there exists a unique p-current T on X whose restriction to U, is equal to T, for each 1 E L.

We shall not write out the proof in full detail; it is based, step by step, on the proof of (1 3.1.9), with the obvious modifications. We begin by writing a p-form u E gP(X)as C ai , where ui = hi a and Supp(hi) c U,, for a suitable i

li; for this it is sufficient to invoke (16.4.2) instead of (12.6.4). It follows that T is unique and necessarily given by T ( a ) = T,,(ai), and the proof of the i

fact that this formula does define a linear form on BP(X)(in other words, that the number T(a)does not depend on the particular decompositionof a chosen)

4 LOCAL DEFINITION OF A CURRENT

245

goes over without change. It remains to show that if a sequence (ak)tends to 0 in QP(X; K), then T(ak)+ 0. We may take the same finite sequence (hi)for all the a k , and each of the sequences (TA,(hiak))kbl then tends to zero by virtue of (17.1.4); this proves our assertion. (17.4.3) It follows in particular from (17.4.2) that if the restriction of a current T to each member of a family of open sets U, is zero, then the restriction of T to the union of the U, is also zero. Hence there is a largest open subset V of X such that the restriction of T to V is zero; the complement S = V is called the support of T, and is denoted by Supp(T). A point x E X belongs to the support of a p-current T if and only if, for each neighborhood V of x, there exists a p-form a E gP(X)with support contained in V and such that T(a) # 0. If T,, T2 are two p-currents, it is clear that

c

SUPP(T1 + T2) = SUPP(T1)

“ Supp(T2),

and that if o E &,(X) is a q-form with q 5 p, then Supp(T A o)c Supp(T) n Supp(w). If n : X + X’ is a proper mapping of class C‘, then for any current T on X of order S r - 1 , we have Supp(n(T)) c n(Supp(T)). If n is a diffeomorphism of X onto X‘, then Supp(n(T)) = n(Supp(T)). If Y is a closed submanifold of X, and j : Y -+ X the canonical immersion, then Supp(j(T)) = Supp(T) for any current T on Y. When p = 0, we recover the definition of the support of a measure on X (1 3.19), by virtue of the fact that each space X ( X ; K) is contained in the closure of 9 ( X ; K’) in X ( X ; K’), whenever K’ is a compact neighborhood of the compact set K. (17.4.4) Let X, X‘ be two pure differential manifolds of the same dimension n, and let n : X’ + X be a local diffeomorphism(16.5.6). Then for each current

T on X there exists a unique current T’ on X’ with the following property: for each open subset U’ of X‘ such that the restriction nu, : U’ + n(U‘) is a diffeomorphism,we have nu,(TLt) = Tn(,,.). For there exists an open covering (U;) of X such that each of the restrictions nu;. is a diffeomorphism; if n ~ :is the inverse diffeomorphism, put Ti = X;:(T~(~;)). For any two indices I, p, the mappings nui and nu; agree by definition on U; n UL, hence n;; and n;: agree on n(U;) n n(UL). This implies that T i and TI have the same restriction to Ui n U;, and the existence and uniqueness of T’ therefore follows from (1 7.4.2). The current T’ is called the inverse image of T by n, and is denoted by ‘n(T). For example, if X‘ is a universal covering of X (1 6.29), then it follows from the definitions that the fundamental group of X leaves invariant the inverse image on X’ of every current on X. Conversely, every current T’ on X’ having this property is the inverse image of a current T on X. To see this, we take a

246

XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

covering (U,) of X by connected open sets over which X' is trivializable, and we define T, to be the image by the canonical projection n of the restriction of T' to any one of the connected components of x-'(U,). More particularly, taking X = R" and X = T",the inverse images on R" of currents on T" are precisely those which are invariant under the group Z" (acting on R" by translations); such currents are said to be periodic with Z" as group of periods. (17.4.5) Let A be the support of a pcurrent T. We shall show that it is possible to attach a meaning to T(a)for some pforms a E &,(X) which are not compactly supported: it is enough that A n Supp(a) should be compact (which will always be the case if A = S u p p Q is compact). For if h : X + [0, 11 is a C"-mapping which is equal to 1 on some compact neighborhood of A n Supp(a), and is compactly supported (16.4.2), then T(hcr) is defined because ha has compact support. Furthermore, if h, is another function having the same properties as h, we have T(h,a) = T(ha),because there exists an open neighborhood V of A n Supp(a) on which h(x) = h,(x), and the support of (h - h,)a is contained in Supp(a) n 6 V, and therefore does not intersect A. We may therefore define T(a)to be T(ha)for any function h with the properties stated above. It is immediate that the set of pforms a E &,(X) such that A n Supp(a) is compact is a vector subspace of &,(X), and that a H T ( a ) is a linearform on this vector space. Next, consider a sequence (a,,) of pforms in &,(X), such that (i) all the sets A n Supp(a,) are contained in ajixedcompact set K; (ii) the sequence (a,,) tends to 0 in &,(X). Then T(a,,)+ 0. For if we choose as above a function h which is equal to 1 on some compact neighborhood of K, then the sequence (ha,,) tends to 0 in g,(X; V) (17.1.4); since T(a,) = T(ha,,) for each n, our assertion follows immediately.

5. CURRENTS ON A N O R I E N T E D M A N I F O L D . D I S T R I B U T I O N S ON

R"

(17.5.1) Consider now an oriented pure differential manifold X of dimension n. We have defined in (16.24.2) the notion of an integrable differential n-form v on X,and its integral, denoted by v or v. Now consider a locally integrable differential (n - p)-form p, where 0 5 p 5 n; for each pform a E 9$'"(X; K), the n-form /3 A a is locally integrable and has support contained in K, hence is integrable. We shall show that the linear form a H /3 A a on g?)(X) is a pcurrent (or order 0). The proof reduces immediately to the situation where X is an open set U in R",and then we have

I Ix

s

5 CURRENTS ON AN ORIENTED MANIFOLD

247

where the bH(resp. the u1-H) are the coefficients of B (resp. a) relative to the canonical basis of the %‘(U)-module &‘$?,,(U) (resp. &‘r)(U)),and in the summation I = {1,2, ...,n} and H runs through all subsets of n - p elements of I. Then wehavetoshowthateachofthelinearmappingsoI-H-/ bH(x)a,-H(x) dx is continuous on each of the Banach spaces X(U; K), where K is any compact subset of U; and this follows from (13.13) because each of the functions bH is locally integrable. Let Ts be the pcurrent so defined. If we denote by &,.-,, ,oc(X)the vector space of locally integrable differential (n - p)-forms on X, then we have a linear mapping B H T ~of 8,,-p,loc(X)into 9:’)(X). From (13.14.4) it follows immediately that the kernel of this mapping is the subspace of negligible (n - p)-forms. Since the support of a Lebesgue measure on X is the whole of X, the restriction of the mapping B-Ts to the space &‘$?,,(X) of continuous differential (n - p)-forms is injective, so that such a form may be identified with a pcurrent of order 0. Under this identification, the notions of support are the same for the continuous (n - p)-form fl and the pcurrent T, with which we have identified it. For, by reducing as above to the case where X is an open subset U of R”,if xo E Supp(fl), then there is an index H such that bH(xO) # 0; we can then choose a 1 - H such that the integral

s

b,(x)a[-H(x) dt’ dt2 * ’ ’ dtn

is #O, and such that Supp(a1-H) is contained in an arbitrarily small neighborhood V of xo ; defining a1-H’ to be 0 for H’ # H, we obtain a form a with support contained in V and such that B A a # 0, which proves the assertion.

I

In particular, for each locally integrable n-form u, the mapping f w / f u is a meusure T, on X, which is positive if and only if u(x) 2 0 almost everywhere (relative to the orientation of X) (13.15.3). (1 7.5.1 .I) Again, iff is any locally integrable complex function on X, then the mapping U H fu is an n-current T, on X, of order 0. If U is any open is the characteristicfunction of subset of X, the n-current Tq, on X, where qPu U, is called an open n-chain element on X, and linear combinations of open n-chain elements are called open n-chains on X. By abuse of notation, we shall often write U in place of T,, , and C Aj Ui in place of AjTPu,.

s

i

i

(17.5.2) We retain the notation and assumptions of (17.5.1). Let y be a continuous differential q-form, where q S p ; then the (n - p + q)-form B A y is locally integrable, and it follows immediately from the definitions that (17.5.2.1)

Tshy = TB A Y.

248

XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

The left-hand side of this formula is meaningful if we suppose only that y is measurable and locally bounded, or, on the other hand, that fl is measurable and locally bounded and that y is locally integrable. Under these conditions we dkfine the right-hand side of (17.5.2.1) by this formula, and then we have (1 7.5.2.2)

T,, A

p = (-

1)4(n--p)T,

h

y.

Next, let z : X + X be an orientation-preserving diffeomorphism. Then the pcurrent n(T& is the linear form u ‘ w J X f l ~ ‘ z ( a ’ )on 9$‘)(X’). By (16.24.5.1) we have

and therefore (1 7.5.2.3)

11(Tg) = Ttn-i(B).

If fl is a locally integrable differential (m - p)-form on an oriented submanifold Y of X, of dimension m, then clearly we can form the imagej(T,) of T, under the canonical immersion j : Y + X, and j(T,) is a p-current. However, when m < n, this current is not in general expressible in the form T, for some locally integrable (n - p)-form y on X, even if j? is of class C”. For example, if p = 0, the support of the measure j(T,) is the submanifold Y, which is negligible with respect to any Lebesgue measure on X, so that a measure of the form T, with support Y is necessarily zero, whereasj(T,) # 0 in general (cf. (17.10.7)). Finally, let z : X + X be aproper mapping of class C‘ with r 2 1 (17.3.7), the manifold X’ being not necessarily orientable. Then for each locally integrable complex-valued functionfon X, n(Tf) is an n-current of order 0 on X‘ (which therefore vanishes if dim X‘ n), defined by the formula

-=

(17.5.2.4)

where CL’ is any continuous, compactly-supported differential n-form on X’. In particular, if we take f to be (px, the constant function equal to 1 at all points of X, then x(T,+,,) is called the n-chain element without boundary on X (cf. (17.15.5)) defined by the proper mapping z, and we write J n cd in place of

IX

%(a’). When n = 1, this is an integral along a particular type of

unending path ” (1 6.27). If X is a closed submanifold of X’ and z is the canonical injec“

5 CURRENTS ON AN ORIENTED MANIFOLD

tion, we shall sometimes write X in place of n(Tqx),and

IX

249

u' in place of

Jx %(a'); but it must be borne in mind that this number depends not only on the manifold X but also on its orientation, and changes sign when the orientation is reversed. (17.5.3) If we fix a C" differential n-form uo belonging to the orientation of X, then every differential n-form on X can be written uniquely asfuo, wheref is a complex-valued function on X. The formfu, is locally integrable if and only iffis locally integrable; in other words, the mappingf-fu, is a linear bijection of the space 2,0c(X)of locally integrable complex-valued functions on X onto the space &,, Ioc(X).We shall write Tf in place of Tfuoand identifv the function f with the corresponding distribution T, . (17.5.3.1) The choice of uo allows us to identifv n-currents and distributions (i.e., 0-currents), because gt+guo is an isomorphism of the FrCchet space 9(X; K) = .9,(X; K) onto 9,,(X; K), for each compact subset K of X; this follows immediately from (17.1.4). Hence every n-current is uniquely expressible as gu, -T(g), where T is a distribution. We shall denote this n-current by Tluowhen it is necessary to avoid ambiguity; in this notation, we have (T,>,uo= T,,, for a l l f e 2Ioc(X). (17.5.3.2) In future we shall make these identifications only when X is an open subset U of R",endowed with the canonical orientation and the canonical n-form uo = dtl A dt2A * * A dt" restricted to U. Then no risk of confusion arises except as regards the image of a current under a diffeomorphism x of U onto an open subset U' of R".If T is a distribution on U and n(T) its image on U', then the image x(Tlvo)is given by

-

(17.5.3.3)

where J(x-l) is the Jucobiun of the inverse diffeomorphism n-l, by virtue of (1 6.20.9.4). In particular, for each functionfE 210c(U)we have (17.5.3.4)

4Tf)

= T,,

Y

where f' is the mapping x' ~ f ( n - ' ( x ' ) ) J ( n - ' ) ( x ' ) . (17.5.3.5) It is clear that the kernel of the linear mappingfHT, of 210c(U) into W(U) is formed by the negligible complex-valued functions on U (relative to Lebesgue measure). Passing to the quotient, it follows that the space Lloc(U) of classes of locally integrable functions on U (13.13.4) may be

250

XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

canonically identified with a subspace of the space of distributions on U. A fortiori, we may identify the spaces L'(U), L2(U),and L"(U) with subspaces See (17.8) for questions relating to the topologies of these spaces. of 9'0. (17.5.4) Questions of a local nature concerning currents reduce to the case where the manifold X under consideration is an open subset U of R",and it is this case that we shall be mainly concerned with in the remainder of this chapter. Each of the spaces SF)(U) (resp. Sp(U))is then an S(r)(U)-module

(resp. an b(U)-module) which is free of rank basis for the p-forms: (H = d = . If the Dirac measure E, on G is identified with an n-current, then %r(ee) is identilied with the measure tr defined in (a).

6. REAL DISTRIBUTIONS. POSITIVE DISTRIBUTIONS

(17.6.1)

Let X be a differential manifold. For each p the vector bundle

P

AT(X)* may be identified with a subbundle of the real vector bundle

( A T(X)*)o,,

P

P

which is the direct sum A T(X)* 63 i A T(X)* (16.18.5). It follows that C?~, 0 such that I c,,/Atl 5 l/n2, namely any integer k such that kp - 0 2 2. Then it is enough to show that the series (1 7.8.8.2)

converges in 9'(R), because if T is its sum, then the series (17.8.8.1) will converge to DkT,by virtue of the continuity of differentiation; but the series of continuous functions (17.8.8.2) is normally convergent in R, hence converges also in 9'(R) if its terms are regarded as distributions (17.8.4).

PROBLEMS

Let z, be a nonzero tangent n-vector at a point x E R",and let (V,) be a fundamental system of bounded open neighborhoods of x in R".Show that the sequence of n-currents (open n-chain elements (17.5.1.1)) (&(V,)-'VJ tends to the limit cz,, where c - l = , uo being the canonical n-form on R". Let (V,) be a fundamental system of bounded open neighborhoods of the origin in R"-l. Let uk be the locally integrable differential (n - 1)-form on R"= R x R"-' which is equal to &-l(Vk) -I d p A . . . Ad@ on R x V, and is 0 elsewhere. Show that, as k + 00, the sequence of 1-currents T,, (17.5.1) tends to the 1-chain element without boundary R x {0}(17.5.2), R being canonically oriented.

+

Let T be a distribution on R.For each h # 0 in R, put A,, T = y(h)T - T, and A n = Ah(A;- 'T) for all integers p > 1. (a) Show that as h+O the distribution (I/hp)At;T tends to DPT. (b) Let f be a continuous function on R. The function f is said to be conipletely monotone if, for each integer p 2 1, A p f ( x ;h, h, . . ., h) has the same sign as hp for all h # 0 (Section 8.12, Problem 4). Show that f is then analytic. (Use (a), Problem 5 of Section 17.5, and Problem 7(c) of Section 9.9.) (S. Bernstein's theorem.)

266

XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

(c) Suppose that, for some integer p 2 1, Apf(x; h, h, ...,h ) / P +0 as h +0, for all x E R.Show that there exists a dense open subset U of R such that, on each connected component of U, f is equal to a polynomial of degree s p - 1. (Use(a), and observe that if a sequence (fJ of continuous functions converges pointwise to 0, then sup (1 f.I ) I

is finite at each point; then apply (12.16.2).) If also there exists a Lebesgue-integrable function g 2 0 such that 1 Apf(x; h, h, ..., h)/hp1 6 g(x) for all h # 0, then f is a polynomial of degree at most p - 1. Show that a distribution on R ' which is invariant under all translations of R" (i.e., is equal to its image under each translation) is a constant function (or, more accurately, a distribution T,, where c is a constant function on R").(Use(17.8.2.1)J What are the translation-invariant n-currents on R"? (b) A l-current on lo, co[ which is invariant under all homotheties hA:x-hr (where > 0) is of the form Tcu0,where c is a constant. A distribution on 10, co[ invariant under all hAis of the form T,, wheref(x) =cx-I, c being a constant. (Usethe isomorphism XH log x of lo, co [ onto R,and (a) above.)

4. (a)

+

+

+

5. (a) Letfbe a holomorphic function on an open set U c C, and let zo E U. Let S c U be a closed annulus with center zo , defined by rl 5 Iz - zo1 5 rz . Also let u be a Cm-

function on R, everywhere LO, with support contained in [rl,rzl, and such that

fu(t) dt

= 1. Show that

+

where r = Iz - zo I = ((x - XO)' 0,- YO)^)"^. (b) Let U be an open disk in C with center xo E R,let U+be the intersection of U with the half-plane fz > 0, and let f be a holomorphic function on U+,regarded as a distribution belonging to P ( U + ) . Show that f is the restriction to U+ of a distribution belonging to 9'(U) if and only if, for each compact interval K C R n U with center xo,there exists an integer k > 0 and an interval J = 10, c[ in R,such that SUP

XCK.)~J

lY'f(x+iy)l 0 and a multi-index a

(al,az)such that

+

for all u E P(U). Let zo = xo iyo E U+,and let S be a closed annulus with center zo contained in U+ . For each z = x iy E U+ such that the line passing through z and zo meets the real axis at a point t E K, let S . be the annulus with center z which is the image of S under the homothety with center t and ratio y/yo. Then by (a) abovewe have

+

9 EXAMPLE: FINITE PARTS OF DIVERGENT INTEGRALS

267

where

Then use the inequality (l).) (c) Show that the extension property in (b) is equivalent to the following: in every open interval I c U n R with center x o , the functions X H ~ ( X iy) converge to a distribution in o'(1) as y tends to 0 through positive values. (For the necessity of the condition, use Section 17.12, Problem 6.) (d) Show that, for each compact interval K in R and each function f e &(R), there exists a sequence (f.)of polynomials such that, for each integer p 2 0, the restrictions to K of the functions DpLconverge uniformly on K to Dpf(cf. (14.11.3)). If T E B'(R)

+

s

has support contained in K,show that the function u defined by u(z) = (x - z) dT(x) for z E C - K is analytic on this open set and that for each polynomial f, we have

where y is a suitably chosen circuit in C - K. Conclude that T is the limit in &(R) of a family of functions of the form X H F(x iy), where F is holomorphic in the half-plane 4 z > 0, and y tends to 0 through positive values.

+

9. EXAMPLE: FINITE PARTS O F DIVERGENT INTEGRALS

(1 7.9.1) Let X be an oriented pure differential manifold of dimension n, and let F be a redvalued continuous function on X. Suppose that the open set Uo= (x E X : F(x) > 0} is not empty and that the frontier P of U, (where F(x) = 0) is negligible with respect to Lebesgue measure on X. The function F-'cpuo, which is equal to 0 in X - Uo and coincides with F-' on Uo,is not in general locally integrable in a neighborhood of P, because F-' is not in general bounded in such a neighborhood. If uo is a differential n-form on X belonging to the orientation of X, then the mapping f~ ~uoF-'fvois a measure on X - P, zero on X (P u Uo),which in general cannot be extended to a measure on X. In this section we shall indicate methods of wide applicability of constructing an extension which is a distribution on X (more precisely, this distribution will extend to 9(X)the restriction of the measure fwsuoF-tfo, to 9 ( X - P)).

-

juo

A first method consists of considering the integral Ffi, (where as usual t c means eclog' for real numbers t 0), which is an analytic function of 5 in the half-plane Eo :95 > 0 in C, for all functions f E X(X)(13.8.6). In other words, on restricting to 9(X), we obtain a distribution T( FJf.o

=-

:f-lU

268

XVll

DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

on X, which is a weakly analytic function (12.1 6.6.1) of 1; on E, ,and which we seek to analytically continue to a larger open subset of C. It may happen that such an analytic continuation exists on an open set containing the point ( = - 1, for each f E X(X),in which case its value at 1; = - 1 will still be a measure on X. Another case which occurs frequently (see the examples below) is that in which an analytic continuation [ H Tp) exists on E, - { - l}, where E, is a half-plane BY > a,with u < - 1, and TI")has a simplepole at the point - 1;in other words, for each function f E 9(X), we have

where A is a linear form on 9 ( X ) and, for all in a neighborhood V c E, of - 1, B, is a distribution on X and the function 1 ; B, ~is weakly analytic in V. It follows then (12.1 6.6.1) that A is a distribution. Moreover, if Suppcf) n P = @, then we have A(f) = 0; for [H T&f) is then an entire function of 1;, hence coincides with 1 ; ~ T ? ) ( f )in E, - {- l}; consequently, as 1; + - 1, Tr(f) and Bscf) both tend to finite limits, whence the assertion follows. Hence we have Supp(A) c P. Consequently B-l is an extension to 9(X) of the distribution f-/uo F-lfu, defined on 9(X - P), and is called the finite part of this integral. (Of course, there are infinitely many such extensions, obtained by adding to B-, any distribution with support P.) Examples (17.9.2) SO

Take X

= R" (n

2 l), and let F be the function r(x) =

(;I,

(5')'

,

that Uo = R" - (0) and P = (0). Take uo to be the candhical n-form

d t ' A d t Z A -*.Adc;".

If t~ is the "solid angle" differential (n - 1)-form on the unit sphere S,,-', then by (16.24.9) we may write (17.9.2.1)

TC(f) =

for 91;> 0 and any f E X(R"), and the function (1 7.9.2.2)

M,(d

=

Is.-.

f(PZ)tJ(Z)

is continuous for p 2 0, compactly supported, and of class C" on 10, + a[ (1 3.8.6). Hence we see already that, forfE X(R"), the function ryis integrable

9

EXAMPLE: FINITE PARTS OF DIVERGENT INTEGRALS

269

not only for Wt:> 0, but for Wt: > -n (13.21.10), and Tc is therefore a measure on R" for all t: in this half-plane. For each integer m > 0 and for f E 9(R"), we shall define an analytic continuation [H TY)(f) of the function t:t+TS(f), as follows. Replacefin (17.9.2.2) by its Taylor series up to order 2m (8.14.3):

+

where ( p , z) ~ g ( pz), is continuous on [0, co[ x Sn-l and has support contained in a set of the form [0, po[ x Sn-l.Hence, by splitting the integral into two parts, we have f o r at:> - n

where

1

(17.9.2.5)

= Jsn-*

- zVa(z). v!

Now, on the right-hand side of (17.9.2.4), the last integral is an entire function of (13.8.6); the first is an analytic function of in the half-plane at:> - n - 2m; and therefore the right-hand side of (17.9.2.4) is a meromorphic function of in the half-plane Wc > - n - 2m, having at most simple poles at the points of the form - n - k,where 0 5 k 6 2m. It is this function which is the desired analytic continuation T:")(f). Since T:"'+')(f) and T:")(f) coincide with TC(f) for W c > -n, they coincide throughout the domain of definition of TY)(f) (9.4.2). We shall therefore denote by TC(f)the function, meromorphic in the whole complex plane C , which coincides with each T',"')(f) in the domain of definition of the latter. We shall now determine the residues of T,(f) at its poles. Notice first that the symmetry Z H -z multiplies the form CJ by (- 1)" and preserves (resp. reverses) the orientation of Sn-l if n is even (resp. odd) (16.21.10). It follows therefore from (17.9.2.5) and (16.24.5.1) that c, = 0 for

c

270

XVH DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

all multi-indices v of odd total degree. On the other hand, if Iv I = 2k is even, then the residue of TS(f) at the point -n 2k has the value

-

as follows from substituting the expansion (1 7.9.2.3) in the expression for

M,(p). To calculate this number, we introduce the Lapluciun, which is the following differential operator on R":

(17.9.2.7)

A = D:

+ D; + - .. + D: .

Using the formulas

we obtain (17.9.2.8)

Art = c([

+ n - 2)r5-2

for all [ E C and x E R" - {O}. This implies, by (17.5.5.1), that

for W[ > -n first of all; but since both sides of this relation are meromorphic functions of [,it follows that it remains valid for all [ which are not poles of Tc,., i.e., [ # - n - 2k. By iteration we obtain, under the same conditions,

and since none of the linear factors on the right-hand side vanishes when [ = -n - 2k, the residue of T,(f) at this pole is given by res-n- 2kT&f) = ((-2)(-4)

- - .(-2k)(-n)(-n

- 2)

.-.(-n

- 2k + 2))-' res_,,Tc(AkS).

Now apply the formula (17.9.2.6),replacingfby A'ffandk by 0; since

9

EXAMPLE: FINITE PARTS OF DIVERGENT INTEGRALS

271

by (17.9.2.2), we obtain finally (17.9.2.11)

1 res-n-ZkT,(f) = -DZkM,(O) (2k)!

(when k = 0, the denominator is to be replaced by 1). Let C, denote the constant which appears in the right-hand side of this formula. We can now, following the method described in (17.9.1), define the finite part Pf(rr) for all c E C.For values of c other than the poles of Tc, we define Pf(rr) = Tr, and for [ = - n - 2k (17.9.2.12)

(Pf(r-"-2k),f) = lim

{+-n-Zk

(T,(f)

- ck A"f(O)(C+ n + 2k)-').

The formula (17.9.2.9) gives the Laplacian of Pf(rs) for [ not equal to a pole of TC.To obtain its values at the poles, we may proceed as follows. For not equal to a pole of T, , write


0 tends to 0, the double integral

+

(where z = x iy) tends to a limit, for each function f~ 9(R2). (Reduce to the case where f (x, y) = z P B and change to polar coordinates.) Show that the mapping

is a distribution on RZ of order m - 1. This distribution is denoted by P.V.(l/z"). Prove the formulas

3. Show that for 5 = n - 2 - 2k which is not of the form -2rn (where rn is an integer ZO), the support of Zc(17.9.4.5) is the cone defined by to20,s(x) = 0. 4. Let f be a holomorphic function on an open set in C containing the closed unit disk D : 111 5 1, and let fD be the function which is equal to f in D and vanishes outside D.

Show that the distribution fD on R*has derivative afD/aZ equal to the distribution

where

E

is the circuit tt-+e'* ( O $ t $ 2 n ) .

(16.24.11).) 5.

(Use the elementary Stokes' formula

Express in terms of Cauchy principal values (Problem 1) the distributions on R defined by the formulas

and

6. For

> 0 and 9 p > 0, consider the distribution TA,,,on 10, 1[ defined by

Show that (~,p)nT,,,,(f)\extends to an analytic function except for A = - n or p = -n, where n EN,and determine the form of this function near these singular points. Outside the singular points, TAspis a distribution on 10, 11, denoted by P f ( X y ' ( 1 - x)";1).

10 TENSOR PRODUCT OF DISTRIBUTIONS

277

10. T E N S O R P R O D U C T O F D I S T R I B U T I O N S

(17.10.1) Let U be an open subset of R",let T be a distribution of order s m on U , let E be a metric space and f a mapping of U x E into C.

(i) Suppose that there exists a compact set K c U and a neighborhood V of a point zo E E such that (1) f( ,z ) E 9(m)(U;K ) for all z E V ; (2) (x, Z)H DVf(x, z ) is continuous on U x V ,for each multi-index v such that I v I 6 m. Then the function 2- F(z) = = ( S n 0 T n

9

h(f 0 m>>

for all n. Now apply (17.10.4(iii)). It should be remarked that the conclusion of (17.11.9) may be false if the supports of the distribution S, are not all contained in a fixed compact set. For an example we may take G = R and S, to be the Dirac measure E - , at the point -n, and for T the measure

n=l

nE, defined by a mass n at each integer

point n > 0. Then the sequence (S,) converges weakly to 0, but the measure S, * T has mass n at the point 0, and therefore does not converge weaklytoo.

286

XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

(17.11 .lo)

Let n : G + G be a homomorphism of Lie groups, and let S , T be two distributions on G. Suppose that either (1) rc is proper (17.3.7) and S, T are strictly convolvable, or (2) rc is arbitrary and S , T have compact support. Then n(S) and n(T) are strictly convolvable, and we have ~17.11.10.1)

n(S * T)

= n(S)

* rc(T).

In case (2), n(S) and n(T) are compactly supported (17.4.3), hence are strictly convolvable. In case (1) it is enough, by virtue of (17.4.3), to show that for each compact subset K’ of G’ the relations x E Supp(S), y E Supp(T), and n(x)n(y)E K’ imply that the pair (n(x), n(y)) belongs to a compact subset of G x G . However, since n(x)n(y)= n(xy) and since n-’(K’) is compact by hypothesis, the point (x, y ) belongs to a compact subset of G x G, where the result follows. For each function f E Q(G’),we have then J f ( z 7 4 4 s * T))(z’)= =

sssfo) f

(n(z))4 s * T)(z) dSW

= Ssf(n(x)n(y))W x ) = SdT(Y) =

jf ( X ’ 4 Y ) ) d(n(S))(x’)

Sd(n(SNx7 sf(”.y.) d(n(T))(y’)

which proves (17.11.I0.1). Remark (17.11.10.2) If S and T are distributions with support {e},then n(S) and n(T) have support {e’}, and the formula (17.11.10.1) remains valid, with the same proof, when n is a local homomorphism (16.9.9.4): for we need only consider functions f whose support is contained i n a neighborhood V of e such that n(xy) is defined and equal to n(x)n(y) for all x, y E V.

In the important case where G = R”,convolution of distributions behaves as follows relative to the operation of differentiation :

11 CONVOLUTION OF DISTRIBUTIONS ON A LIE GROUP

287

(17.11.11) If S and T are strictly convolvable distributions on R", then for 1 5 k n the distributions D, S and T (resp. S and DkT)are strictly convolvable,

and we have

Dk(s * T) = (Dk s) * T = s * (DkT).

(17.11.11.I)

=sf(.

For each function f E 9(R"), put g(x) + y) dT(y), so that by (17.10.1) the function g is indefinitely differentiable, and

s

Dk).(g

=

Since by definition (Dk(S * T),f) = 0, the sequence (p,, ,(T)),? converges to 0. In particular, the summable distributions of order 0 are precisely the bounded measures (13.20). (Argue by contradiction to show that the condition is necessary.) Every derivative DYTof a summable distribution is summable. (b) Suppose that U = R“and that K, is the ball llxll m. Let hl :R”+ [0,1]be a C”-mapping which is equal to 1 on K1and is 0 outside K2, and put hm(x)= hl(x/m)for all m 2 2. If T is a summable distribution, show that for each function f E P(R”) the sequence (T(h,f)),L1 tends to a limit, which we denote by T(f). In this way T is extended to a continuous linear form on the Frkchet space S(R”). (c) Show with the help of (b) that the convolution of two summable distributions on R“can be defined, and that this convolution is also a summable distribution. 2.

(a) With the notation of (17.9.3) show that Y . * Y a= Y o + afor any two complex numbers a,p. (Show that it is sufficient to prove the result for Ba > 0 and Bfi > 0.) If T is a distribution on R whose support is bounded below, then Y-k * T = D‘T for all integers k > 0, and Yk * T is the kth primitive of T whose support is bounded below. By extension, for each complex number 5, the distribution Y, * T is called the primitive of order 5 of T, and the distribution Y-4 * T is called the derivative of order 5 of T. (b) Let a,p, y be complex numbers such that 9 p > 0 and 9~> 0. The hypergeometric function F(a, fi, y ; x ) is defined on the interval 1- 1, 1[ of R by the formula

Show that, for y complex and # -n, where n f N, the function can be extended to all values of fi E C in such a way that

(change the variable to w = tx). In particular, for j3 = - k (where k E N) we have

Deduce that

(Jacobipolynomiaf).(Expand the distribution (1 - x ) ”Dkeo as a sum of point-distributions with support lo}.) (c) For each p E C such that 9 p > -4, the Bessel function of order p , defined for x E R, is given by the formula

Show that the function can be extended to all complex values of p in such a way that, for u > 0, we have

*

2 ~ 7 r ” ~ U ” ~ J P ( U= ’ ~ Yp+1,2 ~) (u -It2

cos u1’2).

12 REGULARIZATION OF DISTRIBUTIONS

289

Deduce Sonine's formula

(Convolve with Yq+l.) 3. With the notation of (17.9.4), show that 2. * Z, = Z.+# for all complex numbers a,8. (Same method as in Problem 2.) 4.

Consider B'(R") as a vector space of linear forms on B(R"), and endow b'(R")with the corresponding weak topology (1 2.15.2). For each distribution S E O'(R"), show that the mapping T H S * T of 8'(R") into O'(R") is continuous.

5.

Let u be a continuous linear mapping of B'(R") into

a'@).

(a) Show that the following two properties are equivalent: (1) y(h)u(T) = u(y(h)T) for all T E B'(R") and all h E R"; (2) D, u ( T ) = u(D, T) for all T E B'(R") and 1 5 j In. (Use formula (17.8.2.1) and consider, for each f ~ 9 ( R " ) , the function h ++ ; calculate its partial derivatives.) (b) If u satisfies the equivalent conditions of (a), show that u is necessarily of the form T- S * T, where S E W W ) .(Consider the linear mapping R Hu(R * T) - R * u(T) of b'(R") into O'(R"), for a fixed distribution T E B'(R"), and show that its kernel is the whole of B'(R"). For this purpose, observe that the Dirac measures E~ (x E R")form a total set (12.13) in B'(R"), by using Problem 13 of Section 12.15; then remark that the E* belong to the kernel of the linear mapping in question.) 6. Let G, G' be Lie groups and let S,T (resp. S', T') be strictly convolvable distributions on G (resp. G'). Show that S @ S' and T OT'are strictly convolvable distribution on

G x G' and that

(S @ S')

* (TOT') = (S * T) @(S' * T').

12. REGULARIZATION OF DISTRIBUTIONS

(17.12.1) Let p , m be two integers 20 such that p 2 m. IfT E W("')(R")and (resp. i f T E B'("')(R")andf E C~'(~)(R")), then the distribution T * f may be identijied (17.5.3) with a function in L?(P-m)(R") such that, for each x E R",

f E @')(R")

(17.12.1.1)

(T * f ) ( ~=)( T , j ( ~ ) y )=

s

f(X

- y ) dT(y).

The 'fact that the function x w J f ( x - y) dT(y) belongs to C?(~-"')@V') follows from the hypotheses and from (17.10.1). Next, if S = T *ft then we

290

XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

have by definition, for any u E 9(R"),

s s

= dT(x)

f ( M x + Y ) d&y)

= (T €3 1,g ) ,

where A is Lebesgue measure on R", and g(x, y ) =f (y)u(x+ y). However,

-

hence (S, u ) = (T €3 (u A), h), where h(x, y ) =f(y

= which proves (17.12.1.I).

s s UCv)

d4Y)

- x ) ; and therefore

f (Y - 4dT(x),

In particular: (17.12.2) If T E C'(R") and f E B(R"), or if T E 9'(R") and f E 9(R"), then T * f E b(R").If T E b'(R") and if (f,) is a sequence of functions in b(R") converging to 0 in this space, then the sequence (T * f,) tends to 0 in &(R").

Let A be the (compact) support of T, let V be a compact neighborhood of A, and let K be any compact subset of R". If we put g,(x, y) =f,(x - y), then the sequence of elements g,(x, .) of b(R") converges t o 0 in this space, uniformly with respect to x E K. For the partial derivative with multi-index v of the function y~ g,(x, y) is (- l)lvlDvfp(x- y). Since the sequence (Dvf,(z)) converges uniformly to 0 in the compact set K + (-V), the sequence of functions (x, y) H D'f,(x - y ) converges uniformly to 0 on K x V; hence, by the definition of distributions, the sequence of functions X H h,(x) = /&(x - y ) dT(y) converges uniformly to 0 in K, and the same is true of the sequence of partial derivatives X H Dvh,(x) for any multi-index v , by virtue of (17.10.1).

For each open subset U of R", the set 9 ( U ) of Cm-functions on U with compact support, ident$ed with a space of distributions on U (17.5.3), is weakly dense in 9'(U).

(17.12.3)

Let (K,) be a fundamental sequence of compact subsets of U, and let h, be a function belonging to 9(U) which is equal to 1 on K, . Since for each u E 9(U) the reexists an integer m such that K, is a neighborhood of the support of u, it follows that for each distribution T E 9'(U) we have

o

D"F; in which case To+= n!A. (To show that the condition is necessary, consider a compact interval J = [Ba, a], where 0 < 0 < 1, contained in 1; by the result of Problem 16, there exists an integer n, an open neighborhood V of J and, for each sufficiently small h, a continuous function GAon V such that the restriction to V of Tnis of the form D"G, , and such that the functions Gnconverge uniformly to n !Ax" inV. Deduce that there exists a continuous function G on 10, b[ such that the restriction of T to 10, b [ is equal to D"G. For all sufficiently small h, we can write G(h)- hlGn(x)= wA(x),where wA is a polynomial of degree - 1, with coefficients depending on A. Now apply (a).) (c) Deduce from (b) that if S is a distribution such that DS = T, and if the limit To+ exists, then so does the limit SO+ . Also, for each function f~ 9 ( R ) , the limit (f.T)o+ exists and is equal tof(0)To+ . (d) If a > 0 and ji?> 0, show that the function x* sin(x-9, which is defined and continuous for x > 0, extends to a distribution T on R for which the limit To+ exists and is equal to 0.

sn

13. DIFFERENTIAL OPERATORS A N D FIELDS O F POINT-DISTRIBUTIONS

(17.13.1) Let X be a differential manifold and E, F two complex vector bundles over X. We have seen that the vector spaces T(X, E) and T(X, F) are canonically endowed with structures of separable complex Frtchet spaces (17.2.2). A C" linear diyerential operator from E to F (or simply a differential

296

XVll

DISTRIBUTIONS A N D DIFFERENTIAL OPERATORS

operator, or even an operator if there is no risk of ambiguity) is by definition a continuous linear mapping f++P * f of the Frkhet space T(X, E) into the FrCchet space T(X, F) which satisfies the following condition:

(L) For each open subset U of X and each section

f 1 U = 0, we have P - fI

U = 0.

fE

T(X, E ) such that

In other words, if two sections f , g of E over X are equal on an open set U, then so are their images P * f and P * g. An equivalent way of stating this is to say that P is an operator of local character. Let (V, cp, n) be a chart of X such that E and F are trivializable over V. If n', d'are the projections of the bundles E, F and if N', N" are their respective ranks over V, then there exist diffeomorphisms ZH (q(n'(z)),v(z))of n'-'(V) onto q ( V ) x C" and zt+(cp(n"(z)),w(z)) of n"-'(V) onto q(V) x C"' such that v (resp. w) is a linear isomorphism of each fiber n'-'(x) (resp. d""(x)) onto C" (resp. C"'). The mapping f w v f o cp-' (resp. f H w f o cp-') is then an isomorphism of T(V, E) onto (&'(cp(V)))" (resp. of T(V, F) onto (tP(q(v)))"'). If P is a differential operator from E to F, then for each section f E T(X, E) the value P f IV depends only on f 1 V, and there is therefore a well-defined continuous linear mapping gt+ Q * g of (&'(cp(V)))" into (&(cp(V)))"' such that 0

0

-

The linear mapping Q is said to be the local expression of the operator P corresponding to the chart (V, cp, n) and the mappings v and w. (17.13.3) In order that a linear mapping P ofT(X, E ) into T(X, F ) should be a differential operator, it is necessary and suflcient that for each x E X there should exist a chart (V,cp, n) of X at the point x , such that E and F are trivializable over V and such that the corresponding local expression of P should be of the form (17.13.3.1)

where, for each multi-index v such that I v I 5 p , the mapping y w A , ( y ) is a C"-mapping of cp(V) into the vector space Hom,(C", C"') (which can be identified with the space of N" x N' matrices over C).

The condition is sufficient. First, it is clear that it implies the condition (L). Second, since each compact subset of X admits a finite covering by

13

DIFFERENTIAL OPERATORS

297

domains of definition of charts which satisfy the condition of the statement of the proposition, it is enough by virtue of (17.2) and (3.13.14) to verify that the mapping (17.13.3.1) is continuous, and this is a direct consequence of (17.1.3) and (17.1.4). To show that the condition is necessary, we may clearly assume that X is an open subset of R” and that E = X x C”, F = X x C”’, so that T(X, E) = (&(X))“ and T(X, F) = (Cp(X))”’. Replacing P by p P o j , where j is a canonical injection of one of the factors of (&(X))” into this product, and p is a canonical projection of the product (Cp(X))”’ onto one of its factors, we reduce further to the case where N’ = N” = 1. Replacing X if necessary by a relatively compact open set, we may suppose, by virtue of the definition of the topology of &(X) (17.1), that there exists a constant c and an integer p such that for all x E X and all f E &(X) we have 0

(17.13.3.2)

This shows that for each x E X the linear form f H( P .f ) ( x ) is a distribution of order S p on X; furthermore, if x $ Supp(f), then by hypothesis we have ( P .f ) ( x ) = 0, so that the support of this distribution is {x). Hence (17.7.3) it is of the form f H

c a,(x>D>vf(x),

IVlSP

where the a,(x) are scalars. Replacing f successiveiy by monomials xu, we see that for functions

S p the

JtlJ

are of class C“. It follows easily by induction on I vI that all the a, are of class C“, and the proof is complete. For each differential operator P from E to F, and each point x E X, the order ofP at x is defined to be the largest of the integers I v I such that A v ( x )# 0 in a local expression of P in a neighborhood of x . It follows immediately from the rule for differentiating composite functions and from Leibniz’s formula that this number cannot increase when we pass from one local expression to another, and hence it is independent of the particular local expression chosen. By virtue of (17.13.3), a differential operator of order 0 may be written (1 7.1 3.3.3)

fHA.f,

298

XVll

DISTRIBUTIONS A N D DIFFERENTIAL OPERATORS

where, for each x E X, A(x) is a linear mapping of the fiber Ex into the fiber F, , in other words an element of Hom,(E, , F,), and A :x ~ A ( xis) a section (of class C") of the vector bundle Hom(E, F) (which may be identified with the tensor product E* @ F). Such a section may also be identified with a linear X-morphism A : E + F ; for each section f of E, A * f is the section A f of F (16.16.4). 0

(17.13.4) If P is a differential operator from E to F, then for each open subset U of X we may define the restriction of P to U, which is a differential operator PI U from E I U to F I U, as follows : for each section f of E over U and each point x E U, there exists a C"-function h on X with support contained in U, which is equal to 1 in a neighborhood of x; hfextended by 0 outside Supp(h) is a C"-section of E over X; hence P (hf) is defined. The value (P* hf)(x) is independent of the function h chosen, and if we denote thisvalue by (P f)(x), it is immediate from (17.13.3) (or directly from the condition (L)) that P f i s a section of F over U and that f H P * f is a differential operator from El U to F J U . If now (U,) is an open covering of X and if, for each 1,we are given a differential operator P, from E I UAto F I U, , so that for each pair of indices 1,p the restrictions of PAand P,,to U, n U,, are equal, then it follows immediately from (1 7.13.3) that there exists one and only one differential operator P from E to F such that PI U, = PAfor each 1.For each section f E r(X, E) and each x E X, we define (P . f ) ( x ) to be the commonvalue of (PA ( f l U,))(x) for all indices 1 such that x E U, .

-

-

-

-

(17.13.5) If P is a differential operator from E to F and if h is a C"-function on X, it is clear that the mapping f H h(P f ) is also a differential operator from E to F, which we denote by h P (17.1.4). The set of differential operators from E to F is therefore an g(X)-module. Let El, E, , E, be complex vector bundles on X, and let PI : El -,E, and P , : E, + E3 be differential operators. Then it is clear that P , Pl is a differential operator from El to E, . Furthermore, from the local expressions of PI and P, it is immediately seen that if P, is of order p and P , of order q at a point x E X, then Pz PI is of order $ p q at x. 0

0

+

(17.13.6) An important particular case is that in which E and F are both equal to the trivial complex line bundle X x C,so that T(X, E) = T(X, F) = b(X). The local expression of a differential operator P from X x C to X x C is then of the form (17.13.6.1)

13 DIFFERENTIAL OPERATORS

299

where the mappings y w a,(y) are complex-valued C"-functions defined on an open subset cp(V) of R".The operator P is of order p at the point x E V if and only if at least one of the numbers a,(cp(x)) for J v I = p is nonzero. The differential operators from X x C to X x C clearly form a C-algebra with respect to the composition defined in (17.13.5); we denote this algebra by Diff-). We have already seen (17.13.3) that for each x E X the mappingfw (P . f ) ( x ) is a distribution with support contained in { X I ; this distribution is denoted by P(x), so that we have

Thus an operator P E Diff(X) is a C"-jield of point-distributions. (17.13.7) Let u : X + Y be a diffeomorphism. For each differential operator PeDiff(X) we transport P by means of u to a differential operator u*(P)E Diff(Y) as follows: for each functionfe S o , we have (17.13.7.1)

u*(P) * f = (P'( f o

or, in other words, for each x

24))

0

u-l

E X,

which shows immediately, bearing in mind the definition of the image under u of a distribution on X (17.3.7). that for each x E X we have

From (17.13.7.1) it follows immediately that if P,,Pz E Diff(X), then

in the algebra Diff(Y). Further, if u : Y + Z is another diffeomorphism, (17.1 3.7.5)

(u 0 u)* = u*

0

u*

,

which shows that u* is an isomorphism of the algebra Diff(X) onto the algebra Diff(Y). With this notation, if (V, cp, n) is a chart of X, it is clear that the local expression of an operator P E Diff(X) is cp*(PI V).

300

XVll

DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

(17.13.8) Let P E Diff(X). For each compact subset K of X, the image under P of 9(X; K) is contained in 9 ( X ; K). If T is a distribution on X, we may therefore consider, for each functionfe 9 ( X ; K), the value (T, P f), and it is clear that the linear mapping ft-+(T, P * f ) of 9 ( X ; K) into C is continuous. Hence we have a distribution 'P * T, defined by the relation

-

for all f E 9 ( X ) (12.15.3). It follows from this definition that Tt-+'P * T is a linear mapping of 9'(X) into itself, and that if the restriction of T to an open set U c Xis zero, then also the restriction of 'P * T to U is zero. In other words, we have Supp('P * T) = Supp(T), and for each open U c X and each T E 9'(X) we have

'(PIU) * (TIU) = ('P * T)I U. If (V, cp, n) is a chart of X for which P has the local expression (17.13.6.1), then, for each function f E 9(cp(V)), (17.13.8.2)

('P'T, f

0

cp) =

C

I4 5 P

(-l)lv'(Dv(avcp(TIV)),f)

having regard to the definition of the derivative of a distribution on an open subset of R" (17.5.5). Remark (17.13.9) Let P : E F be a differential operator. From the local expression (17.13.3.1) and from Leibniz's rule it follows that, for each section f E T(X, E), the mapping ot-+P(of), where u runs through the set of C" scalar-valued

functions on X, is a differential operator from X x C to F ; moreover, if P is of order p at a point x , and if f(x) # 0, then at-+P(uf) is also of order p at x , by virtue of Leibniz's formula. In particular, in order to verify that a differential operator P is of order 0, it is enough to show that P(af) = uP(f) for all u E &(X) and all f E T(X, E). (17.13.10)

The notion of a real differential operator is defined in exactly the same way, by replacing in (17.13.1) complex vector bundles by real vector bundles, so that T(X, E) and T(X, F) are real Frechet spaces; we have only to replace C by R throughout in the developmentsof this section. In particular, to say that P E Diff(X) is a real differential operator signifies that, for each

13 DIFFERENTIAL OPERATORS

301

C" real-valued functionf on X, the function P .f is also real-valued. For each real distribution T on X, 'P * T is then also a real distribution. If E, F are real vector bundles on X and E(,), F(,) their complexifications, then every real differentialoperator P from E to F extends uniquely to a complex differential operator P(,,from E(,) to F(,), because T(X, E,,,) = T(X, E) BRC. (17.13.11) The results of this section are easily extended to continuous linear mappings of I"')(X, E) into T("(X, F) (17.2), where r, s are integers 2 0 . For such an operator one obtains a local expression (17.13.3.1), in which necessarily p S r - s and the A, are assumed only to be of class C'. We leave it to the reader to modify appropriately the other results of this section.

PROBLEMS

1. With the notation of (17.13.1), let P be a linear mapping (not assumed to be continuous) of r(X, E) into r(X, F), satisfying the condition (L). Then P is continuous, and hence is a differential operator (Peetre's theoremLBegin by showing that, for each open subset U of X, the restriction of P to U may be defined as in (17.13.4). This allows us to reduce to the case in which X is an open set in R"and E = X x C", F = X x C"'; we may then assume that N = N" = 1, so that r(X, E) = r ( X , F) = B(X). Then proceed as follows: (1) For each x E X, if f e 8(X) is such that D"f(x) = 0 for each multi-index v, then also Dv(P . f ) ( x )= 0 for all v. (Argue by contradiction, using Section 16.4, Problem 1: There exists a function such that vfis of class C", equal to 0 for all y E X such that 9 - '5 5 Ofor 1 sj 5 n, and equal tofforall y E Xsuch that $ - 5' 2 Ofor 1 5j 5 n. Derive a contradiction by considering P . ( 2 ) A point x e X is said to be regular for P if there exists an integer k,> 0 with the following property: For each function f e B(X) such that D'f(x) = 0 for I v I < k,, we have (P. f ) ( x ) = 0. Show that if U is an open set in X, all points of which are regular for P , then P I U is a differential operator. (Prove that the k , are bounded on each compact K c U ; for this, argue by contradiction and use Problem 2 of Section 16.4. In each relatively compact open subset V of U we may then write ( P . f ) ( x ) = a,(x)D.f(x); prove as in (17.13.3) that the a, are of class Cm.)Deduce that the set

v

(vf).)

IVlSP

S of nonregular points contains no isolated points. (3) Show that the set S is empty. (Prove that there cannot exist a sequence (xk)of distinct points of S, converging to a point x, by using Section 16.4, Problem 2 again.)

2. If P : r(X, E) --f F(X, F) (notation of (17.1 3.1)) is a linear mapping, show that the following conditions are equivalent:

(a) P is a differential operator of order 5 m. tb) For each functionfE cf(X), the linear mapping S"P.(fS)-fP.

is a differential operator of order

5 m - 1.

s

302

XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

(c) For each family (fi)ldtdm+l of rn we have

+ 1 functions in 8(X),and each S E r(X,E),

where H runs through all subsets of {1,2,

...,rn + 1). (Useinduction on rn.)

3. Let X be a differential manifold, E and F two complex vector bundles over X. Show that the differential operators of order s r from E to F may be identified with the linear X-morphisms of the bundle F a , E ) of jets of global sections of E into the bundle F.

4. Let X be a differential manifold. For each point x E X and each integer r >= 0 and let TP(X) denote the vector space of real distributions of order s r with support contained in {x}. For each real-valued C"-function f on X and each real point-distribution S, E T$), the value + (df, [ X Yl> = e x * (df, y > = e x * (6, * f ) for any C” vector field Yon M ; and since

(df, [ X , Yl> = Orx, Y] * f= e x * (6, . f ) - 4 . (6, . f ) by virtue of (17.14.3), we have ( 6 , . df, y > = 8, * (6,

.f)

= (d(&

*f),y > ,

which proves (1 7.14.10.1). If X , Yare C“ vector fields on M, then (1 7.14.10.2)

8,

0

i , - i y 0 8, = i r x ,

,,,

both sides being considered as operators on d ( M ) . For since i , is an antiderivation of degree - 1 of d ( M ) (1 6.18.4), the same is true (A.18.7) of the left-hand side of (17.14.10.2). Since the question is local, it follows as above that it is enough to verify that the two sides of (17.14.10.2) take the same values for functions f E €(M) and their differentials df E €,(M); but the operators i y vanish on tP(M), and by (16.18.4.6) we have i , * df = (df, Y ) = 8 , .f.Hence, bearing in mind (17.14.10.1), the verification that the two sides of(17.14.10.2) take the same value for df reduces to (17.14.3). (17.14.11) Although, for a function f E €(M), the value of 8, * f a t a point x E M depends only on the value of X ( x ) , the same is not true for the value of 8, * Y if Y is a vector field. It may be shown that it is not possible to

define “intrinsically” a vector in T,(M) which should be the “derivative” of Y at the point x in the direction of a given tangent vector h, (Problem 2; cf. (1 8.2.14)).

14 VECTOR FIELDS AS DIFFERENTIAL OPERATORS

311

(17.14.12) The results of this section may be generalized without difficulty to vector and tensor fields of class C', where r is an integer 2 0. If X is a vector field of class C', then 8, * Z is defined for tensor fields Z of class C', where s 2 1, and is a tensor field of class Cinf(r,s-l).All the formulas proved in (1 7.14.4)-(I 7.14.7') remain unchanged. (17.14.13) We shall also leave to the reader the task of transposing the definitions and results of this section and the preceding one to the context of complex-analytic manifolds. Here we remark only that differential operators can no longer be defined by a local property; it is necessary to define them by means of their local expressions (relative of course to the charts of a complex-analytic atlas), and C"-functions and sections are replaced everywhere by holomorphic functions and sections. PROBLEMS

1. Let D be a derivation of the ring b(M) of C"-functions on a differential manifold M. Show that there exists a unique Cmvector field X on M such that D = 8,. (First show that the condition (L) of (17.1 3.1) is satisfied. Then either use Problem 1 of Section 17.13, or else give a direct proof with the help of Section 8.14, Problem 7(b).) 2. Show that there exists no linear mapping h,wDh, of the tangent space T,(M) into Hom(fb(M), T,(M)) which is not identically zero and satisfies the following conditions:

(1) for each vector field Y E fA(M) and each functionfE b(M),

Dhx. ( f y ) =f(X)Dhx

'

y + (ohx 'f)y(X);

(2) for each diffeomorphism u of M onto M, DTx(u).hx' (T(u) ' y ) = TdU) ' (Dhx. y). (Use Problem 11 of Section 16.26.)

3. Consider the nz vector fields t'Dk (1 5 j , k 5 n) on R".Show that the vector space they generate is a Lie algebra o and that [e, 01 # g (cf. (19.4.2.2)). 4.

Let X be a C" vector field on a differential manifold M. Show that if X(x) # 0,there exists a chart c of M at the point x such that X i s equal to the vector field XIin the domain of definition of c, where the notation is that of (16.15.4.2).

5.

Let M be a differential manifold and let JE" be a subset of the algebra &)(M), where r is an integer > 0.Show that the subalgebra generated by JE" is dense in b c r ) ( M )if and only if the following three conditions are satisfied: for each x E M there existsf€ 2 such thatf(x) # 0; (2) for each pair of distinct points x , y E M there exists f~ 2 such that f(x) #fO.); (3) for each tangent vector h, # 0 in T(M), there exists f~ % . ? such that Ohx. f# 0. (1)

31 2

XVll

DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

(To show that the conditions are sufficient, consider a compact subset K of M and a relatively compact open neighborhood U of K. If dimx(M) = n, there exist n tangent vectors h, at x and n functions f, E 2' such that Oh, .f,= a,, . We can therefore cover 0 by a finite number of open neighborhoods V, and define on each V, functions gi, ( I 5 j 6 n) belonging to S and such that the mapping x++(gll(x), . . ,gl.(x)) is a homeomorphism of V, onto an open subset of R",which together with its inverse is of class C'. By considering the compact set L which is the complement of the union of the sets (V, n 0) x (V,n 0 ) in fi x 0,show that there exist a finite number of functions hk E 2 ' such that, for each (x, y) E L, we have hr(x) # hk(y) for some index k. Finally, there exist a finite number of functions fi E &' such that, for each x E 0,we have fr(x) # 0 for some index 1. Deduce that there exist N functions F, E 2' such that the mapping CP : x-(F,(x), . . . , FN(x)) is a homeomorphism of U onto an open subset @(U) of R" whose closure does not contain 0, and such that both CP and CP -'are of class C'. Hence, for each function f e B(')(M), there exists a function p E &(.)(RN)such that p(0) = 0 andf(x) = p(F,(x), . . , FN(x))for all x E K. Finally, use the Weierstrass approximation theorem.)

.

.

6. Let 0, be the Lie algebra of compactly supported C" vector fields on a differential manifold M.

(a) If a # aCis an ideal in 0 =show , that there exists a point xo E M such that X(xo) = 0 for all X E a. (Argue by contradiction: If X E a is such that X(x) # 0, then for each field Z E (Ye there exists a field Y E such that 2 and [X,Y] coincide in a neighborhood of x, by using Problem 4. Then cover the support K of Z by a finite number of suitably l of class chosen open neighborhoods V, (1 5 i 5 m);take a partition of unity ( j J O s5m C" subordinate to the covering of M formed by M - K and the V1,and by considering successively the vector fields Z1=f1Z9

Zz=f2(Z-Z1),

Z3=h(Z-Z1-Z2),

-..,

prove that 2 can be written in the f o r m z [ X , , Y,],where XiE a and Y rE (3, .) i

(b) Let xo E M be such that X(xo) = 0 for all XEa, and let e = (U, p, n) be any chart of M at xo . Show that the local expressions of the vector fields X Ea relative to the chart c have all their derivatives of all orders zero at the point p(xo). (Suppose that the , X I , where X,are result is false for some field X E a, and consider the Lie brackets [X, the fields associated with the chart c (16.15.4.2)). (c) Deduce from (a) and (b) that the maximal ideals of the Lie algebra Bcare the & o , where for each point xo e M the ideal 3xo consists of all X EBcsuch that, for some chart c = (U, p, n) of M at xo , all derivatives of X of all orders vanish at the point p(xo)(in other words, such that X and the zero vector field have a contact of infinite order at the point xo : cf. Section 16.5, Problem 9). (d) Let X o E Bcand xo E M. Show that [Xo , (Yc] &o = if and only if XO(xo)# 0. (Use Problem 3 to show that the condition is necessary.) (e) Let M, N be two compact differential manifolds. Show that if there exists an isomorphism of the Lie algebra FA(M) onto the Lie algebra Y;(N), then M and N are diffeomorphic and every isomorphism of FA(M) onto YA(N) is of the form

+

XH T(u) . X, where u is a diffeomorphism of M onto N. (First observe that there is a canonical bijective correspondence between the closed subsets of M and the ideals of the Lie

15 EXTERIOR DIFFERENTIAL O F A DIFFERENTIAL p-FORM

313

algebra J t ( M ) : to each ideal a corresponds the set of all x E M such that X(x) = 0 for all X E a. Deduce that an isomorphism v of 9A(M) onto F t ( N ) defines a homeomorphism u of M onto N; then show with the help of (d) that if x,, E M is such that X(xd # 0, then we have (o(X))(u(xo))# 0, and deduce that if f~ b ( M ) , we have f o u-' E &N), by considering the vector fieldfXand its image under v . ) 15. T H E EXTERIOR DIFFERENTIAL O F A DIFFERENTIAL p-FORM

Let M be a dserential manifold. The mappingf t+ df (1 6.20.2) is a real differential operator of order 1 from the trivial line bundle M x R to the cotangent bundle T(M)*, as follows immediatelyfrom its local expression. We shall now define, for each p 2 1, a differential operator of order 1 from

(1 7.15.1)

T(M)* to KT(M)*. (1 7.15.2) Let M be a differential manifold. For each integer p 2 0, there ex-

ists a unique real differential operator d of order 1 from A T(M)* to satisfying the following conditions: P

(i)

A T(M)*,

P+ 1

I f a is a C" p-form and p a C" q-form on M, then

(1 7.15.2.1)

d(a A

p) = (dcr) A /3

+(-1)"~

A

dp

>= 0) (in other words (A.18.4) d is an antiderivation of the algebra of differential forms on M). (ii) When p = 0, d is the differential f Hdf (1 6.20.2). (iii) For each function f E b ( M ) , we have d(df) = 0.

(p 2 0 q

In the domain of definition V of a chart of M, every differential p-form may be written as the sum of a finite number of forms of the type f d g , A dg, A - - - A dg,, where f and the gk are real-valued functions of class C" on V. Condition (i), applied by induction on p, together with conditions (ii) and (iii), shows that we must have dcfdg,

h dgz A

A

dgp)= d f A dg,

A

dg2 A

...

dgp, which proves the uniqueness of d. By virtue of this uniqueness, we need only establish the existence of d in the domain of definition U of a chart (U, cp, n) of M. Then a p-form a is uniquely expressible as (1 7.15.2.2)

a= il

*.*

dg, A . -.A dg, +f(e, .dg,) A - - - A dg,, + ..- +f d g , A - . A (ex . dg,),

ex

(fdg,

so that the result follows from (17.14.10.1). To prove (17.15.3.4) we observe that, since i, and d are antiderivations, i, d + d ix is a derivation (A.18.7), and by virtue of (17.14.7.5) it is therefore enough to verify that 8, and this derivation take the same values for functions fof class C" and differential I-formsfdg. Since i, * f = 0, the first property is nothing but the definition of 8, f = (df, X ) = i, df. Also, by (17.14.10.1) and (1 7.14.7.5), we have 0

0

-

8,

*

c f d d = (0,

-

-

*f)dg + f @ x

d d = (0, .f)dg + f d @ , g)

and ix

*

dg) = (df, X ) dg - (dg, X ) df = (ex *f)dS ix * ( f d g ) =f (dg7 X > = f ( 4 .g). (&A

- (8,

. S ) df,

Consequently

d(i, * ( f d g ) ) = (0,

g ) df

*

+fd(&

*

9)

and (17.15.3.4) is verified for a =f dg. (iv) The proof is by induction on p. For p = 0, the formula (17.15.3.5) reduces to (df, X) = 8, * f for f E b ( M ) , which is just the definition of the operator 8,. If p > 0, by (1 7.1 4.7.7) we may write

0. (Take u' to be

Me1 - 1)h((e2)'

+ .. +

df' A dtzA

*

... A&,

where h is a nonnegative C"-function on R with arbitrarily small support, and such that h(0)> 0. Define U" similarly.) (b) Let M, N be oriented connected differential manifolds of the same dimension n, and let f be a proper (17.3.7) C"-mapping of M into N, such that the inverse image ' f ( u o ) of an n-form uo belonging to the orientation of N is 2 0 at all points of M (relative to the orientation of M). If f i s not surjective, show that all points of M are critical points o f f (16.23). (The set f(M) is closed in N. Suppose that there exists a point y1 E N -f(M) and a point x, E M which is not critical for f, and let y z = f(.x2)E ~ ( M ) Then . there exists an open subset U C N, diffeomorphic to R" and containingyl and y, (16.26.9). Use (a) to show that there exist two n-forms u', u" on N and an (n - 1)-form u such that du = u' - u", the forms u', u" and u being of class C" and compactly supported, and such that

'f(u")> 0, contradicting

(17.15.5.1).)

M be a pure differential manifold of dimension n. Let = - d K M be the canonical 2-form (17.15.2.4) on the cotangent bundle N = T(M)*. For each z E N, the mapping h,wi(hz). n(z) is an N-isomorphism of the tangent bundle T(N) onto the cotangent bundle T(N)*. For each C" differential 1-form a on N, there exists therefore a unique C" vector field X, such that a = ix,. 0. (a) If a,,!? are C" differential 1-forms on N, their Poisson bracket {a,,!?}is defined to be the differential 1-form - icx,,xgl . 0.Show that

9. Let

{a,/?I = -Ox,

. /? + Ox, .a + d(ir, . (ix, . a)).

For any three 1-forms a,p, y on N, show that { a ,{P, YH

+ {P, {Y,4 )+ {Y,I@,PI} = 0.

IfJ g are real-valued C"-functions on M,their homogeneous Poisson bracket is defined to be the function

{ f , s= } - i X d / .( j X d g . - -'.%'d/

'

= exdB

'

f.

Show that d{f,g} = {dJ dg} and that

{f,{g,hH + 19, {h,fH + v, {J9 ) )= 0, { J gh}= Nf,d + df,h3 for any three functions f,9, h.

324

XVll

DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

With the notation of (16.20.6), if the local expression of

+

n

i s c dt' A d v i , then I s 1

((ti),

(b) Replace n by n 1 and denote a point of N = T(M)* by (7,)) (0 $ i 5 n), where M is an open set in R"+'.Given any function F(z, x', ...,xn,y , , ...,).y defined on M x R",construct the function

defined on the open subset U : vo # 0 in N. If G is another function on M x R"and g the corresponding function on U, and if F, G are both of class C", then the function TO{ f,g} corresponds to the function

which is denoted by {F, G} if there is no risk of ambiguity, and is called the nonhornogeneous Poisson bracker of F and G. If w is the differential I-form

then we have

16. C O N N E C T I O N S IN A VECTOR BUNDLE

(17.16.1) Let G be a Lie group. We have seen (16.15.6) that for each pair of points x,y of G there is a well-determined isomorphism hx-yx-' . h, of the vector space T,(G) onto T,,(G), which depends only on the points x, y ; but the existence of such an isomorphism depends essentially on the supplementary Liegroup structure on the differentiable manifold G. In the absence of such a structure, there is no canonicalisomorphism of the tangent space at a point x of a pure manifold M onto the tangent space at another point y, i.e., no isomorphism which is determined uniquely by the manifold structure of M and the two points x, y. The notion of a connection in a vector bundle E over M is the mathematical expression of the idea of defining, for each point x E M, a procedure for providing an isomorphism of Ex onto E,, for points y "infinitely near" to x . Since the question is local, we shall consider first the case of an open subset U of R" and a trivial bundle E = U x RP.Let x be a point of U, and

16 CONNECTIONS IN A VECTOR BUNDLE

suppose that for each vector h rs R" such that x isomorphism (17.1 6.1 .I)

(x, U) H ( x

+ h, F(h)-'

*

325

+ h E U we have a linear U)

+

of the fiber Ex= { x } x Rp onto the fiber Ex+* ={x h} x RP,so that h H F(h) is a mapping of a neighborhood V of 0 in R" such that x V c U, into the vector space Y ( R p )of all endomorphisms of RP (a space which is isomorphic to Rp2).Suppose that F(0) = I,, and that F is of class C" in V, so that the mapping (h, U ) H ( X

(17.16.1.2)

+

+ h, F(h)-' - U)

of V x RPinto U x RPis indefinitely differentiable. Its derivative at the point (0, u) is the value at the point (x, u) E Exof the linear isomorphism "infinitely near" to the identity which we wish to consider; this value is, by virtue of (8.1.5), (8.9.1), and (8.3.2), (k, v ) H ( ~ ,v - (DF(0) * k) * u),

(1 7.1 6.1.3)

a linear mapping of R" x RPinto itself; DF(0) belongs to S(R"; S(Rp)), and therefore (5.7.8) the mapping (k, u) H (DF(0) * k) . u is a bilinear mapping of R" x RP into RP,which we shall denote by

Conversely, if we prescribe arbitrarily such a bilinear mapping and put

+

F(h) := I,, T,(h, .), then F(h) is an automorphism of RPfor all sufficiently small h (8.3.2), such that (DF(0) * k) * u = Tx(k, u).

(17.1 6.2) Let us now interpret these remarks in terms which are independent of the trivialization of E chosen. Since E = U x Rp and since (x, u) is a point

of the fiber Ex,the tangent space T(x,u)(E)may be identified with

TAU) x T,(RP), and therefore with ({x} x R")x ({u} x Rp).To a pair of vectors ( x , k) E Tx(M) and ( x , u) E Ex we associate, by means of (17.16.1.4), the vector (17.1 6.2.1)

CAX, k), (x, u)) = ((x, u), (k, - u k , UN)

326

XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

Moreover, T(E) is a vector bundle over E (16.15.4); if oE: T(E) + E is its projection, then we have oE((x,u), (k, v)) = (x, u). Finally, ((x, k), (x, u)) may be identified with the vector (x, (k, u)) of the fiber over x of the vector bundle T(U) @ E over U (16.16.1). Hence the mapping C, defined by (17.16.2.1) satisfies the following conditions:

the mapping

is a linear mapping of T,(U) into T(,, u)(E);and the mapping

is a linear mapping of Exinto T(n)-'(x, k). It is now easy to give the definition of a connection (or linear connection)in an arbitrary vector bundle E over a differential manifold M. We have merely to replace in the definitions and conditions of (1 7.1 6.2) the vector (x, u) by a vector u, E Ex,and the tangent vector (x, k) by a vector k, E Tx(M). We denote again by n : E + M, 0, : T(M) + M, and oE : T(E) E the canonical projections, and we observe that T(E) is a vector bundle over T(M) with projection T(n) (1 6.1 5.7). Finally, the composite mappings n oE and oY 0 T(n) of T(E) into M are equal, and if w denotes this mapping, then the triple (T(E), M, w ) is again aFbration, but not a vector bundle (16.15.7). Having said this, a connection (or linear connection)in the vector bundle E is defined to be an M-morphism (17.16.3)

0

(17.16.3.1)

C : T(M) @ E + T(E)

of fiber bundles over M (1 6.12.1), having the following properties:

the mapping (17.16.3.3)

k,

H C,(k

9

u3

16 CONNECTIONS IN A VECTOR BUNDLE

327

of T,(M) into Tux(E)is linear; the mapping

is a linear mapping of Exinto (T(E)), ,the fiber over k, of T(E) considered as a vector bundle over T(M) with projection T(n) (16.15.7). In particular, for each scalar c E R, we have

where m, is the mapping u, H c * u, of E into itself. We remark that the conditions (17.16.3.2) imply that the linear mappings k, H C,(k, , u,) and u, HC,( k, , u,) are injective. For each u, E Ex, the subspace of Tux(E)which is the image of T,(M) under the mapping k,

H C,(k

3

4)

is supplementary to the subspace TUx(n)-'(O,) formed by the vertical tangent vectors to E at the point u, (16.12.1). This supplementary subspace is sometimes called the space of horizontal tangent vectors to E at the point u,, relative to the connection C. For each vector field X on M, the horizontal lifting (relative to C) of X is a vector field on E, denoted by relc(X), and defined by (1 7.16.3.6)

reMX)(u,)

= CX(X(4, UX).

(17.16.4) Let (U, rp, n) be a chart of M such that E is trivializable over U, and let (n-'(U), 6, n + p ) be a corresponding3bered chart of E (16.15.1), with 6(n-'(U)) = q ( U ) x Rp. Then, for each point x E q(U)and each vector (k, u) E R" x RP,we have (1 7.16.4.1)

cq-l(x)(Tx(v-l). (x, k), O-'(x, u)) = q,, u)(e) . ((x, u), (k, - u k , u)),

where (k, u) H r,(k, u) is a bilinear mapping of R" x RPinto RP (1 7.16.1); the mapping (k, u ) H ( ~ ,-T,(k, u)) is called the local expression of C corresponding to the fibered chart. More explicitly, this mapping may be written as

(k',

..., k " , u l , ..., u P ) w ( k l ,..., k", - r i ( k , u ) , . . . ,

(x E rp(U)), where each ri is a bilinear form (1 7.16.4.2)

the

Tt(k,

U)

=

being C"-functions on rp(U).

1l-L(x)khu', h, 1

-rXp(k,u))

328

XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

Now consider another fibered chart, corresponding to a chart (U, rp’, n) defined on the same open set U. The transition diffeomorphism $ : rp(U) x Rp +. rp’(U) x RP is of the form (1 7.16.4.3)

( X l , . . . , 2,U 1 ,. . . ,UP) H(@(x), A(x) * u),

where @(x) = (Q1(x), . ..,@,,(x)) and A(x) = (uij(x)) is a square matrix of order p, so that the derivative D$ is the linear mapping

(k’, . ..,k”, d , . . . , v p ) ~ ( k ‘ ’ , . . . , k‘”, of‘, ..., dP) given by

2 Dh@,g kh, dY= uYld + 2 DhuYI khU’ I I

k” = (I7.16.4.4)

h

*

h,

((8.9.1) and (8.9.2)). Now let (k”,

. . ., k”’, u”, .. ., U ’ ~ ) H ( ~ ’.’.., , k‘”, -rk?(k’, u’), .. ., -T!$(k;

u’))

(x’ E rp’(U)) be the local expression of the connection C relative to the second chart, and let

(17.1 6.4.5)

T!Jk’,

u’) =

2 rG(X’)k’’U‘’.

8. Y

Also put @ - l ( x ’ ) = (ai1(X’), . . . , ai“(X’)),

Then we have from above, for x’

A-’(@-’(x’)) = (Hay(X’)).

-

= @(x) and u‘ = A(x) u,

also the first of the formulas (1 7.1 6.4.4) gives kh= 1 D, aih(x’) * k‘8 8

and we have UI =

c HIY(X’)U’Y, Y

so that finally we obtain the following expression for the T;(x‘):

16 CONNECTIONS IN A VECTOR BUNDLE

329

We see from this that, contrary to what one might have expected from (17.16.1), the presence of a connection on M does not enable us to define, for all x E M, a bilinear mapping rxof T,(M) x Ex into Ex; the mapping rxcor-

responding to a trivialization of E depends on this trivialization. In particular, to say that a connection is “zero” has no meaning, because all the ril can vanish without the l?g vanishing. (17.16.5)

Since, for each vector (k,, u,) in T(M) 8 E, the value C,(k,, u,) of a connection belongs to the tangent space TUx(E),it follows that the sum of the values of two connections at the point (kx, u,) is defined, as is also the product of C,(k, , u,) by a scalar. We may therefore form linear combinations Cj of connections in the bundle E, the coefficients being real-valued

cfi

fi

J

(2%’-functionson M; but in general the M-morphisms so obtained are not connections, by reason of the first condition (1 7.16.3.2). Nevertheless,there are two important particular cases. In the first place, if C and C’ are two connections in E, their difference C‘ - C is no longer a connection, but is an M-morphism (17.16.6)

(1 7.16.6.1)

A : T(M) @ E + T(E)

such that (1 7.1 6.6.2)

T(n) * A,@, , u,) = O x , o,(A,(k,,

u,)) = U,

.

The first of these relations shows that A,(k,, ux) is an element of Tux(E,), the tangent space to the3ber Ex at the point u, , identified with the subspace of *‘ vertical ” tangent vectors in Tux(E). Furthermore, k, I+ A,( k, , u,) is a linear mapping of Tx(M) into TUx(Ex),and u, H A,(kx, u,) a linear mapping of Ex into (T(E)),x, the fiber of T(E) over the point 0, E T,(M) relative to the fibration (T(E), T(M), T(n)), which is isomorphic to the tangent bundle T(E,), identified with Ex x Ex. Also we have a canonical isomorphism T,_ : Tux(E,) --* Ex (1 6.5.2) which, modulo the preceding identifications, is the restrictiop to Tux(E,) of the second projection. It follows that, if we put

330

XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

B, is a bilinear mapping of T,(M) x Ex into Ex.In other words, we have defined a bilinear M-morphism B : T(M) 0 E +E, which, by abuse of language, is called the diference of the connections C' and C. Relative to a local trivialization of E in which C and C' correspond, respectively, to the bilinear functions r and r' (17.16.1), the morphism B corresponds to -r'+ r. Conversely, if B : T(M) 0 E + E is any bilinear M-morphism, then if we put A,(k,,

u 3 = Gx1(Bx(kx9UX)),

C f A is a connection, for every connection C in E.

(17.1 6.7) Let (U, ,cp, , n) be afamily of charts of M such that E is trivializable over each U, , and such that the family (U,) is locallyfinite. For each a, let C, be a connection in the vector bundle n-'(U,), and let vh> be a Cw-partitionof unity subordinate to (U,) (16.4.1). Then cf.C, is a connection in E (with the understanding that we replace &(x)(C,),(k, space TUx(E)for each point x 4 U,).

, u,) by the origin of the tangent

Since for each point x E M there is a neighborhood of x which meets only a finite number of the U,, and since c & ( x ) = 1, the only point that needs to be checked is that the functionsfa C, are Cw-mappingsof the whole of T(M) 8 E into E. If x is a frontier point of some U, ,there is a neighborhood V of x in M which does not meet the support of fa, and therefore, for each point (k,,, u,) of T(M) 0 E lying over a point y E V, by definition f.( y)(C,),,(k, , u,) is the origin of the vector space TUy(E).The assertion now follows by taking a trivialization of T(E) (considered as a bundle over M) in a neighborhood of x. In particular : (1 7.16.8)

There exists a connection in any vector bundle.

Choose a family of charts of M having the properties stated in (17.6.7), and define each connection C, by taking a particular trivialization of n-'(U,) and taking the corresponding mapping (17.16.1.4) to be, e.g., zero for all x E (PU(U3.

(17.1 6.9)

If (f,g) is an isomorphism of a vector bundle E over M onto a vector bundle E' over M', then every connection C in E is transported by (A g) to a connection C' in E', such that

17 DIFFERENTIAL OPERATORS ASSOCIATED WITH A CONNECTION

331

17. DIFFERENTIAL OPERATORS ASSOCIATED WITH A C O N N E C T I O N

(17.17.1) We take up again the situation considered in (17.16.1). Let Z be an open subset of some Rq,and let G be a C"-mapping of Z into the bundle E, so that we may write G(z) = (f(z), g(z)), wheref(resp. g) is a C"-mapping of Z into U (resp. into Rp). Let W be a neighborhood of 0 in R4 such that z W c Z, and for each w E W put h(w) =f(z + w) -f(z). Since the point G(z + w) lies in the fiber Ef(,)+,,(+, we may consider the point

+

(f(z), F(h(w)) . g(z + w)), which lies in the fiber EfCz,,and we are thus led to take as "derivative" of G the derivative of the mapping wHF(h(w))

*

g(z

+

W)

at the point w = 0; bearing in mind that F(0) = 1R P , and using (8.1.4) and (8.2.1), this gives us the following linear mapping of Rqinto RP: (17.17.1 .I)

WH

Dg(z) * w

+ (DF(0)

*

(Df (z) * w)) * g(z).

Since DG(z) . w = (Df(z) * w, Dg(z) * w), the right-hand side of (17.1 7.1 .I) can also be written as

-

DG(4 . w - (DfW w, -rf(z)(Df(z) * w, g(z)>) in which appears the value Cf(z)(Df(z)* w, g(z)) of the connection C . (17.17.2) We shall now give an intrinsic definition of the covariant derivative relative to a connection C in a vector bundle E over M. Let N be a differential manifold and let G be a Cr-mapping (r 1 1) of N into E. It follows from (17.16.3.2) that, for each z E N and each tangent vector h, E Tz(N),the vector Tz(G)

. hz - Cn(G(z))(Tz(B G, . hz

9

G(z))

belongs to the tangent space TG(Z)(En(G(z))) to the fiber through the point G(z:)of the bundle E. Since this fiber is a vector space, we may apply the canonical translation zG(z)(16.5.2) to the above vector: the vector so obtained in tliefiber Es(G(=))

is called the covariant derivative of G at the point z (relative to the connection C ) in the direction of the tangent vector h, .

332

XVll

DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

When M and N are open sets in R" and R" (a situation to which we can always reduce by a suitable choice of charts), G is a function of the form ZH(~(Z), g(z)) and the connection C is given by (17.16.2.1). We have then, with h, = (2, h),

or, expressing everything in terms of coordinates as in (1 7.16.4),

for the ith component of V h z G in Rp. It is clear that h, HV h z * G is a linear mapping of T,(N) into If GI, G, are two mappings of class C' (r 2 1) of N into&, such that A Gl = II G2 (in other words, two liftings to E of the same C'-mapping of N into M), then we have 0

0

For every scalar function CJ of class C' on N, the mapping z Ho(z)G(z) is a C'-mapping of N into E which we denote by aG; clearly 7t 0 (aG) = A G. Bearing in mind the definition of (17.14.9), it is easily seen that 0

by reducing to the case where the covariant derivative is given by (17.17.2.2). Finally, if u : N1 H N is a mapping of class C', then for each point z, E N, and each tangent vector h,, E TzI(Nl),it follows immediately from the definitions that (1 7.17.2.6)

where z

= u(zl)

VhZl

*

(G 0 U ) = Vhz . G,

and h, = T,,(u) . h,, .

Remark By virtue of (17.17.2.1), the relation v h z * G = 0 signifies that the tangent vector T,(G) . h, is horizontal (17.16.3) at the point G(z).

(17.17.2.7)

(17.17.3) Having defined the covariant derivative of the mapping G : N + E at a point z E N in the direction of a tangent vector h, E T,(N), it is now easy

18 CONNECTIONS ON A DIFFERENTIAL MANIFOLD

333

to define the covariant derivative of G (relative to C ) in the direction of a vectorjield 2 E T;(N);this is the mapping V, * G of N into E which at each point z E N has the value VZ(=). G. The results of (17.17.2) give rise to the formulas

for two vector fields Z,, 2, on N;

V,

(17.1 7.3.2)

*

G = c(VZ * G )

for every scalar function cr of class C' on N;

for two liftings G,, G, to E of the same mappingf: N-, M of class C'; (17.1 7.3.4)

v, - ( c r ~=) (e, - c r ) +~ cr(~, - G)

for every scalar function cr of class C' on N. In the particular case where N = M and n G = 1, (that is, where G is a section of E), we see that for each C" vector field Xon M, the operator V x is a real differential operator of order 6 1 from E to E. 0

18. CONNECTIONS ON A DIFFERENTIAL MANIFOLD

(17.18.1) Given a differential manifold M, a connection (or linear connection) on M is (by abuse of language) a connection in the tangent vector bundleT(M), that is to say an M-morphism of T(M) @T(M) into T(T(M)) satisfying the conditions of (17.1 6.3), with E replaced by T(M). Given such a connection C we define, for each vector h,ET,(M) and each vector field Y on M, the covariant derivative (relative to C ) VhX* Y of Y at the point x in the direction of h, (17.1 7.2.1). Hence, for each vector field X on M, we have a differential operator YHV, . Y from T(M) to T(M) (17.17.3), with the following properties: (17.1 8.1 .I)

v,,, v,

(17.1 8.1.2) (17.1 8.1.3)

V,.(Y,

*

Y = v,,

Y + v, . Y ,

*

Y = O(V,

*

+ Y,)=Vx-

Y),

Y , +VX' Y , ,

XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

334

vx - (or)= (ex

(17.18.1.4)

Y

+ o(vx- Y ) ,

where Q is any C" scalar function on M. In particular, if U is an open subset of M over which T(M) is trivializable, so that the d(U)-module FA(U) of C" vector fields on U admits a basis Y,, ... , Y,,,then the above formulas show that a knowledge of the vector fields VYi. Y j on U for 1 S i,j 6 n determines Vx Y completely for any two vector fields ,'A Y E FA(U). In particular, if (U, cp, n) is a chart of M, and if (Xi)lsi6nis the basis of FA(U) over B(U) associated with this chart (16.15.4.2), then it follows from (17.1 7.2.3) that

-

(17.18.1.5)

VX,

*

XI, =

i

r;kXi,

which shows that if two connections give rise to the same covariant derivative, they are identical (cf. Problem 2). (17.18.2) Let C be a connection on a differential manifold M , and let f : N + M be a mapping of class C" (m 2 1). For each pair of integers r 2 0, s 2 0, let W : ( f ) denote the vector space of all C" liftings o f f to the bundle T:(M). Then, for each Z E N and each tangent vector h,ET,(N), there exists a unique linear mapping G w V h Z G of W',(f) into the fiber (T:(M)),.(,, which: (1) for r = s = 0 coincides with Oh, (17.14.1); (2)for r = 1 and s = 0 coincides with the covariant derivative defined in (17.17.2.1); (3) satisjies the following conditions :

-

for G'

E

Wi:( f ) and G" E W$(f);

(17.18.2.2)

ohz

(G, G*) = (vhz

for G E W:(f) and G*

E

. G, G*(Z)) + (G(Z), v h ,

*

G*)

Bs(f).

The proof follows that of (17.14.6) step by step, replacing the vector field X by the vector h, , and tensor fields on M by liftings off. (It is also possible to obtain (17.14.6) and (17.18.2) simultaneously as corollaries of the same algebraic lemma: see Problem 1.) (17.18.3) If now 2 is a C" vector field on N, we define V, * G as in (17.17.3) for a lifting G E a:(f ) , by putting (V, * G)(z) = VZ(,) . G for all z E N; it is a lifting off to T:(M), of class C'"-'. The formulas of (17.17.3) remain valid without any change.

18 CONNECTIONS ON A DIFFERENTIAL MANIFOLD

335

(17.18.4) Consider the particular case where M = N and f is the identity mapping. The liftings off are then tensorfields on M . If U E F : ( M ) is a tensor field, the mapping (V*, X)H (V, * u,V * )

of F:(M) x FA(M) into I ( M ) is b(M)-bilinear by virtue of (17.17.3.2). Consequently this morphism defines a tensorfieldon M, belonging to F I +l(M), which is called the covariant derivative ofU (relative to the connection C) and is denoted by VU or VcU.Thus we have (17.1 8.4.1)

(VU,v* 0 X )

= (VX

- u,V * )

and it is clear that U H VU is a diferential operator of order II from T:(M)to T:+l(M).If o E d ( M ) is any scalar function, then

V ~= T do

(17.1 8.4.2)

by virtue of (17.14.1.1) and (17.18.2). Furthermore, we have (17.1 8.4.3)

V(&)

= .(VU)

+ U 0 do.

For with the notation introduced above, we have

(v, . (q, v * ) = o(v,. u,v*> + (ex . by use of (17.17.3.4); but 8,

*

v*)

o = (do, X), and

(do, X)(U, V * )

= (U 0 do,

V* 0 X )

by the definition of duality in tensor products (A.II.I.4); (17.18.4.3) now follows.

the formula

Let M be a diferential manifold and N a closed submanijold of M . Then every connection on N is the restriction of a connection on M. (17.1 8.5)

There exists a locally finite denumerable covering of M by open sets which are domains of definition of charts of M for which the conditions of (16.8.1) are satisfied for the submanifold N. Using a partition of unity subordinate to this covering and (17.16.7), we reduce to the situation in which N is an open subset of R"'and M = N x P, where P is an open subset of R"-",so that T(N) may be identified with N x R", T(P) with P x R"-", and T(M) with T(N) x T(P). Then if y E N and h', k' E R", the connection C' given on N may be written (17.1 6.2.1) in the form C:(h', k') = (h', -l-i(h',

k')),

336

XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

where r6 is a bilinear mapping of R"'x R" into R"',and y H r; is of class C" on N. We then take for C the connection on M given by Cdh, k) = (h, -r;,ix(Prl

where x

E

h, pr, k)),

M and h, k E R" (considered as the product R"' x R"-"').

(17.18.6) Let f:M' + M be a local difleomorphism (16.5.6) and let C be a connection on M. Then there exists a unique connection C' on M' such that, for each open subset U' of M' for whichf 1 U' is a diffeomorphism of U' onto the open subsetf(U') of M, the restriction of C tof(U') is the image underf of the restriction of C' to U' (17.16.9). For this requirement determines C;, at every point x' of M, because if U', V' are two open neighborhoods of x' such that f 1 U' and f 1 V' are diffeomorphisms onto f(U') and f(V'), respectively, then f l (U' n V') is a diffeomorphism onto f(U' n V'). The connection C' is said to be the inverse image of C underf. This remark may be applied in particular to the situation in which (M', M,f) is a covering of M. If x i , x; are two distinct points of M' such that f ( x ; ) = f ( x ; ) = x E M, then there exist disjoint open neighborhoods U; of x i and U; of x; in M' such that f(Ui) =f(U;), and a diffeomorphism g : U; --f U; such that f I U; = g (fl U;). With the notation used above, if C; and C; are the restrictions of C' to U; and U;, respectively, then C; is the image of C; by g . Conversely, if a connection C' on the covering space M' has this property, it is immediately seen that we may define a connection C on M by taking (with the notation of the beginning of this subsection) the restriction of C tof(U') to be the image underfl U' of the restriction of C' to U'. The above condition on C' then ensures that C is unambiguously defined. 0

PROBLEMS

1. Let A, A be two commutative rings; p : A + A a surjective ring homomorphism; E, F two free A-modules with bases (eJl r j n and (A)] 1 6 n ; Q a bilinear form on E x F such that cD(ei,fi) = S1,. Also let E , F be two free A-modules with bases (e;)lsisn and (f;)]blJn, and W a bilinear form on E X F such that W(e;J;)= Si,. Let pb : E -+ E' be the A-module homomorphism defined by pb(el) = e; (1 6 i 5 n), and py : F + F the A-module homomorphism defined by p?(h) =f,'(l 5 i 5 n). Suppose also that we are given a mapping L : A + A such that L(ab) = L(a)p(b) p(a)L(b), and a linear mapping LA : E + E such that Lb(ax) = L(a)pb(x) + p(a)Lb(x) for all a, b E A and x E E.

+

(a) Show that there exists a unique mapping Lp : F -+ F such that

+

LB(ay) = L ( ~ ) P ? ( Y )p(a)L?(y)

19 THE COVARIANT EXTERIOR DIFFERENTIAL 337

for all a E A and y

E

F, and

L(@(X,Y)) = (P'(Lb(x), pP69

for all x E E and y

E

+ @'(pb(x), LPQ)

F.

(b) Show that for each pair (p, q) of integers 2 0 there exists a unique mapping L; : E@P@F@q+E'@P@F'@q

such that L%a4= L ( a ) p ! m

+ p(a)Y(z)

for a E A and z E E@P0Feq,where p; is the canonical extension of pb and p? to the tensor product); L:,+:(u

0v ) = L W 0pXv) + p%u) 0LXv)

for u E E m P@ Faqand v E E@'@ F*s; and L ( W , v*))

= WLZ(U),

p%*))

+ @'(p:(u), Y ( v * ) )

and u* E Eeq 0Fa', where (9 and (9' are the canonical extensions of for u E E e P @ F@* the bilinear forms to the tensor product. (Follow the proof of (17.14.6).) 2.

Let E be a vector bundle over M, and let Diff,(E) denote the B(M)-module of differential operators of order sl from E to E. Show that every B(M)-linear mapping X - P X of Y;(M) into Diff,(E) such that P x .( a V ) = (Ox . u)V u(Px . V ) for all u E B(M) and all Cm-sections Y of E, is of the form XHV, relative to a unique connection in E. (Show first that if X vanishes in an open set U, then Px I U = 0.)

+

3. Generalize the result of (17.18.1) to linear connections in an arbitrary vector bundle E over M. Consider in particular a Cm-sectionG* of the dual E* of E, and associate with it. the scalar function on E given by u,~S2(u,) = (u,, G*(x)). Show that

4.

.

With the notation of Section 16.19, Problem 1Is, how that a linear connection in E is a mapping C : E x.T(B)+T(E) such that po C = lexsT(.), and such that C is a bundle morphism of E x T(B) into T(E) both as bundles over E and as bundles over T(W.

5. With the notation of (17.18.4), show that V(c$U) = c;(VU) for any contraction c$.

19. T H E COVARIANT EXTERIOR DIFFERENTIAL

(17.19.1) The formula (17.15.3.6) enables us to calculate at each point x E M the value (d(w(x),h , k,) ~ of the exterior differential of a I-form w by considering two C" vector fields X , Y on M, such that X ( x ) = h, and Y(x)= k,, and calculating the value of the right-hand side of (17.15.3.6) at

338

XVll

DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

the point x . It is remarkable that although each of the terms on the righthand side depends on the values of the fields X, Y in a neighborhood of x and not merely at the point x , yet the left-hand side does not. We shall see that an analogous phenomenon presents itself when the Lie derivative is replaced by the covariant derivative relative to a connection. (17.19.2)

In detail, let E be a vector bundle over M, and let f be a C"P

mapping of a differential manifold N into M. If o :/\T(N)+E is a C"mapping such that (f, o)is a oector bundle morphism (1 6.15.2), then 0 is said to be a C" differential p-form on N with values in E (relative to the mapping f)(cf. Section 20.6, Problem 2). If Z1, Zz , ... , 2, are C" vector fields on N, then for each point z E N the element o(Zl(z) A Z,(z) A * A Zp(z)) belongs by definition to E,,,,; moreover, it is immediate that the mapping Z H O(z,(Z) A

zz(Z) A

''* A

zp(Z))

is a C"-lifting off to E, which we denote by

o (2,A &

A

~ 2 ~ ) .

(1 7.1 9.3). Now suppose that we are given a connection C on E, and consider first the case p = 1. Let 0 be a differential 1-form of class C" on N, with values in E, and let X , Y be two CcO-vector fields on N. By analogy with (17.15.8.1) we form the C"-lifting off to E (17.19.3.1)

v, - (0

*

Y )- v

y

*

(0* X

) - 0 * [X, Y].

We shall show that the value of this mapping at a point z E N depends only on the values X(z) and Y(z) of the3elds X , Y at the point z. For this it is enough to show that if we replace X (resp. Y ) by O X (resp. aY), where a is a scalar function of class C" on N, the value of (1 7.1 9.3.1) is obtained by multiplying by a(z) the value for X and Y. For, reducing to the case where M, N are open subsets of RP, Rq and E = M x R", we have X(z) = ( z , g(z)) and Y(z) = (2,h(z)), and the formula (17.17.2.2) shows that (17.19.3.1) is a bilinear function of the vectors (g(z), Dg(z)) and (h(z), Dh(z)); hence, by virtue of (8.1 .4), the condition stated above is necessary and sufficient for this function not to depend on Dg(z) (resp. Dh(z)). Now we have vex.(0 * Y) = aV* (a* Y ) by (17.17.3.2), and

-

0

-

*

(ax) = a(o * X),

vy(o (ax)) = (e,

- a)(O - x)+ o(v, - . x)) (0

19 THE COVARIANT EXTERIOR DIFFERENTIAL

339

by (17.17.3.4), and finally

by (17.14.4.2). Hence the result. Since (17.19.3.1) is an alternating bilinear function of ( X , Y ) , there exists a unique C" differential 2-form on N with values in E, called the covariant exterior differential of o (relative to C ) and denoted by d o , such that for any two C" vector fields X , Yon N we have (17.19.3.2)

d o * ( X A Y ) = V , * (o Y ) - V , * (o* X ) - o * [ X , Y l .

(1 7.19.4) This result generalizes easily to differentialp-forms on N with values in E, where p > 1. We take the analog of the formula (17.15.3.5) by proving that if o is a C" differential p-form on N with values in E and if Xo , XI, .. ., X p are p + 1 C" vector fields on N, then the C"-lifting off to E (1 7.19.4.1) A

8, A

has at each point z E N a value which depends only on the values X j ( z ) of the vectorJields X j at z (0 5 j 5 p). The method of proof is exactly the same: we replace each X j successively by axj. In this way we establish the existence and uniqueness of a C" differential (p + ])-form d o on N, with values in E, such that d o ( X , A ... A X,) has as its value (17.19.4.1); d o is said to be the covariant exterior diRerentia1 of o (relative to C). 0

Finally, since by convention A T(N) = N x R (1 6.16.2), a C" differential 0-form on N with values in E is identified with a C" lifting G off to E. For each C" vector field X o n N, the value of V , * G at a point z E N depends only on X(z) by definition (17.17.3). Hence there exists a unique differential I-form of class C" on N, with values in E, which is called the covariant exterior differential of G and is denoted by dG, such that (1 7.19.4.2)

dG-X=V,.G.

340

XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

(17.19.5) With the notation and hypotheses of (17.19.2), let u : N, -+ N be a C"-mapping, and consider the composite mappingf, = f o u : N, + M. It is

clear that the pair (A,a,),where o1= o 0 A T(u), is a morphism of vector bundles. The differential p-form o1on N, with values in E is called the inverse image of o by u. Now suppose that we are given a connection C on E. Then the covariant exterior differentials relative to f and tof, = f o u satisfy the relation P

(17.19.5.1)

d(o

P

A

P+ 1

A T(u)) = (do) o A T(u)

for any C" exterior differentialp-form o on N with values in E. We shall give the proof only for p = 0 and p = 1 (cf. Problem 1). For p = 0, in view of the definition (17.19.4.2), the formula (17.19.5.1) reduces to (17.17.2.6). For p = 1, since the question is local with respect to N, N,, and M, we may assume that M yN, N, are open subsets of R"', R",R"', respectively, and that E = M x Rq. Then o may be written as (z, h)w(f(z), A(z) h), where ZHA(Z) is a C"-mapping of N into 9 ( R " ; Rq); this implies that DA(z) is an element of 2Z2(R"; R4) ((5.7.8) and (8.12)). It now follows from the formula (17.17.2.2), the definition (17.1 9.3.2), and the rules for calculating derivatives in vector spaces (Chapter VIII) that d o is the mapping

-

(17.19.5.2) (2,

h A k)H(f(Z), DA(z) * ((h, k) - (k, h)) + r,(z)(Df(z) * h, A@) * k) - I-/(&?f(4 * k Y 4 4 * h)).

(Incidentally, this * calculation provides another proof of the fact that (17.19.3.1) depends only on the values X(z) and Y(z).) Likewise, o1may be written as (z,,hl)i+(fl(zl), A,(z,) . hl), wheref, = f o u and

A,(z,) = A(u(z1)) O Du(z1). We have then Dfl(z,) h, = Df(u(z,)) * (Du(z,) * h,); also the mapping h, H DA,(z,) (h,, k,) is the derivative of z1wA,(z,) k, ; consequently, in view of the definition of Al, we have

-

-

-

DA,(z,) (hl, k,) = DA(u(z1)) * (Du(z1) * h,, Du(z1) * k,) + A(u(z1)) * (DZU(Zl)* (hl9 kl)). Inserting these values of Df,(z,) and DA,(z,) into the expression for do, analogous to (17.19.5.2), we obtain, using the symmetry of D2u(z,) (8.1 2.2),

(do,)(z,, h, A k,)

= (do)(z,

h A k),

where z = u(z,), h = Du(z,) * h,, k = Du(z,) k,. This proves (17.19.5.1) in the casep = 1.

19 THE COVARIANT EXTERIOR DIFFERENTIAL

341

PROBLEMS

1. Prove the formula (17.19.5.1) for arbitraryp, as follows. To calculate the value of the

-

+

P+ 1

left-hand side at a ( p 1)-vector ko A kl A * * A k, E A Tsl(Ni), consider separately the cases where the vectors T.,(u) * k, (0 sj s p ) are linearly independent or linearly dependent. In the first case, reduce to the situation where Nl is a submanifold of N of dimension p 1 by use of a chart, and use the fact that in calculating the value of (17.19.4.1) we may assume that the X , are fields of tangent vectors to N1. In the second case, we may assume that T,,(u) ko= 0 and that the fields X, such that X,(z,) = k, for 0 s j s p are such that [X,,X,]= 0; use the formula (17.17.2.6).

+

2. Let M be a differential manifold. Given two integers p 2 1, q 2 0, and M-morphisms P

4

P : A T(M) + T(M), Q : A T(M) +T(M), we define an M-morphism

by the formula (P

Q) * (Xi

A Xz A

s . 0

A XP+,-,)

(When q = 0, Q is identified with a vector field X , and Q * (Xu(i)A A Xu(,))has to be replaced by X.) Likewise, for each scalar-valued differentialp-form a on M, we define a (p q - 1)form a K Q by the formula -1.

+

(with the same convention for q = 0). Extend this definition to the casep = 0 by putting f ilQ = 0 for all functions f~ b(M). Show that: (a) If p 2 1 , q 2 1 and if a is a scalar-valued r-form, we have (a K P ) K Q - a 7 1 ( P 7 T Q ) = ( - l ) c p - ' ) t q - 1((a ) 71 Q) 7 i P - a

K (Q FPN.

(b) ~ T ( M ) K Q = Q ; P K ~ T ( M ) = P P (c) If (I,v are endomorphisms of T(M) (1-forms with values in T(M)), or tensor fields of type (1, l), then (I

~ v = u o v ,

for any vector field X. 3. The notation is as in Problem 2.

.71X=u*X

XVll DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

342

(a) The mapping a-a iT Q of the exterior algebra at of scalar-valued differential forms on M into itself is an untiderivation of degree q - 1 which is zero on Qo This antiderivation is denoted by i Q . Conversely, every antiderivation of at of degree q - 1 which vanishes on do is of this form. (Observe that such an antiderivation is a

.

9

differential operator from T(M) to A T(M).) (b) The mapping D : a*(da) i T Q + ( - l ) q d ( a T Q ) of at into itself is an antiderivation of degree q such that

D d = (-1)gdo D.

(1)

0

(We have D = iQ 0 d - (- 1)q-ld iQ .) This antiderivation is denoted by dQ . Conversely, every antiderivation D of d of degree q which satisfies condition (1) is of the 0

Q

form da . (Observe that D is a differential operator from M x R into A T(M).) (c) Every antiderivation D of d of degree r can be written uniquely as D = ip r+l

r

+ dQ,

where P (resp. Q) is an M-morphism of A T(M) (resp. A T(M)) into T(M). (Determine Q by the condition that dQ coincides with D on 80.)(Frohlicher-Nijenhuis theorem.) (d) If Q is a vector field X (i.e., if q = 0), then iQ coincides with the interior product ix If Q is an endomorphism u of T(M) (i.e., if q = l), then

.

and, in particular, ilTcM) * a =pa. If Q is a vector field X,then dQ coincides with the Lie derivative O x . We have d l T ( M ) - d (exterior differentiation in 4. (el With themme notation, there exists a unique M-morphism [P,Q ] of T(M) such that [dP

(2)

We have

9

P+4

A T(M) into

~ Q= I d[p,Q l .

[Q,pI = (-l)pq+’[p, Q1, / ~ T ( M )9

Ql

= 0,

+

(-1)“[P, [Q, Rl1+ (-1)‘”[Q, [R,PI] ( - 1 ) ” W

[ P , Q11= 0.

(f) If P = X and Q = Yare vector fields, then [P,Q ] is the usual Lie bracket [ X , Y ] . For each Q , [ X , Q ] is the Lie derivative Ox . Q (where Q is identified with a tensor field of type (1,q)). If P = u and Q = u are endomorphisms of T(M), then [U, U ] *

-Id

*

+

+

+

Y ] [u-X,U.Y ] . - ( a . [ X , Y ] ) u * ( u - [ X , Y ] ) [u * x,Y ] - D * [U . x,Y ] - U * [X, 0 . Y ] - D . [X, U .Y ]

( X A Y )= [ u . X ,

U

S

and, in particular, &[Id, U ]

*

( X A Y ) = [ u . X , U ‘ Y ]+ U ’ ( u . [ X , Y ] ) -

(the Nijenhuis torsion of u). (9) Show that IiQ, dpl= d

U .

[ u . X, Y ] - U

K Q + (- I)%P. Q I .

*

[X,U * Y ]

20 CURVATURE AND TORSION OF A CONNECTION 4.

343

Let M be a pure differential manifold of dimension n. Consider the canonical exact sequence (Section 16.19, Problem 11)

o

--f

T(M) x T(M) LT(T(M))1:T(M) x T(M) +0.

p is called the certical endomorphism of T(T(M)). Its local The endomorphism J = expression relative to a chart of M at a point x is 0

(X,

h,, Ux,khx)-(X,

hx,0, Ux).

It is a T(M)-morphism for the vector bundle structure on T(T(M)) with projection not for the bundle structure with projection T(oM).It is of rank n at every point, and we have J J = 0.

o ? ( ~ )but ,

0

(a) With the notation of Problems 2 and 3, show that J TJ=O, [ i j ,d j ] = 0,

[J,J]=O, d j d j = 0.

[d,d j ] = 0,

0

(b) Show that, for all vector fields Z, Z'on T(M), we have

+

.

[ J .z,J Z ]= J . [ J .z,2 1 J * [Z,J . 2 1 , [ i ~ , i ~ -]i J= . z .

20. CURVATURE A N D TORSION OF A C O N N E C T I O N

Let E be a vector bundle over M, let C be a connection in E, let E a C"-lifting off. Since dG is a C" differential 1-form on N with values in E, we may consider the differential 2-form d(dG) on N with values in E. By contrast with the case of the exterior differential (17.1 5.3.1), however, d(dG) is not identically zero in general; but, for all h, , k, in T,(N), the vector d(dG) * (h, A k,) E Eft,, depends only on the value G(z) E E,,,, of G at the point z (and not on its values in a neighborhood of z). To see this, let X , Y be two C" vector fields on N such that X ( z ) = h, and Y(z) = k,. Then the vector d(dG) . (h, A k,) is by definition ((17.19.3) and (17.9.4)) the value at z of the following lifting o f f t o E: (17.20.1)

f:N -+ M be a C"-mapping and G : N

(17.20.1 .I)

V,.

(Vy

*

G) - V y . (V, . G) - Vr,,

yl

*

G.

By the same argument as in (17.19.3), it is enough to show that if o is any C" scalar-valued function on N, the value of (17.20.1.1) for oG is obtained by multiplying the corresponding value for G by o(z). Now, by virtue of (17.17.3.4), we have

vx . (vy. ( 0 ~ )=) v, . ((e, . o ) +~o(vy. GI) = (ex . (e, . O ) ) G + (e, o)(v,

+ (0,

+

*

a)(Vy * G)

+ o(VX

*

G)

(Vy

. G)).

344

XVll

DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

Interchanging X and Y in this formula, and remembering that V [ x ,Y] * (4= ( e [ x , Y] * 4 G + d v f x , rim G),

it follows that our assertion is a consequence of the definition of [ (1 7.14.3). The expression (17.20.1.1) is clearly a linear function of G. Hence there is an endomorphism R, (h, A k,) of Eft,, such that

(17.20.2)

d(dG) (h,

A

k,) = R,(h,

A

-

k,) G(z).

Furthermore, it is immediately seen that the mapping

r f : h, A k, H R,(h,

(17.20.2.1)

A

k,) 2

is such that (f, r,) is a vector bundle morphism of A T(N) into Hom(E, E) = E * @ E (in other words, r, is a diferential 2-form on N with values in E* 8 E). (17.20.3) Now let u : N, + N be a C"-mapping. By applying the formula (17.19.5.1) for p = 0 and p = 1 we obtain (17.20.3.1)

d(d(G u)) = (d(dG)) o 0

2

A T(u).

In the notation introduced in (17.20.2), this takes the form (17.20.3.2)

for if G, : N,

+E

is any lifting of f o u, and if z1E N,, then there exists a

lifting G o f f to E such that G(u(z,)) = G,(z,) ((1 6.15.1 -2) and (1 6.19.1)). (17.20.4)

Consider in particular the case where N = M and f M-morphism r l Mis denoted simply by

=

1,.

The

2

r : AT(M)+E*@E

and is called the curvature M-morphism (or simply the curvature) of the connection C in E. Knowledge of this morphism determines all the differentials d(dG), by virtue of (17.20.3.2); in other words, with the notation of (17.20.1), if G is any (?-lifting off, we have

20 CURVATURE AND TORSION OF A CONNECTION

345

When M is an open set in R" and E = M x R4,then we have h, = (x, h) and k, = (x, k) with h, k E R" for all x E M ; if u, = (x, u) is any element of Ex,an easy calculation using (17.1 7.2.2) and (17.1 9.5.2) gives

(Since x H r , is a mapping of M into B2(R", R4;Rg), it follows that Dr, belongs to S?(R"; Y,(R", R4; R4)), identified with the space of trilinear mappings Y,(R", R", R4; R4).) (17.20.5) Consider in particular the case where E = T(M), so that C is a linear connection on M. Then the curvature morphism r of C defines a bilinear M-morphism (hx, k,)wr * (h, A k,) of T(M) 0 T(M) into T:(M) and hence may be identified with a tensorfield r of type (1, 3) (16.1 8.3), called the curuature tensorfield (or, by abuse of language, the curvature tensor) of the connection C. If (U, cp, m) is a chart of M, and ( X i ) t j i J mthe basis of FA(U) over b ( U ) associated with this chart (16.1 5.4.2), then from (17.20.4.2) we have (17.20.5.1)

from which we obtain the corresponding components of the curvature tensor r: (17.20.5.2)

ar:, art..

r!. = -- 2 + C (rLi lJk

ax)

axk

h

-

ri,).

(17.20.6) Again assume that E = T(M). The identity mapping 1,(,, of T(M) can be considered as a drflerential 1-form on M with values in T(M). Its covariant exterior differential r = d( 1T(MJ is therefore an M-morphism of 2

A T(M) into T(M), which is called the torsion M-morphism (or simply the torsion) of the linear connection C on M. So, by definition (17.19.3), we have

(17.20.6.1)

t.(X/\Y)=V,- Y - V y '

x- [ X , Y ]

for any two C" vector fields X , Y on M. This morphism defines a bilinear M-morphism (h,, k3-r * ( h , ~k,) of T(M) @T(M) into T(M), and hence may be identified with a tensor$eld t of type (I, 2), called the torsion tensor

346

XVll

DISTRIBUTIONS AND DIFFERENTIAL OPERATORS

field (or, by abuse of language, the torsion tensor) of the connection C. If (U, cp, m)is a chart of M and ( X i ) l 6 i 6 mthe basis of 9-:(U) over B(U) associated with this chart (16.1 5.4.2), then from (17.1 8.1.5) we have

from which we obtain the corresponding components of the torsion tensor

t:

Iff: N + M is any C"-mapping, Tcf) is a lifting off to T(M),and we may write it as T ( f ) IT(M). The formula (17.19.5.1) consequently shows that, for any two vector fields X , Y of class C" on N, we have 0

Let C', C" be two linear connections on M , t' and t" their respective torsions. If B is the bilinear M-morphism of T(M)@ T(M) into T(M) which is the difference of C' and C" (17.1 6.6), then we have

(17.20.7)

t" * ( X A Y)

(17.20.7.1)

- t'

*

(XA Y ) = B(X, Y ) - B(Y, X ) .

If we denote byVLx and VLx the covariant derivatives relative to C' and C", respectively, then it follows immediately from (17.1 7.2.1) that

VkX* Y - VLX * Y

= ~y(,)(

- C:(h,,

= -B,(h,,

Y(x))

+ C:( h, , Y ( x ) ) )

Y(XN

from the definition of B. Hence we have (17.20.7.2)

V i * Y -V'*

Y

=

-B(X, Y )

and the formula (17.20.7.1) follows immediately from this and the definition (17.20.6.1) of the torsion.

APPEND1 X

MULTILINEAR ALGEBRA

(The numbering of the sections in this Appendix continues that of the Appendix to Volume I.)

8. MODULES. FREE MODULES

(A.8.1) None of the results of (A.l .I)-(A.3.5) inclusive involves the field structure of K, and therefore all these results remain valid without modification when K is replaced by an arbitrary commutative ring A (with identity element). In place of K-vector spaces we speak of A-modules (in (A.2.3), h, is bijective if and only if 1 is invertible in A). By abuse of language, the elements of an A-module are sometimes called vectors and the elements of A are called scalars. (A.8.2) The definitions of a free family and of a basis of a vector space, given in (A.4.1) and (A.4.4), require no modification for an A-module, and the same is true of (A.4.3). On the other hand, the condition given in (A.4.2) for a family to be free is no longer valid in general (in the Z-module Z, for example, the number 1 does not belong to the Z-module 2 2 generated by 2, but (1, 2) is not a free family). An A-module possessing a basis is said to be free. (A.8.3)

Everything in (A.5.1)-(A.6.6) inclusive, on determinants and matrices, also remains valid when the field K is replaced by an arbitrary commutative ring A. In (A.6.4) it is merely necessary to choose the formf, so that it takes the valuef,(b,, ..., b,) = 1 for a basis (bi) of E. We remark that these results prove that any two bases of the same A-module E (assumed to have a 347

348

APPENDIX:

MULTILINEAR ALGEBRA

finite basis) necessarily have the same number of elements (which one might call the “dimension” of E, but it should be realized that most of the results concerning dimensions of vector spaces do not generalize to free modules). The determinant calculations in (A.6.8) and (A.7.4) are also valid for an arbitrary commutative ring A.

9. DUALITY FOR FREE MODULES

(A.9.1) If E is an A-module, the A-module Hom(E, A) of linear forms on E (A.2.4) is called the dual of the module E and is often written E*. If x E E and x* E E*, we shall often write (x, x*) or (x*, x) in place of x*(x). The mapping (x, x*)H(x, x*) is a bilinear form, called the canonicaZ bilinear form, on

E x E*. For each x E E, the mapping X*H (x, x*) of E* into A is a linear form on E*, in other words an element c ~ ( xof ) the bidual E** = (E*)* of E, and the mapping cE(called the canonical mapping) of E into E** is linear.

(A.9.2) Suppose that E is a free module having a j n i t e basis (ei)l,i6n (also called a ,finitely-generatedfree module). For each index i, let e: be the linear form on E (called the ith coordinatefunction) such that (ei ,eT ) = dij (Kron-

ecker delta). Then for each x =

n

i=1

tiei E E, where ti E A (1

S i 5 n), we have

(x, e?) = ti. Hence (A.5.1) (e:)lsisn is a basis of the dual A-module E*, called the basis dualto the basis (ei). From this definition it follows immediately that if (e?*)l is the basis of E** dual to ($1, then we have cE(ei)= e:* for all i, and hence cEis an isomorphism, by means of which we shall identifv E** with E, so that each element x E E is regarded as a linear form on E*, namely the form x* H(x, x*).

(A.9.3) Let E and F be two A-modules and u a linear mapping of E into F. Then for each linear form y* E F*, the function y* o u is a linear form on E, hence an element of E*, and it is immediately verified that the mapping y* H y* 0 u of F* into E* is linear. This mapping is called the transpose of u and is denoted by ‘u. Clearly we have (A.9.3.1)

‘(U1

+

u2)

= ‘U1

+ ‘U2,

‘(nu) = 1 . ‘24

for all u l , u 2 , u ~ H o m ( E ,F) and AEA. If G is another A-module and u : F -P G a linear mapping, then (A.9.3.2)

‘(0 0 2.4) = ‘u 0 ‘27.

9

DUALITY FOR FREE MODULES

349

The definition of the transpose is contained in the fundamental duality formula:

for all x E E and y* E F*. If u is an isomorphism of E onto F, then ‘u is an isomorphism of F* onto E*. Its inverse ‘u-’ (which is also the transpose of u-’) is called the isomorphism contragredient to u. It satisfies the relation (u(x), ‘u-’(x*)) = ( x , x * )

(A.9.3.4)

for all x E E and x*

E

E*.

Suppose now that E and F are free modules with finite bases (aJlsisn and (bJlslSm, respectively. If U = (orji) is the matrix of u with respect to these bases (A.5.2), then orji = (u(ai),b;), and the formula (A.9.3.3) therefore shows that ( a i , ‘u(b;)) = aji . Since (ai) is the basis dual to (a:), the matrix of ‘U with respect to the bases (b;) and (a:) is obtained by interchanging the rows and columns of the matrix U of u (it is called the transpose of U and is denoted by ‘ U ) . Further, it is immediately clear that (A.9.4)

(A.9.4.1)

‘(54)

= u.

(A.9.5) Suppose that the finitely-generatedfree A-module E is the direct sum M @ N of two finitely-generated free A-modules (A.3.1). To the canonical projections p : E + M, q : E + N (A.2.3) there correspond by transposition canonical injections * p : M* + E*, ‘q : N* E*, such that E* is the direct sum

of the submodules No = ‘p(M*) and Mo = ‘q(N*).To see this, we take a basis of E consisting of the elements of a basis (ai)’ ism of M and a basis (bj), of N. It is then immediate that if (a:) and (b;) are the bases dual to (ai) and (bj),respectively (A.9.2), the elements ‘p(a:) and ‘q(br)form the basis of E* dual to the chosen basis of E. The submodule Mo (resp. No)can also be defined as the set of linear forms x* E E* such that (x, x*> = 0 for all x E M (resp. all x E N), and is called the annihilator of M (resp. N) in E*. By reason of the identification of a finitely generated free module with its bidual, it is clear that (M’)’ = M and (No)’ = N. Finally, note that if i : M -+ E and j : N --t E are the canonical injections (A.2.3), their transposes ‘i : E* + M* and 7 : E* + N* are the canonical projections, when we identify E* with the direct sum M* @ N+. (A.9.6) If E is a finite-dimensional vector space over a (commutative) field K, every vector subspace M of E admits a supplement N in E, and the results of

350

APPENDIX: MULTILINEAR ALGEBRA

(A.9.5) can be applied to the direct sum decomposition M 0 N of E. The annihilator Mo of M in E* does not depend on the choice of the supplementary subspace N. It is clear that (A.9.6.1)

codim Mo = dim M,

(A.9.6.2)

(M0)O = M.

Also, if M,,M, are two subspaces of E, (A.9.6.3)

(Ml

+ M,)'

= MY n Mg,

( M , n M,)'

= MY

+Mi;

this is easily seen by taking a basis of E as in (A.4.12). Let F be another finite-dimensionalvector space over K, and let u : E + F be a linear mapping. Then, with the same notation, we have (A.9.6.4)

(Ker u)'

= Im('u),

(Im u)' = Ker('u)

as is easily shown by decomposing E into the direct sum of Ker(u) and a supplementary subspace, and F into the direct sum of Im(u) and a supplementary subspace. It follows that (A.9.6.5)

rk('u) = rk(u).

10. T E N S O R P R O D U C T S O F FREE M O D U L E S

(A.IO.1) Let El, .. ., E, be A-modules, and for each index j let xi* E E? be a linear form on Ej . Then the mapping

of El x E, x . . * x E, into A is an r-linear form (A.6.1), which is called the tensor product of the forms x:, . ..,x,* and is denoted by (A.10.1 .I)

x:: @ x; @ *

* *

0 x:.

The set 2',(El, . .., E, ;A) of all r-linear forms is an A-module, and it follows directly from the definition that the mapping (A.10.1.2)

of Er x Ef x

. ..,X,*)HX:: 0 x: 0 x: into Y,(E1,..., E, ;A) is r-linear.

(x?, x:,

- .-x Ef

*.-

10 TENSOR PRODUCTS OF FREE MODULES

351

(A.10.2) Suppose now that each Ej is a free A-module with a finite basis ( e j k ) l s k j n jThen . it follows from (A.6.2) that the mapping (A.10.1.2) is bijective and that the elements

form a basis consisting of nlnZ * * . n, elements o f the A-module 9 , ( E 1 , .. . , E, ;A). This A-module is called the tensor product of E:, ..., E,*, ET 0 * * * @ E,* . and is denoted by E: @ A E: @ A * * @ A E,*, or simply E: (A.10.3) Since, under the hypotheses of (A.10.2), Ej is identified with Ej** (A.9.2), it follows that we may define in the same way the tensor product El @A E, @ A .* . oAE, (or El @ E, @ * . . @ E,) which has a basis consisting

of the nln,

*

n, elements

(A.10.3.1)

el, kl

8 eZ,

k2

@

'.

'

@ er,

k,

(1 5 k j S n j , 1 5 j 6 r). The fundamental property of this A-module is the following:

x E , into an arbitrary For every r-linear mapping u of El x E, x A-module F , there exists a unique A-linear mapping v of El @ E, @ * * * @ E , into F such that

for all xi E Ej (1 5 j 5 r). For if u(el,k , ,e,, tions

k2,

..., e,, k,)

= C k l k 2 ...k , E

v(el,kl Be,,,, 8 * * * 8 er,k,)

F , we can define u by the condi= Cklk2...kr)

and the mapping u so defined clearly has the required properties. If E and F are two finitely-generated free A-modules, and if (ei)l i s m and are bases of E and F, respectively, then the ei @fi form a basis of E 63 F, and hence every element of E @ F is therefore uniquely expressible in the form z = C t i j e i@fi. This expression can also be written as

(fi), s j g ,

iJ

(A.10.3.3).

z=

n

m

j=1

i= 1

cxj@fj= 1eioyi,

where the xi (resp. y i ) are elements of E (resp. F) uniquely determined by z.

352

APPENDIX:

MULTILINEAR ALGEBRA

(A.10.4) The above definition implies the existence of canonical isomorphisms between tensor products of finitely-generated free A-modules. If El, E, , E, are three finitely-generated free A-modules, there is a unique isomorphism (the associativity isomorphism) (A.10.4.1)

(El @Ez)8E3+El 8 Ez B E 3

which maps (x, @ x,) 8 x, to x, @ x, 8 x 3 .It can be dejned by this property for the basis elements (el,k,@ e2,k2)8 e,,k3 of the left-hand side. Likewise, there is a unique isomorphism (the distributivity isomorphism) (A.10.4.2)

(E,

eE,) 6E,

-+

(E, 8 E,)

e(E, 8 E,)

which maps (xi @ x,) @ x3 to (x, @ x,) 8 (x, @ x,); it is defined in the same way as before. (A.10.5) We have already seen (A.10.3) that there is a canonical isomorphism of Hom(El @ E, , F) onto the A-module Y,(E,, E2; F) of bilinear mappings of El x E, into F. Moreover, there is also a canonical isomorphism (A.10.5.1)

Hom(E,, Hom(E,, F)) -+ Hom(E, 8 E, , F).

For if x, H V , , is a linear mapping of El into Hom(E,, F), then (xl, x,)Hu,,(x,) is a bilinear mapping of El x E, into F, and we apply (A.10.3).

Now let El, E, , F,, F, be four finitely-generated free A-modules. To each pair of A-linear mappings u1 : El + F,, u, : E, + F, we associate the bilinear mapping (x,, x2)wu1(x1) @ uz(xz)of El x E, into F, 8 F, ;to this bilinear mapping there corresponds (by (A.10.3)) a linear mapping ui

+Fi OF2

Ei

such that (A.10.5.2)

(4 0 UZ)(Xl@

X2)

= Ul(X1) @ uz(xz). @ u, is bilinear; hencz (A.10.3)

Furthermore, the mapping (u,, U,)HU, there corresponds to it a linear mapping (A.10.5.3)

Hom(E,, F,) 0 Hom(E,, F,)

-+

Hom(E, 63 E, , F, @ F,),

which is in fact an isomorphism (and therefore the fact that the symbol u, @ u, has different meanings in the two sides of (A.10.5.3) is unimportant). If (ai), (bj),(c,,), (dk)are bases of El, E, , F,, F, ,respectively, if uih E Hom(E,, F,) is defined by the conditions U i h ( a i ) = ch, and Ui,,(am)= 0 for m # i, and if likewise

10 TENSOR PRODUCTS OF FREE MODULES

353

wjk E Hom(E, , F2) is defined by wjk(bj) = dk, Wjk(b,) = 0 for n # j , then it is immediate that the linear mappings uih@wjk form a basis of Hom(E, €3 E2 , F, 0 F2). In particular, taking F, = F, = A, since A 0 A is canonically isomorphic to A (considered as the free A-module with basis consisting of the element l), we obtain a canonical isomorphism

If on the other hand we take F,

= E, = A, we

obtain a canonical isomorphism

E* @ F + Hom(E, F)

(A.10.5.5)

under which x* @ y (where x* E E* and y mapping XH ( x , x * ) y of E into F.

E

F) corresponds to the linear

(A.10.6) Let B be a commutative ring containing A which is a finitelygenerated free A-module and has the same identity element as A. If E is any finitely-generated free A-module, there is a unique B-module structure on E OAB such that (x @/?)A = x @ (/?A) for all x E E and /?,A E B. For if (ej)lsi4n is a basis of E, every element of E @AB is uniquely of the form e, @ pi with pi E B, and the B-module structure required may be defined by i

e, €3 /?. A C ei 8 (Bin.This B-module is said to be obtained from E by (T extension of the ring of scalars to B and is denoted by E(B). The elements =

1)

i

g j 5 n) form a basis of E(B), and E may be identified with the sub-A-module of E(B)generated by these basis elements, by means of the canonical injection XH x 8 1. Every A-linear mappingf of E into a B-module G extends uniquely to a B-linear mappingfof E(B)into G, such thatf(x @ fl) = f ( x ) / ? .We may definef by the conditionsf(ei 0 1) =f(ei) for 1 5 i 5 n. In particular, if F is another finitely generated free A-module and if j : F -+ F(B)is thecanonicalinjection, thentoeachA-homomorphismu : E + F there corresponds the extension o f j 0 u to E(B), which is a B-linear mapping u ( ~ :, E(B)+ F ( B ) such that u ( ~ ) (0 x p) = U(X) @ p. In this way we define a canonical isomorphism ei €3 I (1

(A.10.6.1)

F))(B)

+

HomB(E(B)

9

which, in particular, gives an isomorphism (A.10.6.2)

(E*)(B)

+

(E(B))*.

F(B)),

354

APPENDIX:

MULTILINEAR ALGEBRA

Finally, there is also a canonical isomorphism (A.10.6.3)

E(B) @B

F(B)

--*

(E @ A F)(B)3

where E, F are finitely generated free A-modules; the element (x @ B) @ (Y 0 B'>

(x e E, Y E F, B, B'

E B)

is mapped to (x 0 Y ) 0 (BP'). 11. TENSORS

If E is a finitely generated free A-module, we denote by E@"or T"(E) or T:(E) the tensor product of n copies of E, for n 2 2. This A-module is called the nth tensor power of E. We also define TA(E) to be E itself and T:(E) to be the ring A, considered as an A-module. Likewise we denote by T;(E) the tensor product (E*)@", with the convention that Ty(E) = E*. Finally, if p and q are two integers >O, we denote by T:(E) the tensor product (E*)@4' @ (EBP). The elements of T",E) (resp. T;(E)) are called n-foid contruvariant (resp. n-fold covariant) tensors; the elements of T:(E), for p > 0 and q > 0, are called mixed tensors of type ( p , q), and p (resp. q) is the contravariant (resp. covariant) index. It follows from (A.10.5.4) and (A.10.5.5) that T:(E) may be canonically identified with Hom(EB4, EBP). Hence by (A.10.5.3) we have a canonical isomorphism (A.ll.l)

(A.ll .I.I)

T:(E) 0 TXE) -,T::S(E)

-@ in which the product u @ u of a tensor u = x ~ @ * * * @ x ~ @ x l @ - -xp and a tensor v = y : 0 * * . 0 y*, @ y1 0 .* * 0 y, corresponds to the tensor XT

0. * * 0 x:0

y r 0

..* 0 ys*@ X I 0 . * .0 xpOy , 0 ... 0 y , .

When p = q = 0 or r = s = 0, the isomorphisms (A.ll .I.I)are the linear mappings corresponding to the bilinear mappings (A, Z)HAZ and (2,A)- Az, respectively. With these definitions it is immediate that for any three tensors u, u, w we have (A.ll .I.2)

(u 0v) 0 w = u 0 (v 0 w).

Again, by reason of the identification of a finitely generated free module with its bidual, and the canonical isomorphism (A.10.5.4), we have a canonical isomorphism (A.11 .I .3)

(T,P(E))*

+

T:W

11 TENSORS

355

such that, if the dual of T:(E) is identified with T:(E) by means of this isomorphism, we have

(A.11.2) If (ej)l elements

ism

is a basis of E and (e?) the dual basis of E* (A.9.2), the

of T:(E), where the indices ih and jk run independently through the set {1,2, . .., m}, form a basis of T,P(E)which is called the basis associated to (ei). A tensor belonging to T:(E) then has a unique expression of the form

C cx{,$:i2e71

.

* *

ei*,

ej,

* * *

0 eip

. .

the sum being over all mp+qfamilies of indices ( j l , .. ., j q , zI, . . ., i,). (A.11.3) Given two indices i, j such that 1 5 i 5 p and 1 5 j 5 q, there exists a unique linear mapping c j : T,P(E) 4T:I i(E),

called the contraction of the contravariant index i and the covariant index j such that, for xlr . . .,x p E E and x:, ...,x: E E* ,

wherein the circumflex accent signifies that the term underneath it is to be omitted from the tensor product. The mapping ci may indeed be defined by this formula for the basis elements (A.11.2.1). In particular, taking p = q = I , we have c:(x* O x) = (x, x*) E A. The elements of E* @ E correspond canonically to the endomorphisms of E (A.10.5.5); the value of the contraction c : for the tensor corresponding to an endomorphism u is called the trace of u and is denoted by Tr(u). If (with the above notation) u corresponds to the tensor eT @ e i , i.e., if u is the endomorphism XH (x, e 7 ) e i , then its trace is d i j (Kronecker delta). It follows

356

APPENDIX:

MULTILINEAR ALGEBRA

easily that if U = ( a i j )is the matrix of u with respect to the basis (e,), then (A.11.3.2)

Tr(u)

= i

aii,

the sum of the diagonal elements of the matrix U ; this is also called the trace of the matrix U and is denoted by Tr(U). It is immediately verified that, for any two endomorphisms u, u of E, we have Tr(u 0 u ) = Tr(u 0 u)

(A.11.3.3)

(it is enough to consider the case where u, u correspond to "decomposed" tensors a* 8 a and b* 8 b).

12. SYMMETRIC A N D ANTISYMMETRIC TENSORS

From now on we shall assume that the ring A contains the field Q of rational numbers, so that for each a E A and each integer m # 0 the element m-'a belongs to A and is the only element E A such that m5 = a. (A.12.1) Consider the A-module T"(E) of n-fold contravariant tensors over a finitely-generated free A-module E. We define an action of the symmetric group 6, on Tn(E)as follows. For each permutation u E 6, , the mapping ( X I > x2

9

* * *

9

xn) Hxu- '(1)

8 x u - '(2) 8 . ..8 x u - '(n)

of E" into T"(E) is n-linear, hence factorizes as (XI, x2,

.

-,X,)HXI 0 x2 0 . * . 0 X,HO.

(xi 0 x 2 0

Ox,),

where u is an endomorphism of the A-module T"(E), defined by

From this definition it follows immediately that, if 6,T are any two permutations in G,, we have (A.12.1.2)

T

. ( 0 . Z) = (TO) . Z

for all z E T"(E). A tensor z E T"(E) is said to be symmetric (resp. antisymmetric) if o * z = z (resp. CT * z = E,,z, where E, is the signature of the permutation a) for all

12 SYMMETRIC AND ANTISYMMETRIC TENSORS

357

6,. If we take the basis of T"(E)associated with a basis (ei) of E (A.11.2), a tensor is symmetric if and only if its components satisfy the conditions 0E

(A.12.1.3)

and antisymmetric if and only if (A.12.1.4)

for all indices il, i2, .. ., inand all CT E G,,. It is sufficient that these relations should be satisfied for all transpositions T E G,,. (A.12.2) If z is any tensor belonging to T"(E), we obtain from z a symmetric tensor (called the symmetrization of z) (A.12.2.1)

s*z=

c

OE

b ' Z

Gn

and an antisymmetric tensor (called the antisymmetrization of z) a .z =

(A.12.2.2)

c

&&.

z).

O€G,

It is evident that s . z is symmetric; to show that a * z is antisymmetric, we observe that, for any p E 6, , p

* (@

*

2)

=

c CAP

oeQ,

O

(a . z ) ) = & p

*

c

ae6.

&pO((Pc) . 2)

= Ep . ( a * z ) .

If z is already symmetric, then (A.12.2.3)

s .z = n ! z ,

and if z is already antisymmetric, (A.12.2.4)

a * z = n !z.

The n-linear mapping (x,, .. . , x,,)++s(x~Q Q x,,) of E" into T"(E) is symmetric, and the n-linear mapping (x1, . ,x,,)++u(x, Q 0 x,,) of Eninto T"(E) is alternating (or antisymmetric). In particular, if xi = xi for some pair of distinct indices i, j , then a(xl Q * . . Q x,,) = 0.

..

APPENDIX: MULTILINEAR ALGEBRA

358

If z E T"(E) and z* E T"(E*),then for each permutation a E 6,we

(A.12.3)

have

( 5 . z,

(A.12.3.1)

z*) = (z, a-l * z*)

by virtue of the formula (A.ll .I.2), because

If we identify covariant tensors with multilinear forms on E (A.10.3), we have therefore (a

-

z*)(x1,

. ..,x,)

= Z * k ( l )7

a .

-

Y

xu,,,);

consequently, symmetric (resp. antisymmetric) covariant tensors may be identified with symmetric (resp. antisymmetric or alternating) multilinear forms (A.6.3).

13. T H E EXTERIOR ALGEBRA

All the tensors considered in this section are contravariant. (A.13.1) Let E be a finitely generated free A-module, (eJlsiSma basis of E. The antisymmetrization a(ei, @ ei2€3. * 69 ein)is zero whenever two of the indices ik are equal (A.12.2). On the other hand, if the indices ik are all distinct (which requires that n 5 m), there is a unique permutation a E G, such that < iu(z)< * * . < iU(,,).For each subset H = { i l , i z , ..., in} of n elements of the set {1,2, .. .,m} such that i, < i2 < * . . < in, the elements eH =

-

a (ei, @ ei2@ .*

Bein)therefore form a basis (consisting of

(3

elements)

of the A-module A,,(E) of antisymmetric tensors of order n over E. (A.13.2)

Given two antisymmetric tensors zp E. AJE), zq E A,(E), their exdenoted by zp A zq ,is defined to be the antisymmetric tensor of q given by the formula

terior product,

order p

+

(A.13.2.1)

1

Zp A Zq = -a ( Z P @ Zq).

p !q !

We shall prove the following two fundamental properties:

13 T H E EXTERIOR ALGEBRA

(A.13.2.2)

359

(Anticommutativity) I f z , E A,(E) and z,, E A,(E), then

.

Z,,A 2, = (- l)”ZP A Z,,

(A.13.2.3)

(Associativity) Ifz, E A,(E), z,, E A,(E), z, E AJE), then Zp A (Zq A 2,)

= (2, A 2,) A 2,.

To begin with, we shall establish two preliminary results: (A.13.2.4)

If t, (resp. t,,) is a tensor of order p (resp. q), then aMr,) 6 t,,) = P!ao, 6 tq), a@, 6 4t,N = q !4,6 t$.

We have a(a(tp) 6 t q ) =

C

C

~u Erg usSP+, roep

*

((7

tp)

6 tq);

but we can identify G, with the subgroup of G,+, which fixes the integers > p in {1,2, . . . , p + q}. We have then

=p !

c

PESP+4

-

E p P ( t p 6 t,,) = P ! U ( t , @ t,,)‘

The other formula is proved in the same way. (A.13.2.5)

Zft, (resp. t,,) is a tensor of order p (resp. q ) then

360

APPENDIX:

MULTILINEAR ALGEBRA

On the other hand, if we put y i = x ~ ( ~then ) , the permutation c’ such that ye,(i)= xer(i)is equal to z-’az, so that E,. = E,, . Consequently

c

&e~e(,,+l)@...@~e(,,+q)@~e(~)@...~~e(,,)

UEQp+*

=

c

E d Ye,(,)

e’ E B p + q

= a(r,

€3* * * @ Y a y q )@ Y e y q + 1) @ * * * @ Y o ’ ( p + q )

@ t,).

The general case now follows by linearity. (A.13.2.6) that

To prove (A.13.2.2) and (A.13.2.3) it is now enough to observe

The relation (A.13.2.2) is then an immediate consequence of (A.13.2.5). As to (A.13.2.3), we have

by the second formula of (A.13.2.4). Similarly the first formula of (A.13.2.4) gives us (Z,,A Zq) A Z,.=

1

p! q ! r !4

z p

@ Zq @ z r )

and (A.13.2.3) is therefore proved. This proof shows, by induction, that for any family of h antisymmetric tensors zpkE A,,(E) (1 5 k 5 h),

In particular, for n vectors xi E E (1

s j 5 n), we have

13 THE EXTERIOR ALGEBRA

(A.13.2.8)

XI A X 2 A

”.

AX,,

= a(Xl @X2 @ *

361

@Xn).

* *

Consequently

for all permutations u E 6,; and if xi = xi for two distinct indices i, j , then XiAX2A

AX,,=O.

*.’

(A.13.3) By reason of the last formula, the module A,,(E) is called the nth exterior power of the finitely-generated free A-module E, and is denoted by n

A E. The basis of A E associated with the basis (ei)of E consists of the fl

elements (A.13.3.1)

eH= ei,

A

ei,

A

* *

. A ein,

where H runs through the set of subsets {il,i,, ..., in} of (1,2, i, < i2 < *.. 0 be chosen sufficiently small that, for each t E ]c, d [ , the open ball with center F,(x,, t ) , and radius6 is contained in U (3.17.11), and then choose I: > 0 such that E O ~ " - " ) < 6. Then it follows from (10.5.6) that for each point z E U such that llz - xoII 5 E , there exists a solution t H u ( z , t ) of the differential equation DJJ = f(y), defined on the i n t e n d ]c, d [ and such that u(z, 0) = z and ilu(z, t ) - Fx(xo,t)II 5 E for all t E ]c,d [ . By (18.2.4) applied to U and M, for each x E M satistjling IIx - xoII 5 E , the function t H u ( x , t ) is an integral curve of X in the interval ]c, d [ that takes the value x at t = 0. Consequently, we have ]c, d [ c J(x) and u(x, t ) = F,(x, t ) for all t E ]c, d [ . Moreover, it follows from (10.7.4) that by replacing the interval Jc, d [ , if necessary, by a smaller interval ]a, h [ such that a < 0 5 t o < b, we may suppose that the function (x, t ) H " ( x , t ) is of class C' in the product of a neighborhood of of L in M and the interval ]a, b[. This completes the proof.

This proposition leads directly to the following corollaries: (18.2.6) For each t E R, thc sct of points x E M such that ( x , t ) E dom(F,)

is open in M. This follows from (3.20.12). X H t+(.r)is lower semicontinuous, and the function is upper semicontinuous on M .

(18.2.7) The ,function XH~-(X)

For the set of points x such that t ' ( x ) > x > 0 is equal to the set of points 2 ) E dom(F,). hence is open by (18.2.6). The first assertion therefore follows from (12.7.2) and the second assertion is proved similarly.

x such that (x,

(18.2.8) Let U be an open set in M arid a a real number > 0 such that

U x ] -a, a [ c dom(F,). Then, ,for each t E ] -a, a[, the mapping X H F,(x, t ) is a homeomorphism (of class C') of U onto an open subset U, of M, and the mapping (of class Cr) x H F,(x, - t ) is the inverse homeomorphism.

8

XVlll

DIFFERENTIAL EQUATIONS

This follows from (18.2.5) and (18.2.3.2). (18.2.9) For a C" vector field X , the numbers t + ( x ) and t - ( x ) may be finite: take, for example, M = R and X(x) = (x, x2). When this is the case, the " global " analog of (10.5.5) is the following proposition: (18.2.10) Let X be a vectorjeld on M of class c'( r 2 I ) , and x a point qf M such that t'(x) < + co. Then, .for each compact subset K of M. there exists E > 0 such that, for each t > t'(x) - E , the point F,(x, t ) does not lie in K . (In other words, the integral curve " ends outside " every conipact subset of M.) The proof is by contradiction. I f the assertion is false, there will exist an increasing sequence (t,) of real numbers strictly less than t + ( x ) , with t + ( x ) as limit and such that F,(x, 1,) E K for all n. Passing to a subsequence, we may assume that the sequence of points F,(.u, t,) converges to a point z E K. By virtue of (18.2.5), there exists an open neighborhood U of z in M and a real number a > 0 such that t + ( y )> a for all y E U. Now, if n is sufficiently large, we have t+(x)< t, + a and F,(x, 1,) E U, and therefore

t'(F,(x,

1,))

> a;

but by (18.2.3.1), t+(F,(x, 1,)) = t+(x)- t,, whence t + ( x ) > t, diction.

+ a, a contra-

There is an analogous result for t - , the statement of which we shall leave to the reader. In particular: (18.2.11) Let X be a vector j7eld qf class C' ( r 2 1) on M, \cith compact support (in particular, this condition will be automatically satisfied if the manifold M is compact). Then J(x) = R for all x E M. Let K be the support of X . For each x 4 K we have J(x) = R, and the integral curve t H F,(x, t ) is the constant function t w x . Hence, if x E K, the function F,(x, t ) takes no values outside K , and therefore by (18.2.10) we have J(x) = R in this case also. If X i s a vector field of class C" with compact support K, then for each t E R we have a diffeomorphism (18.2.11.l)

h, : X H F,(x, t )

of M onto M, such that

(18.2.11.2)

h , + , . = h,

0

11,. = I t , .

0

h,

2

FLOW OF A VECTOR FIELD

9

for all r, r ' E R and such that ho = I M . This follows from (18.2.8) (taking U = M and ] - a , a [ = R ) and (18.2.3.2). The 17, form a group, called the oiie-par.uiiic~/cvgroup of d~fSeotnorpltisms of M dLifi,ied by X . Notice that if .Y $ K , we have h , ( x ) = x for all t E R . Remarks (18.2.12) If M is a real-analytic manifold and X is an analytic vector field on M, then i t follows from the proof of (18.2.5) and from (10.7.5) that the flow F, is aiiul.y/ic in the open set dom(F,). (18.2.13) If I' is a C' solution of (18.1.1.1) in I and if X ( u ( t ) )# 0 at a point r E I, then 1' is an in~mersioiiat 1. But it can happen that D is an injective immersion of I in M but not a n embedding (16.9.9.3). (18.2.14) Suppose that the vector field X is of class C L , and put g,(x) = F,(x, 1 ) . Then y-,(g,(s)) = x for all sufficiently small / E R (18.2.3.2). If Y is any CLvector field on M, put (18.2.14.1)

YAX)

= T q c , J g - , ). Y(y,(.u)),

which is a tangent vector at s and is defined for all sufficiently small t E R. With this notation. we have the following interpretation of the Lie bracket [ X , Y]: (18.2.14.2)

cl

"K Yl(x) = Y,(x)l, = O d/

in the vector space T,(M), endowed with its canonical topology (12.13.2). To prove this. we may assume that M is a n open subset of R"; then the fields X , Ycan be written in the form j ' ~ ( yGO))) , and y ~ ( y H(y)), , where G and H are C' mappings of M into R". Hence, for a fixed x, we have

Consequently. for all sufficiently small (18.2.14.3)

g,(x) = x

r, we may write

+ t G ( x )+ / M I ) ,

where h(/) tends to 0 with / (8.6.2). On the other hand, if Dg,(y) denotes the deribative at J! of the function z ~ g , ( z )then , rt+Dg,(x) is the solution of the linear differential equation

U' = DG(g,(x)) U , (

10

XVlll

DIFFERENTIAL EQUATIONS

which reduces to the unit matrix I at t = 0 (10.7.3). Hence (1 8.2.14.4)

Dyl(x) = I + tDG(x)

+ tW(f),

where the matrix W ( t )tends to 0 with 1. I t follows that the right-hand side of (18.2.14.2) is of the form ( x , V(X)), where

V(X)= lim

I

rzo

r-0,

- (Dg- r(gi(X)) f

*

H(g,(x)) - H(.y)).

Now, for ( t , y ) close to (0, x), we have g-,(g,(y)) = y and therefore DY-,(Y,(X)) O Dy,(x) = 1 by differentiating. Hence V(x) = lim

1-0.1#0

I

- Dg-,(g,(x))* (H(yl(x))-

1

t

H(x)).

But, by virtue of (18.2.14.3) and (18.2.14.4), we have

(18.2.14.6)

+

Dg,(x) * H(x) = H(x) tDG(x). H(x)

+ fo,(t),

where o , ( t )and o,(t) tend to 0 with r. Since Dg-,(g,(x)), the inverse of Dy,(x), tends to I as t -+ 0, we obtain V(X) = D H ( x ) * G(x) - DG(x) H(x),

which proves our assertion (17.14.3.2). More generally, if Z is any C' tmsor.fir/dof type ( r , s) on M, and if we put

then we have the formula (1 8.2.14.8) i n the vector space (T:(M)), endowed with its canonical topology. This follows immediately from the uniqueness statement in (17.14.6), since the right-hand side of (18.2.14.8) evidently satisfies the conditions of (17.14.6) by virtue of ( 8.1.4).

2

FLOW OF A VECTOR FIELD

11

PROBLEMS

1.

Let F be a closed set in R" and

a frontier point of F . A vector u f 0 is said to be a n ( I if there exists a point b := a 4 pu. with p > -: 0, such that the (Euclidean) open ball with center h a n d radius p is contained in the complement of F. A vector v t T,,(R") is said to be tangent to F if (T,(V)~u) = 0 for all outward nornials u to F at a. A vector field X defined on an open neighborhood U of F is said to be rang~vit/o F a/ong F if, for each frontier point a of F, the vector X ( a ) is tangent to F. (1

oirtwrrrl normti/ t o F at the point

(a) Let I * s ( I ) he a C 1 curve, i.e., a C' mapping of an open interval I c R into R". For each I c I let 8(/)denote the (Euclidean) distance d(.v(t),F), and let y be a point of F such that i ~ . v ( t ) - y -c/(.r(t), F). Show that, if x ( t ) $ F and if u is a unit vector proportional to .Y - jf,then

( I f (/I,,) is a sequence of real numbers converging to 0 and if yc is a point of F such that d ( . t ( t t /iJ,J',,) c/(x(t 1 / I , , ) , F), observe that 6 ( t ) , i x ( t )- y , , ~ . ) (b) Let X be a Lipsc4iit:irrn vector ficld dcfincd o n an open neighborhood U of F (we identify T,(R") with R" by means of T.,).Suppose that X i s tangent to F along F. Show that there exists a constant h- ,.O such that, with the notation of (a),

-:

for each integral curve tp - x ( t ) of the vector field X. (Use (a), a n d the definition of a tangent vector to F at the point j,.) (c) Show that every integral curve of X which meets F is contained in F. (Argue by contradiction,and suppose that a n integral curve / - + x ( / ) satisfies x ( / , ) E F a n d x ( / ) 4 F for t n , / ( I )- cft sI)for sl t 5 s2.) ~

2.

Let X I , Xz be two C" vector fields on an open set U c R" (a) Let h , , hL be two real-valued functions of class C' o n U, a n d let Z(x) =h,(x)X,(x) I

A*

(xW2

(x),

Suppose that a n integral curve /-..u(t) of the vector field Z is defined for 0 5 t 5 1 and that .r(O) -~ . y o . For each positive integer n , consider the continuous function t + z n ( / )

12

XVlll

DIFFERENTIAL EQUATIONS

defined on [O. I ] by the following conditions: Z"

(0) xo 1

which is (0 5 k 2 n). Show that z. +x uniformly on [0, I]. (Consider the function affine-linear on each interval [ k / n , ( k t I ) / n ] and such that y n ( k / n ) z , ( k / n ) for 0 5 k 5 n, and use (10.5,1).) (b) Let Z - [ X I , X,]. Suppose that an integral curve t - - x ( t ) of the vector field Z is defined for 0 5 t 5 I and that ~ ( 0=) x, . For each positive integer I f , consider the continuous function t i t z , ( f ) defined on [0, I]by the following conditions: :

4k t I - = Vcp-') * ((cp

O

o>(t),

Wcp

O

W)).

If u is of class Cz, this local expression shows that u' is a mapping of class C' of I into T(M) and is a lifting of u ; hence we may define the vector u"(t), which belongs to the tangent space Tv,,,)(T(M)) to the manifold T(M), and thus U"

18

XVlll

DIFFERENTIAL EQUATIONS

is a continuous mapping of I into the differential manifold T(T(M)). In this way we can define successively the higher derivatives of u ; the derivative u(‘) is defined if u is of class c‘,and is a continuous mapping of I into T‘(M), where T‘(M) is the manifold defined inductively by the conditions T’(M) = T(M) and T‘(M) = T(T‘-’(M)) for r > 1.

(18.3.2) In order to define an (autonomous) second-order differential equation on M, we must therefore start with a vector field Z of class C ‘ ( r 2 0) on the tangent bundle T(M), and consider mappings u of class C2 of an open interval I c R into M which are such that the mapping t H u’(t) of I into T(M) is an integral curve of the vector field Z, or in other words satisfies the equation (18.3.2.1)

v”(t) = Z(u’(t))

for t E I. However, this is possible only if the vector field Z satisfies a supplementary condition. For u’ has to be a Ijfting of [;; that is to say, u ( t ) = o,(u’(t)); differentiating this relation, we obtain T(u) * E ( t ) = T(oM) . (T(u’) . € ( t ) ) , which may be written as u’(t) = T(oM) * u”(t), and by virtue of (18.3.2.1), this gives v ’ ( t ) = T(oM) +

Z(U’(t)).

Since we wish to have solutions satisfying arbitrary initial conditions, u’(t) must be able to take all values in T(M). Hence we must impose on the vector field Z the condition

f o r all h, E T(M). A vector field Z on T(M) satisfying this condition is called a uectorfielddejning a second-order diferentialequation. In particular, (18.3.2.2) implies that Z(h,) # OhX if h, # 0,. An (autonomous) second-order equation on M is therefore by definition a differential equation of the first order on T( M), defined by a vector field Z satisfying (18.3.2.2),and a solution of such an equation is a mapping u of an open interval I c R into M, of class C2, satisfying (18.3.2.1)for all t E I. In terms of a local chart (U, cp, n) of M, the tangent bundle T(M) is identified locally with q(U) x R”, and a vector field Z satisfying (18.3.2.2)has a local expression

wheref: cp(U) x R” +R”is a mappingofclassC‘;if~~isasolutionof(18.3.2.1),

3 SECOND-ORDER DIFFERENTIAL EQUATIONS

the function eq u ii t ion

ii

= ip

I'

:I

-+

(18.3.2.4)

ip(

19

U ) satisfies the second-order vector differential

D'u( I ) = f( i i ( t ), Du(t ) ) .

I t comes to the same thing to say that ;i function u of class C2 on an open iaterval J c R IS ;I solution of the second-order equation defined by Z , or to say that I' = oh., w . where w is an integral curve of the vector field Z , defined i n J . For we have r ' ( t ) = T(o,) . w ' ( t )= T(o,) . Z ( w ( r ) )= w ( t ) by (18.3.2.2). A niu.Yittiu1 solution of the second-order equation defined by Z is a solution which cannot be extended to a solution on a strictly larger interval; or, equivalently, a solulion of the form oM0 w , where w is a maxitnal integral curve of the vector field Z. DiKererential equations of higher orders on M are defined analogously.

(18.3.3) The results of (18.2)can of course be applied to second-order differential equations since what is involved is a particular case of integral curves of vector fields on T(M). Particular interest attaches to the set of solutions of a second-order differential equation for which a(0) is a giaen point U E M and o'(0) takes all possible values in the fibre 'I',(M). From (18.2) we deduce : (18.3.4) Lct Z bc N iwtorjickl of duss C' ( r 2 1) on T(M), dejtiing a secondordm clifliwntiul cqiiution.

( i ) For eaclr a E M, tliere exists a r u l number LY > 0 and a neighborhood U of'0, in T(M ) sucli that,,fbr each point h, E U, there is a solution t I+ y ( t , h,) of tlw stw)rid-ortior equation dejined by tlie vectorjeld Z which is dejned in the open iritcrrlal I-%, a [ c R and is sucli tliut y(0, h,) = x = oM(hx)and Y'(0,h,) = h,. Wc c'an nioreoiw choose a and U suck tliat, ,for each point x E U n M ( ii) (wlicro M is iclrntijiipd "itli the zero section ofT( M)) and each t o # 0 in ] - a , a [ , the t n a p p i q h,, Hy( t , , h,) is a liorncomorphisni of U n T,(M) onto an open neigliborliood V, i?f s in M , this lioi7it~oniorpliisr~i and its iniierse botli being c f l cluss C'. l f ' d is u di3uncc wliicli dejines the topology of M , we inuy assume also thut ( , f i ) rJisc~rlt o ) tlicrc cxisrs p > 0 such that V, contains the open ball with center x uiid rudiiis p,Jor cach x E U n M.

to the integral curves of the vector field Z : the num( i ) We apply (18.2.5) ber a and the neighborhood U are chosen so that U x ] - a , a [ is contained in the open set dom(F,).

20

XVlll DIFFERENTIAL EQUATIONS

(ii) Since the question is local as regards T(M), we may assume that M is an open set in R" and that the vector field Z has the local expression

(x,Y) H((x, Y), (Y1 f(X, Y))), where f is continuously differentiable in a neighborhood of ( a , 0) in M x R". Changing notation, let us write ( x o , yo) in place of h, and (u(t, s o Yo), v(t, xo 3 9

Yo))

in place of y ( r , hJ, so that the vector-valued function t H( u ( t ,xo Yo),

xo 7 Yo))

V(t,

9

is the solution of the system of two vector differential equations Y' = f(x, Y),

x' = y,

(18.3.4.1)

such that u(0, xo , yo) = x o , v(0, x o , yo) = y o , where u and v are of class C' in ] - a , a [ x U. Put &t, xo Yo) = D,f(u(t,xo Yo) v(t, xo Yo)), H i , x o , yo) = D, f(u(t, xo, yo), v(t, x o , yo)). 7

9

2

It follows from (10.7.3) that, for ( t , x o , yo) in a neighborhood of (0, a, 0) in R x R2", the functions (with values in -Y(R'')) a t , xo, yo) = D3 g ( 4 xo Yo), W ( t ,xo Yo) = D3 h(t, xo Yo). 7

7

9

where g(t, xo , yo) = (u(t, xo , yo) - xo) - tY0 h(t, xo , Yo) = V ( t , xo Yo) - Yo

3

1

form (as functions o f t ) the solution of the linear differential system

Z ' = w, (w'= A + B o w + ( t A + B ) , o

z

such that Z(0, x o , yo) = 0 and W(0,x o , yo) = 0. Application of Gronwall's lemma (10.5.1) to this system shows that we may choose go and

u = {(xo

9

Yo) : II xo

- a II < rr /I Yo I1 < r )

such that, for all I t I < a. and (x,, yo) E U, (18.3.4.2)

l l w ( ~ , ~ o ~ Y oS ) lbI I l I ,

I I Z ( ~ ~ ~ v o ~ YSbItI', o)II

where b > 0 is a constant. Now consider the mapping (18.3.4.3)

yo ~

~

( 7 xo 1 90YO)

- xo = 10

YO + g(to xo YO) 9

9

3

SECOND-ORDER DIFFERENTIAL EQUATIONS

for some t, # 0 in ] - Y, Y [ ? where value theorem, we have

Y

21

< a , ) .By virtue of (18.3.4.2) and the mean-

for l/yl11 < r and l/y211 < r , provided that Y < clo has been chosen sufficiently small $0 that brr < 4. Hence we see that (18.3.4.3) is a n injectire continuous mapping of the compact ball B’ : llyoll 5 + r into R”, hence is a honieoriiorphisrn of B’ o n t o its image (3.17.12). It remains to show that this image contains a ball with center 0 and radius i/i&>pcnriwit o / x o (for Ilx0 - a ( / < r ) . For this, we shall show that (10.1.1) can be applied t o the function ( X , Y ) H x - g(fo, xo, fi-IY)

defined in the product of the balls

Now. by virtue of (18.3.4.1), there exists a constant c such that

Take Y < Y, sufficiently small so that we have

CY

< &r. Then, for Y , and Y, in B”,

l I g ( ~ , , - v o . f i ‘ Y , ) -g ( ~ o 3 ~ o 3 ~ i5~3 Y l I Y2 l~-I lY,Il

by virtue of (18.3.4.4), and for X

E

B,

l I x - g ( ~ o , . ~ o , o ) 5l lA r l ~ o l< ? r l t o I .

Hence there exists a continuous mapping X H F( X) of B into B” such that

x = F ( X ) + g(to,xo, tOIF(X)) and this proves that the image of B’ under the mapping (18.3.4.3) contains B. Hence we obtain the assertion (ii) of (18.3.4) by taking p = & r l r o \ , a n d observing that in a nietrizable compact space all distances are uniformly equivalent (3.16.5).

22

XVlll

DIFFERENTIAL EQUATIONS

4. S P R A Y S A N D I S O C H R O N O U S SECOND-ORDER E Q U A T I O N S

(18.4.1) In the theory of second-order differential equations, one is often interested less in the solutions tw u ( t ) (which are iinending p a t h in M (16.27)) than in the iniagcJ of these solutions. These images are called the frujcctorics of the equation or of the vector field Z that delines the equation. The image under 1 1 of a compact interval [a. p] contained in the open interval of definition of 1%is called an arc ojtrujcctor~:and ( ' ( a ) and o ( P ) are called respectively the origiti and endpoint of the arc. The tangent vectors ~ ' ( 2and ) a'(P) at the origin and endpoint of the arc are well-defined. I f z l is a solution defined in an interval 1-3, r [ , then for each positive real number c the function 'c1~: tt-+c(cf), defined on ] - c - ' a , c-'Y[, has the samt iniagt as but is not, in general, a solution of the same second-order equation. In the case where M is an open set in R", t i is a solution of (18.3.2.4), and we have 11,

d ( t ) = CU'(Ct),

so that

11%is

w"(r)

= c2P"(ct),

a solution of the vector differential equation IV"(t)

= C2f(ll.(t), c-'ll*'(t))

for - c - ' a < t < F l u . We are therefore led, in this case, t o consider mappings f satisfying the condition (18.4.1.1)

f(x, ('y) = c2f(x. y)

for all c E R . T o express this condition in an intrinsic form for a n arbitrary differential manifold M, we introduce the mapping ni, : h , H c . h, of T ( M ) into T ( M ) : for a vector field Z on T ( M ) defining a second-order differential equation, the condition corresponding to (18.4.1. I ) is (18.4.1.2)

Z(m,(h,)) = c . (T(m,) . Z ( h , ) .

The vector field Z is said to be isodironoim, or to be a s p r q ' over M. if it satisfies this relation for all h, E T(M) and all c E R. The corresponding differential equation is also said to be isochronous (if the variable t represents time, the equation "does not depend on the unit of tncasiirC of time"). We remark that if Z is an isochronous field, then Z(0,) = OOx for all .YE M and Z ( h , ) # OhX whenever h, # 0,; if z' is a solution of the corresponding equation, then (>it/irr II is constant or v ' ( t ) # O,,,,) throughout ez'ery open interval I of definition of v (and if Z is of class C", then 1' is an itntncrsiorz of I in M). If u is a solution defined in ] - a , a [ , then the function t w r ( - f )

4

SPRAYS AND ISOCHRONOUS SECOND-ORDER EQUATIONS

23

is also a solution defined in the same interval. Since for each c E R the function t H~ ( +t 0 ) is also a solution, it follows that an arc of trajectory with origin a and endpoint b i n M is also an arc of trajectory with origin b and endpoint a, and hence a and h are also called the endpoints of the arc. (18.4.2) Let Z bc N spray o w r M of class C ' ( r 2 0). With the notation of (18.2.3), j i i r [ w l i rral number L' # 0 and cnch i>ectorh, E T( M), we have

(18.4.2.2)

for all t

E

F,(ch,, t ) = c F,( h, , ct) 9

J(ch,)

This follows directly from the definitions. The trajectories,which are the images of the solutions t ~ o ~ ( F ~ ( c th) ), , (where t E J(ch,)), are therefore independiwt ofthe choice of c # 0 for h, # O ; they are called the niuxinial trajcctorics of the second-order equation defined by Z (or of the vector field Z ) passing through the point x and tangent at this point to tlic (lircction defincct bv the cector h, (or any of the vectors ch,, c # 0). (18.4.3) I n what follows, we shall denote by R,, R,(M), or simply R, the set of points h, E T(M) such that the open interval J(h,) contains the closed unit interval [0, I]. I n the notation of (18.2.3), this is equivalent to tf(h,) > 1. Since t + is a lower semicontinuous function on T(M) (18.2.7), the set R is open i n T(M), and it follows from (18.4.2.1) that the relation h, E R implies ch, E R for all L' E [0, I ] (in other words, each of the sets R nT,(M) is star-shapcrl in the fiber T,(M); but it is not necessarily symmetrical with respect to 0, (18.4.9)). The mapping h, HoM(F,( h, , I)) is called the exponential t?iappin~j&fined bj, Z and is written h, Hexp,( h,), or simply h, H exp(h,); its value at h, is the value at t = 1 of the solution u of the second-order differential equation defined by Z which satisfies ~ ( 0=) x, u'(0) = h, E R. (18.4.4) For c w h h, E T(M), the fiinction 1 1 : tHexp(th,), defined in the intiwul J( h,), is thc niu.uinin1 solution of the second-order diflerential equation clefincd b)! Z. nhicli sutisfirs tlic initial conditions c(0) = x, ~ ' ( 0=) h,. For ea(li t # 0 iri J(h,), we h a w (18.4.4.1)

u'(t) = t-'F,(th,,

1).

24

XVlll DIFFERENTIAL EQUATIONS

For it follows from (18.4.2.2) that the function which takes the value h, at t = 0 and the value t-’F,(rh,, 1) at t # 0 in J(h,) is equal to FZ(h,, t ) for all t E J( h,). We recall also (18.2.3.1) that for to E J(h,) we have (18.4.4.2)

J(L”(t0))= J(h,)

+ (-to).

(18.4.5) Let Z be a spray oiler M of class C’, nhcre r 2 1. Then the exponential map is a C’ mapping of R, into M. For each x E M, the tangent linear tnapping Tox(exp) at the point 0, of the zero section OM of T(M), when restricted to the tangent space Tox(T,(M)) of vertical cectors (identijed with T,(M)), is the identity mapping, and when restricted to the tangent space Tox(OM) to the zero section is the canonical bijection of this space onto T,(M).

The first assertion follows from (18.2.5), and the others follow from the facts that the restriction of exp to the zero section of T ( M ) is by definition the canonical bijection OM -+ M (the restriction of the projection OM

: T(M) -+ M),

and that, considering the linear mapping r : t H th, of R into the vector space T,(M), we have h, = Z,,,,(r’(t)) and in particular h, = r’(O),so thatTOx(exp)* h, (being equal to v’(O), where u(t) = exp(th,)) is equal to h, (18.4.4). Hence we obtain, by virtue of (10.2.5): (18.4.6) For each a E M and each neighborhood Uo of 0, in T(M), there exists an open neighborhood U c Uo of 0, in T(M) such that the mapping

h,

c-,( x , exp( h,))

is a homeomorphisni of class C‘, nith inverse of class C‘, of U onto an open neighborhood of ( a , a ) in M x M, and such that the mapping ha H exp( ha) is a homeomorphism of U nT,(M) onto an open neighborhood V of a in M. (18.4.7) Let a be a point of M and let U be a neighborhood of 0, it1 T ( M ) such that U nT.(M) is star-shaped and contained in R, and such that the mapping h,wexp(h,) is a homeomorphism of U n T , ( M ) onto an open neighborhood V of a in M. Then, f o r each y # a in V, the arc of trajectory which is the image of [0, I ] under the mappity t ~ e x p ( t h , ) ,bchere ha is the unique solution of exp(h,) = y in U n T,(M), is the unique arc of trajectory with origin a and endpoint y which is contained in V.

4

SPRAYS AND ISOCHRONOUS SECOND-ORDER EQUATIONS

25

An arc of trajectory L with origin a and endpoint is necessarily the image of an interval [0, 33 under a mapping t w e x p ( r h i ) , where h: E T,(M). We shall show that if L c V, then th: E U for 0 5 t 5 a and therefore ah; = h,, which will establish the uniqueness of the arc in question. Now, it is clear that rh:, E U for all sufficiently small t ; if the closed set of t E [0, a ] such that th; 4 U were not empty, it would havea least element /I> 0. But, by hypothesis, we have z = exp(/lh;); if $ : V U nT,,(M) is the inverse of exp, then th; = $(exp(th:)) tends to $ ( z ) E U as t +/l, contrary to the hypothesis that Phi 4 u. --+

It should nevertheless be remarked that in general there will exist other arcs of trajectories with origin a and endpoint a p o i n t y E V, but not contained ~ ~ / ? o /it1/ yV. In other words, the exponential mapping is not necessarily itljectirc on the whole of R. Likewise, we shall meet, in Chapter XIX, examples where it is not surjective; and finally, even if M is compact, the open set l2 is riot tiecessarilj, (lie M~holeofT( M).

Exanip Ies

+

(18.4.8) Take M to be the open interval R r = 10, co[of R, and identify the tangent bundle T ( M ) with M x R. Consider the spray a x , Y ) = ((x, J9, ( Y , J,2/x))

corresponding to the differential equation x" = x"/x in M. Here we have R = T(M), and for each x E M the exponential mapping restricted to TJM) = {x} x R is the diffeomorphism (s.j ' ) c t x e y of T,(M) onto M . This example will be generalized in Chapter XIX, and will justify the nomenclature of exponential mapping. (18.4.9) Let M

=

R and identify T ( M ) with R x R. Consider the spray

Z(x. .I%)= ( ( x ,j,),( 1 9 , .I,')), corresponding to the differential equation x" = .Y" in R. It is easily seen that in this case the set R is the set of points ( x ,y ) with j x < 1 , and that the exponential mapping, restricted to 2 ! n T,(R), is the mapping (.Y, j-)H.Y - log( I which is a diffeomorphism of R n T,r(R)onto R . 13).

(18.4.10) I n the preceding example, the vector field Z was translationinvariant. Consider then the quotient group T = R/Z, for which R is a covering, and let n : R T denote the canonical homomorphism. There exists a unique vector field Z , on the tangent bundle T(T) for which Z is a lifting (relative t o the morphism T ( n ) : T(R) --+T(T)); Z , is isochronous, a n d if I' is any solution of the differential equation defined by Z , then t w r ( r ( t ) ) is a solution of the dilyerential equation defined by Z,. We see therefore that although T is compact, R,, is not the whole of T. --+

26

XVlll

DIFFERENTIAL EQUATIONS

5. C O N V E X I T Y PROPERTIES O F I S O C H R O N O U S DIFFERENTIAL E Q UAT10NS

(18.5.1) Let M be a differential manifold and let Z be a spray of class C' >= 1) over M . The manifold M is said to be conwx relatice to Z if there exists a C' mapping s of M x M into the open set R, in T(M) satisfying the following conditions :

(r

In other words, given any two points x,, x2 of M, the mapping (18.5.1.2) of the interval [0, I ] into M is a path with origin xI and endpoint x2 and is the restriction to [0, I ] of a solution of the differential equation defined by 2 (18.4.4). A manifold which is convex relative to a spray is therefore necessarily connrcted (3.19.3). An open set U in M is said to be coni-ex relutii>eto Z if it is convex relative to the restriction of Z to T( U), or equivalently, if there exists a C' mapping s : U x U + Q z satisfying (18.5.1.1) and in addition such that (1 8.5.1.3) for all xl, x2 E U and all

tE

[0, I].

(18.5.2) Suppose that t k c open set U c M is concex relatire to Z. Then, with t h e notation of (18.5.1), f;)r m c h x E U, tile imuge of U under the niupping J' HS(X, J-) is t h e star-shaped open set T,( M)n Qzu( U), where Z, denotcJs the rc)striction of tlie i w t o r Jield Z to T(U). Tlic niupping y H S ( X , J,) is a homoniorpliisni of U onto this open s e t . und thc inuerse honieomorphisni is h, H exp( h,) ; both homc~oi,rorj~liisiiis ure qf' class C'. In purtirdur, ,for euch J, E U, tlw i i n u p of tlie interi:ul [0, I ] wider t h mapping t H exp( ts(x, J,)) is tlic oiilj>arc of'trujectorji with origin x and endpoint y contuined in U.

Since only the vector field Z , features in the statement of the proposition, we may assume that U = M. The relation exp(s(x, J')) = J! shows that the composition of the tangent linear mappings of h,Hexp(h,) and .YHS(X, J!) is the identity, and therefore each of them is bijectiiv (since the tangent spaces ThX(T, 0

for all

Z E

L.

Indeed, we have s(a, z ) E U, hence t * s(a, z) E U for 0 2 t 2 I , and therefore the function u ( t ) = exp(t . s(a, z)) is defined in an open interval I containing [0, 11. By the definition of L, we have z # a, hence s(a, z ) # 0, and therefore ~ ' ( 0#)0; and because f(x, 0) = 0, we have also v ' ( t ) # 0 for all f E I , since otherwise c would be constant in I . I t follows then from (18.5.3.2) that the function ft-+Q,(u(t)) has its second derivative everywhere > O in I . Since Q,(u(O)) = Q,(a) = 0 and D(Q, ' v ) ( O ) = DQ,(a) . ~ ' ( 0=) 0, we conclude (8.14.2) that Q,(c(r)) > 0 for 0 < t 2 1 , and in particular, that Q,(z) = Q,(u(l)) > 0. Since L is compact, there exists r > 0 and a neighborhood W" c W' of a such that, for all b E W", (18.5.3.4)

Q b ( c )2 r

>0

for all c E L.

5

CONVEXITY OF ISOCHRONOUS DIFFERENTIAL EQUATIONS

29

Now let p be such that 0 < p < r , and let W be the open neighborhood of .Y E W " such that Q(x, x) < p. For each b E W, let C, be the connected component of b in the open set defined by the inequality Q(h, x) < p. It follows from (18.5.3.4) that cb cannot intersect the frontier L of W', and hence (3.19.9) we have C, c W'. Now let xl,x2 be two points of C,. Since C, c W', we have s(x,, x 2 ) E U and therefore t . s(xl, x2)E U for 0 5 t 5 I . The function t H q ( t ) = Q,(exp(t. s ( x , , x2))) is therefore defined in the interval [0, I ] , and its second derivative cp"(t) is 2 0 for all t E [O, I ] by virtue of (18.5.3.2). Hence we have the inequality a consisting of the points

(18.5.3.5)

s

for 0 5 t I . (For the function $(t) = ~ ( t -) ( ( I - t)q(O) + tcp(l)) has second derivative $ " ( t ) = cp"(t) 2 0 in [0, I], and i,b(O) = I)( 1) = 0 ; consequently the derivative $'(t) is increasing, and if we had $(%) > 0 for some x E 30, 1 [, we should have $'(/I) > 0 for s a n e p E [O, X I , whence $'(t) > 0 for t E [p, I ] and so $ ( I ) > $(/I) > 0, a contradiction.) From (18.5.3.5) we deduce that

(18.5.3.6) for all t

E

[0, I ] . by the definition of the mapping s. This implies that

exp(t s ( x , , x2))E

cb

for 0 5 t 5 I . by (3.19.7) and the definition of a connected component, and proves that C, is convex relative to Z . Q.E.D. Propo4tion (18.5.3) has the following corollary: (18.5.4) U i i c l w t k c hypotl1esc.r. qf (18.5.3), if K is an)! conipact subset of M, tlwrr csists u Jiiitc cmwiiiy o/' K bjs open sc'ts C i ( 1 5 i 5 ni) which, together w i t h ull thcir fiiiitr iiitcrscctioiis, ure c o i i w x rclatiw to Z.

Let d be a distance defining the topology of M . For each a E K , i t follows fro01 (18.3.4(ii))and (18.4.7) that there exists r, > 0 such that, for each point .Y satisfying ( / ( ( I , .u) < t r , and each x' satisfying d(x, x') < f r o , there is only one arc of tra.jcctory with origin x and endpoint x' contained in the open ball with center x and radius j r c , , Cover K by a finite number of open balls B, ( 1 5 h 5 i l l ) w i t h centers a, and radii t r a k . Then there exists r' > 0 such that r' < J r o kfor I 5 k 5 111, and such that for each x E K the open ball B(x) with center s and radius r ' is contained in one of the balls B, (3.16.6). By definition, for each point x' E B(x), there is only one arc of trajectory with

30

XVlll

DIFFERENTIAL EQUATIONS

origin x and endpoint x' contained in B(x). This being so, it follows from (18.5.3) that for each b E K there exists an open set Cbcontaining b which is convex relative to Z and of diameter < r'; moreover, if E is the intersection of a finite number of these open sets C,, ( 1 5 k 5 / I ) , then for each k and each pair of points x, x' of E, there exists one and only one arc of trajectory L, with origin ?c and endpoint x' contained in Cbr,(18.5.2); but since all these arcs L, ( 1 2 k _I p ) have diameter < r', they are all the same, and therefore E is convex (18.5.1.3). Finally, K can be covered by a finite number of the open sets Cbh. We shall see in (20.17.5) that there exist functions Q with the properties of (18.5.3) on any differential manifold M.

6. GEODESICS

O F A CONNECTION

(18.6.1) Let C be a linear connection on a differential manifold M (17.18.1). For each x E M and each h, E T,( M), the vector is such that T(o,) . G(h,)

=

h,; furthermore, for each c E R , we have

G(c . h,) = C,(C . h, , c . h,)

=c

*

C,(h,, c . h,)

by virtue of (17.16.3.3), and T(m,.) . G( h,)

= C,(

h, , c . h,)

by (17.16.3.5). I t follows from these formulas that h,-G(h,) is a sprajs over M . This spray is called the geodesic spray of the linear connection C; the sohrioirs of the second-order differential equation on M defined by G are called the geodcsics of the connection C; and the trajectories (resp. arcs qf trajectories) of this equation are called the geodesic trajectories (resp. geodesic urcs) of C. The differential equation of the geodesics of C is therefore (18.6.1.1)

u"(1 ) = C",,,(!If(1 ) ,

tl'( I ) ) ,

which, by virtue of (17.17.2.l), can also be written in the form (18.6.1.2) with the notation of (18.1.1).

V E ' v'

=0

6

GEODESICS OF A CONNECTION

31

Relative to a chart (U, cp, 11) of M for which the local expression of the connection C is given by (17.16.4.1),this equation is equivalent to the following system of (nonlihear) second-order differential equations:

d2ui

(18.6.1.3) where

11 =

--

(It

cp

I'

dLll + C r;&)- =o clt clt dll"

( I 2 i 5 n),

11.1

has components

21' =

r ( I 5 i 2 n).

cp'

(18.6.2) (i) In order that tM'o connections C and C' on M should haile the

sanie geodesic sprq', it is necessary and siificient that the bilinear nrorphisni

B : T(M) 0T(M) + T(M), the d@wnce of C and C' (17.16.6), should be antisjwmctric. (ii) Giren an)' connection C on M, there exists a Liiiique connection C' on M i4hiclt has zero torsion (17.20.6) and the satne geodesic spray as C. From (17.16.6.3) we have BJh, h,) = ~hx(C,(h,, h,) 1

-

C.L(h, h,)) 9

and to say that B is antisymmetric means that B,,(h,, h,) = 0 for all h, E T(M), whence (i) follows. Furthermore, if t and t ' are the torsions of C and C', respectively, then (17.20.7.2) we have t' - t = 2B if B is antisymmetric; hence the only connection C' having the same geodesic spray as C and zero torsion is obtained by taking B = - i t . (18.6.3) The geodesics of a connection C can be interpreted in a different tramport relative to C. Consider a way by introducing the notion of parull~~l path I' : [I,/I] -+ M of class C' in M (resp. an unending path c : 1 + M of class C', where I is an opin interval in R). Then a parallel transport along the path (resp. along t h e unending path L.) is by definition a C' mapping w of [ci, B] (resp. I ) into T(M) such that (18.6.3.1)

oM(~ ( t )=)It([)

and

V E. w

=0

in

[a, /I] (resp.

in I),

which by definition (17.17.2.1) is equivalent to

(18.6.3.2)

o M ( W ( t ) )= Z ) ( f )

and

W'(f) =

Cv(,)(L"(f),W ( f ) ) .

(18.6.4) For cacti path I' of class C' in M, &fined on [a, 83, and each tangent M), there exists a unique parallel transport w along the w c f o r h,,(,, E Tu(a,( path L., such that w(ci) = h,,,, .

32

XVlll

DIFFERENTIAL EQUATIONS

Let U be a relatively compact open neighborhood in M of the compact set L = v ( [ s ( , PI). By virtue of (16.25.1) there exists a n embedding of U in RN for N sufficiently large, and we may assume therefore that M = U c RN. Then by (17.18.5) there exists a n open set V in RN such that M is a closed submanifold of V, and a connection on V which extends the given connection on M . In view of (18.2.4), we are reduced to proving (18.6.4) in the case where M is an open set in R". But then, in the notation of (17.16.2), we may write w ( t ) = ( r ( t ) ,u(t)), where u(t) E R" and u is a solution of the vector differential equation

(the local expression of (18.6.3.2)) which is homogeneous linear, by the definition of a connection. The result therefore follows from (10.6.3). The mapping w satisfying the conditions of (18.6.4) is called the parallrl transport of tke cector h,,,, along tire path and w ( t ) is said to be obtained by parallel transport of the vector h,(,, f r o m z'(r)to ~ ( t along ) 11. The same method proves: L?,

(18.6.5) Zf(ej)ljjjn is a basis of the tangent space TL,(a,iM)and & f o r each index j , w j is the parallel transport of the i'ector ej along the path z', then f o r each t E [ r , 81 the w j ( t ) (1 2 j 2 n ) f o r m a basis ofTUc,,(M).

This is a consequence of (10.8.4). We may therefore say that the wj form a basis of the vector space of parallel transports along 21. (18.6.6) If now we compare equations (18.6.3.2) and (18.6.1.2), we see that every geodesic of the connection C may be defined as a C2 mapping I' of a n open interval I c R into M such that z'' : I + T(M)is a parallel transport along the unending path and that for any two points s < t in I the tangent vector I?'( 1) is obtained by parallel transport of the tangent vector r'(s) along 1'. 11,

Remarks

Let f : M, + M be a local difeonlorphistn and let C , be the connection on M, which is the inverse imageof C under f (17.18.6). Then it follows immediately from the definitions that, if is a n unending path in M I , and wI a parallel transport along v1 (relative to C , ) , then f w , is a parallel transport along f 0 If u I is a geodesic relative to C,, thenf'u 1 9 , is a geodesic relative t o C. (18.6.7)

ill

0

19,.

.6

GEODESICS OF A CONNECTION

33

(18.6.8) . More generally, we may define a tensor parallel transport of type (r, s) along v as a lifting Z of class C1 of v to T~(M), such thai VE · Z = 0 (17.18.3). With the notatjon of (18.6.5), it follows from (17.18.2) that if, for each t e [a, /3], (wJ(t)) (1 ~j ~ n) is the basis of Tv(t)(M)* dual to the basis · (wj(t)). of Tv(t)(M), then the liftings w'!'J1'Cl 'X' w'!' (8) .•• ® }2

w'!' (8) w. Js

·

11

(8) w.12 (8) •.• (8)

w.

Ir

form a basis of the space of tensor parallel transports of type (r~ s) along v.

PROBLEMS

1. Show that, if j is the canonicalinvolution of T(T(M)) (Section 16.20, Problem 2(b)), thenj(v") = v" for every C 2 mapping v: 1 ~ M (where I is an open interval in R). 2.

Let E be a vector bundle with base B and projection .Tr. A real-valued function f on Eis said to be homogeneous of degree r (where r is an integer ~0) ifthe restriction off to each fiber Eb is a homogeneous function of degree r, Le.; if f(c · ub) = c'f(ub) for all ub E Eb and all c E R. A homogeneous function of degree O is therefore a function of the for_m g O 1r, where g is a real-valued function on B. A vector field Z on E is said to be homogeneous of degree r if Z(c · ub)

= c'- 1 (T(mc) · Z(ub))

for all ub E Eb and all c E R. (a)

When Bis an open set in W, and E =Bx Rn, in order that Z(x, y)

= (g(.x, y), f(x, y))

should be homogeneous of degree r, it is necessary and s:i,i~cient that g(x, cy) = c'- 1g(x; y)

and

· f(x? cy) = c'f(x, y).

(b) If Z is a vector field on E, homogeneous of degree rand of class C 0 , and if /is a real-valued function on E, homogeneous of degrees and of class C 1 , then 0z · f is a homogeneous function of degree r + s - 1. Conversely, let Z be a C 0 vector field on E · such that (i) for every C 1 homogeneous function f of degree O on E, the func_tion Oz ·f is homogeneous of degree r - 1; (ii) for~each C 1 section s* of the dual bundle E*, if we d~fine f by (so that/ is of class C1 and homogeneous of degree 1 op. E), then 02 • f is homogeneous · of degree r. Under these conditions, show that Z is homogeneous of degree r. A homogeneous vector field of degree() consist~ of vertical vectors in T(E)._ (c) If Z 1 , Z2 are C 1 homogeneous vector fields on E, of degrees r 1 , r 2 , respectively, · then [Z1, Z2] is homogen;eous of degree r 1 + r 2 - 1. (d) Let g be a B-morphism of E®' into E. With the notation of Section 16.19, Problem 11, show that the mapping · u1, i-+.;\(ub, g(ub ® ub ® · · · ®ub))

xvrn

34

DIFFERENTIAL EQUATIONS

is a field of vertical tangent,veotors on E; homogeneous of degree r. This vector field is denoted by gv. (e) For each ub E Eb, let .

\

H(ub)= T.;/(ub)

E

Tub(Eb);

His a cxi field of vertical vectors on E, homogeneous of degree 1, and is sometimes called the Liouville field. In order that a real-valued function f of class C 1 on E should be homogeneous of degree r, it is necessary and sufficient that 0H · f = rf (Euler's identity)., For a vector field Z of class C 1 on E to be homogeneous of degree r, it is necessary and sufficient that. [H; Z] = (r - l)Z. (f) Suppose that E = T(B). If S : hx~ (hx, hx) is the diagonal n·~morphism of T(B) into T(B) x 8 T(B), then we have H = ,\ 0 8. With the notation and d~finitions of Section 17.19, Problems 2, 3 and 4, show that H7' J=J7' H=O, [J, H] ~ -[H,J] = J, [iJ,0H].=iJ,

3,

[iH,dJ]=iJ, ·[dJ,

0H]=dJ,

[iJ,iH]=O.

Let X be a C 1 vector field on M. Show that there exists a unique vector field X on T(M) such that · 0hx · (0x

·f) = (0.x · df)(hx)

2

for every function f of class C on. M (the notation is that of Section 16.20, Problem 2). X is a homogeneous vector field of degree 1, and we have · X = j O T(X) and T(oM) X = X o oM .. The field .Xis called the canonicallifting of X. 0

4. · Let G be the geodesic spray of a lin_ear connection C on M. Let g be a covariant tensor field of order r and class C 1 on M, and for each hx E T(M) put J(hx) = (g(x), hx@ hx@ · · · @ hx) .

show that (0G ·, i')(hx) "".' (Vg, hx@ hx@ · · ·@ hx)

(where there are r + 1 factors hx in the tensor product). (Consider the particular case where g = w 1 @w 2 @ · · · @w,., the w i being differential 1-forms on M ;) 5.

To

Let M be a differential 111anifold .and g an M-morphism of (M) = (T(M))®' into T(M). Let G be the geodesic spray of a torsion-free connection Con M. In Problem 2(d) we defined the field gv of vertical vectors on T(M), which is homogeneous of degree r. Define also g 8 (h.~)

=

0

ex (hx, g(hx@ hx@ •' • @ hx))

for all hx E T(M); then g 8 is a homogeneous vector field on T(M), of degree r Finally, let r · g denote the M-morphism of T 0+ 2 (M) into-T(M) defined by (r · g)(k 1 @ k2@·::@ k,+2)

=

(r · (g · (k1@ · · ·@ k,)i\ k,+ 1)!_' k,+2.

Show that [G, gY]

=

(Vg)Y - gH,

[G, gH] = (Vg)H

+ (r. g)v.

+ 1.

7

6.

ONE-PARAMETER FAMILIES OF GEODESICS AND JACOBI FIELDS

Let X be a C 1 vector field on a differential:manifold M, and let gt (x) = Fx (x, t), in the I_lotation of (18.2.2). Let C be a linear connection on Mand let Z be a C 1 terisor field on M. For each t E J(x), let Zr be the parallel transport along' the curves 1-+ g. (x) whfch takes the value Z(gc (x)) at ·s = t. Show. that (Vx. Z)(x)

7.

35

d

= -d zt (x)lt=O. t

'

Let M be a differential manifold, C a linear co11nectioh on M; let a be a point of M and let U be an open neighborhood of 0 0 in TO (M) such that the restriction of expa to U is a diffeomorphism of U onto an open neighborhood V o'f a in M (18.4.6). For each vector . h EU, let Zh denote the C 00 vector field on V for which Zh (exp u)--is ·the tangent vector obtained by.parallel tr;rnsport of hfrorn a to exp(u) along the path t 1-+expa(tu) . .for each real-valued function f of class C 1 on M, we have . · · d (0zh · f)(expa th)= dt (f(expa th))

(cf. (18.1.2.5)). Deduce that, if Mis a real-analytic manifold and if /is analytic in a neighborhood of a, then in a neighborhood of t = 0 we have co

f(expa th)=

L

tn

1 (0~h · f)(a). n=on,,

8.

On T(R 2 ) (identified with R 2· x R 2 ) consider the spray Z(x, y) = ((x, y), (y, 0)), the trajectories of which are the lines in R 2 . Let M be the cylinder R 2 /(Z x {O}), of which R 2 is a covering, and let Z O be the unique spray over :tv1 which lifts to Z. Give an example of two open sets in M which are convex relative to Zo but whose intersection is not c;onnected. · ·

9.

Let G be a spray of class C 00 over a differential manifold M, and let B be an antisymmetric bilinear M~morphism of T(M) EB T(M) into T(M). Show that there exists on M a unique linear connection C with torsion equal to B, and geodesic spray 'G. (If C' is a linear _connection on M with torsion equal to B (17.20.7), show. that there exists a symmet~ic bilinear M-morphism A: T(M) EB T(M)-+ T(M) such that G(h.~) - C~ (h.~, hx) = ,\(hx, A(hx, hx)) (in the notation of Section 16.19, Problem 11) ..

7. ONE-PARAMETER FAMILIES 'OF GEODESICS AND JACOBI FIELDS

Let defined by.

(18.7.1)

£1 and

£ 2 denote the vector fields (called canonical} on R 2 ·

Give_n two open intervals I, J iQ. R, a mapping f: l x J ~ M of class C'·(r ~ 1), with values in a differential manifold M, is often called a oneparpmeter family of curves. (Consider for each ~ E J the mapping t~ f(t, ~)

36

XVlll

DIFFERENTIAL EQUATIONS

of I into M ; the function f may be regarded as describing the “variation” of this family of “curves.”) When there is no risk of confusion, we shall write, by abuse of notation (generalizing the notation of (18.1.2.3)) ,f;(t,5 ) = T ( f

(18.7.1.1)

.

t),

f i ( t , 5 ) = T(f)E2(t, 0,

so that the mappings ( f , t ) H f : ( t ,5 ) and (I, t ) w f ; ( t5,) are C - ’ mappings of I x J into T(M) which lift5 (18.7.2) Let C be a connection on M , and for each lifting w : I x J + T ( M ) o f f , of class C‘ ( r 2 3), w i f e (by abuse of notation)

v,

v,.

w=

v,,

V , . (V, . w ) - V , . ( V , . w ) = ( v ( f

A

w = V E ,* w ,

*

.w

(1 7.17.3). Then we haue

(18.7.2.1)

,f;)) . w,

where r is the curvature of C .

This is a particular case of (17.20.4.1) since [E,, E,] = 0. (18.7.3) With the notation of (18.7.1), and assuming M endowed with a connection C , the mappingf is said to be a one-paratnetcr family of geode.sic.r of M iffis of class C‘ with r 2 3 and if, for each 5: E J, t h e mapping t hf ( r , 5 ) is a geodesic. (18.7.4) If the connection C is torsion-free and i f f is a one-parat~ieterfamil~ of geodesics in M, then

v, . (v,.f ; ) = ( r . ( f ; A fi))

(18.7.4.1) for all ( t ,

5)E I

x

Since [ E l , E,]

.f ;

J. = 0,

the formula (17.20.6.4) gives us V ; f ; = v,*.f;

(1 8.7.4.2)

since the torsion is zero. Using (18.7.2.1), we obtain

v, (v,‘ f ; , = v, (v, f i ) = v, (v,‘f;)-k ’

‘

’

‘

( V .

(f;A\:))

‘f;.

But since f is a family of geodesics, we have V , .f;= 0 (18.6.1.2), whence the result.

7

ONE-PARAMETER FAMILIES OF GEODESICS AND JACOBI FIELDS

37

Throughout the remainder of this section we shall assume that the connection C is torsion-free. Under the hypotheses of (18.7.4), fix a value a E J and put v(t) = f(t, a), so that vis a geodesic. If we then put w(t) = f~(t, a), the mapping w of I into T(M) is of class C2 and satisfies the homogeneous linear equation (18.7.5)

Vt · (V 1 • w) = (r · (v'

(18.7.5.1)

w)) · v'.

A

For each geodesic v of C, defined on the interval I, a C 2 mapping w of I into T(M) which satisfies (18.7.5.1) is called aJacobifield along the geodesic v. If v1 = v