Foundations of constructive analysis (McGraw-Hill series in higher mathematics) [1 ed.]

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FOUNDATIONS of

CONSTRUCTIVE

ANALYSIS

Errett Bishop Department of Mathematics University of California, San Diego

McGRAW-HILL New York

San Francisco

BOOK

COMPANY

St. Louzs

Toronto

London

Sydney

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

Copyright © 1967 by McGraw-Hill, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.

Printed in the United States of America 1234567890 Library of Congress catalog card number: 67-22952

Quotation on page i from Lascelles Abercrombie, ‘‘Emblems permission of The Bodley Head, Lid. (John Lane), London.

MP

of

72106987 05470

Love,”

by

Acknowledgments

It is a pleasure to compliment my wife and children

for their patience and forebearance while I wrote this book. I owe much to J. L. Kelley, without whose encouragement I might have abandoned mathematics long ago. Gabriel Stolzenberg has read the manuscript and given me good advice. Joan Lagioe has done a fine job of typing the manuscript. Support for much of the writing of this book wnto 1t was supplied by the National Science for the summers of 1964, 1965, and 1966, the Unwersity of California, tn the form of 1964—1965.

and much of Foundation, and by the a fellowship

the research that went in the form of grants Miller Foundation of for the academic year

PREFACE

Most mathematicians would find it hard to believe that there could be any serious controversy about the foundations of mathematics, any controversy whose outcome could significantly affect their own mathe-

matical activity. Their attitude well represents the actual state of affairs: during a half-century of splendid mathematical progress there has been no deviation from the norm. The voices of dissent, never much heeded, have long been silent. Perhaps the times are not conducive to introspection. Mathematics flourishes as never before. Its scope is immense, its quality high. Mathematicians flourish as never before. Their profession is respectable, their salaries good. Mathematical methods are more fashionable than ever before. Witness the surge of interest in mathematical logic, mathematical biology, mathematical economics, mathematical psychology—in mathematical investigations of every sort. The extent to which many of these investigations are premature or unrealistic indicates the deep attraction mathematical exactitude holds for the contemporary mind. And yet there is dissatisfaction in the mathematical community. vil

viii

PREFACE

The pure mathematician is isolated from the world, which has little need of his brilliant creations. He suffers from an alienation which is seemingly inevitable: he has followed the gleam and it has led him out of this world. . If every mathematician occasionally, perhaps only for an instant, feels an urge to move closer to reality, it is not because he believes mathematics is lacking in meaning. He does not believe that mathematics consists in drawing brilliant conclusions from arbitrary axioms,

of juggling concepts devoid of pragmatic content, of playing a meaningless game. On the other hand, many mathematical statements have a rather peculiar pragmatic content. Consider the theorem that either every even integer greater than 2 is the sum of two primes, or else there exists an even integer greater than 2 that is not the sum of two primes. The pragmatic content of this theorem is not that if we go to the integers and observe we shall see certain things happening. Rather the pragmatic content of such a theorem, if it exists, lies in the circumstance that we are going to use it to help derive other theorems, themselves of peculiar pragmatic content, which in turn will be the basis for further developments. It appears then that there are certain mathematical statements that are merely evocative, which make assertions without empirical validity. There are also mathematical statements of immediate empirical validity, which say that certain performable operations will produce certain observable results, for instance, the theorem that every positive integer is the sum of four squares. Mathematics is a mixture of the real and the ideal, sometimes one, sometimes the other, often so presented that it is hard to tell which is which. The realistic component of mathematics—the desire for pragmatic interpretation—supplies the

control which determines the course of development and keeps mathematics from lapsing into meaningless formalism. The idealistic component permits simplifications and opens possibilities which would otherwise be closed. The methods of proof and the objects of investigation have been idealized to form a game, but the actual conduct of the game is ultimately motivated by pragmatic considerations. For 50 years now there have been no significant changes in the rules of this game. Mathematicians unanimously agree on how mathematics should be played. Accepted standards of performance suffice to regulate the course of mathematical activity, and there is no prospect that these standards will be changed in any significant respect by a revision of the idealistic code. In fact no efforts are being made to impose such a revision. There have been, however, attempts to constructivise mathematics,

PREFACE

ix

to purge it completely of its idealistic content. The most sustained attempt was made by L. E. J. Brouwer, beginning in 1907. The movement he founded has long been dead, killed partly by compromises of Brouwer’s disciples with the viewpoint of idealism, partly by extraneous peculiarities of Brouwer’s system which made it vague and even ridiculous to practicing mathematicians, but chiefly by the failure of Brouwer and his followers to convince the mathematical public that abandonment of the idealistic viewpoint would not sterilize or cripple the development of mathematics. Brouwer and other constructivists were much more successful in their criticisms of classical mathematics than in their efforts to replace it with something better. Many mathematicians familiar with Brouwer’s objections to classical mathematics concede their validity but remain unconvinced that there is any satisfactory alternative. This book is a piece of constructivist propaganda, designed to show that there does exist a satisfactory alternative. To this end we develop a large portion of abstract analysis within a constructive framework. This development is carried through with an absolute minimum of philosophical prejudice concerning the nature of constructive mathematics. There are no dogmas to which we must conform. Our program 1s simple: to give numerical meaning to as much as possible of classical abstract analysis. Our motivation is the well-known scandal, exposed by Brouwer (and others) in great detail, that classical mathematics is deficient in numerical meaning. Some familiarity with Brouwer’s critique is essential. Chapter 1 is primarily devoted to an examination of the defects of classical mathematics, following Brouwer, and a presentation of the thesis that all mathematics should have numerical meaning. Chapter 3 presents constructive versions of the fundamental concepts of sets and functions, and examines some of the obstacles to the constructivization of general topology. The remaining chapters are primarily technical, and constitute a course in abstract analysis from the constructive point of view. Very little formal preparation is required of the reader, although a certain level of mathematical sophistication is probably indispensible. Every effort has been made to follow the classical development as closely as possible; digressions have been relegated to notes at the ends of the various chapters. The task of making analysis constructive is guided by three basic principles. First, to make every concept affirmative. (Even the concept of inequality is affirmative.) Second, to avoid definitions that are not relevant. (The concept of a pointwise continuous function is not rele-

PREFACE

X

vant. A continuous function is one that is uniformly continuous on compact intervals.) Third, to avoid pseudogenerality. (Separability hypotheses are freely employed. The justification for this is discussed in Appendix A.) Chapters 1, 3, 4, 6, and 9 are the core of the book. Of the remaining chapters, Chapter 2 (Calculus and the Real Numbers), Chapter 5 (Complex Analysis), and Chapter 7 (Integration) are of less interest because they represent somewhat routine constructivizations of the corresponding classical theories. The constructivizations involved in Chapter 8 (Limit Operations in Measure Theory), Chapter 10 (Locally Compact Abelian Groups), and Chapter 11 (Banach Algebras), on the other hand, are by no means routine, but these chapters are perhaps of less interest because they are concerned with material of a somewhat more special kind. As already remarked, Chapters 1 and 3 are primarily philosophical. This leaves Chapters 4 (Metric spaces), 6 (Measure theory), and 9 (Banach spaces). These chapters contain the core of constructive abstract analysis.

In a first reading one could well omit Chapters 5, 8, 10, and 11. The book has a threefold purpose: first, to present the constructive point of view; second, to show that the constructive program can succeed; third, to lay a foundation for further work. These immediate ends tend to an ultimate goal—to hasten the inevitable day when constructive mathematics will be the accepted norm. We are not contending that idealistic mathematics is worthless from the constructive point of view. This would be as silly as contending that unrigorous mathematics is worthless from the classical point of view. Every theorem proved with idealistic methods presents a challenge: to find a constructive version, and to give it a constructive proof. Errett Bishop

CONTENTS

PREFACE

CHAPTER 1. 2. 3.

o~ Suds

1

A CONSTRUCTIVIST

MANIFESTO

The descriptive basis of mathematics 1 The idealistic component of mathematics The constructinization of mathematics 6

CHAPTER

NS

VII

2

CALCULUS

AND

THE

Sets and functions 12 The real number system 16 Sequences and series of real numbers Continuous functions 32 Differentiation 39 Integration 46 Certain important functions 561

3

REAL

26

NUMBERS

11

X11

CONTENTS

CHAPTER 1. 2. 3.

METRIC

62

SPACES

5

6

75 76

112

ANALYSIS

COMPLEX

The complex plane 113 Derwatives 114 Integration 117 126 The winding number Estimates of size, and the location of zeros The Riemann mapping theorem 141

CHAPTER 1. 2. 3. 4.

4

Fundamental definitrons and constructions Assocrated structures 82 83 Completeness &8 Compactness Locally compact spaces 102

CHAPTER 1. 2. 3. 4. 5. 6.

SET THEORY

Some basic notrons of the theory of sets 63 Borel sets 62 Netghborhood spaces and function spaces 69

CHAPTER 1. 2. 3. 4. 8.

3

MEASURE

13/

153

Test functions, measures, examples 15/ Measures of sets 159 Measures on R 168 Approximation by compact sets 175

CHAPTER

7

INTEGRATION

1.

Measurable functions

2. 3.

The wntegral 189 Convergence of integrals

4.

The L, spaces

6.

Product measures

2083 206

183 194

182

CONTENTS

X111

CHAPTER

1. 2. 3.

8

LIMIT

OPERATIONS

IN

THEORY

214

Decompositions of measures 215 Derwatives of set functions 221 Upcrossing inequalities 231

CHAPTER

9

NORMED

LINEAR

SPACES

1.

Definitions and examples

2. 3.

The L, spaces 251 The extension of linear functionals

4. 5. 6.

Hilbert space and the spectral theorem Locally convex spaces 284 Extreme points 289

CHAPTER

1. 2. 3. 4.

MEASURE

10

244

LOCALLY

CHAPTER

11

APPENDIX

A

METRIZABILITY

APPENDIX

B

ASPECTS

REFERENCES SYMBOLS

INDEX

363

366

COMMUTATIVE

361

2568

COMPACT

Haar measure 299 Convolution operators 313 The character group 316 Duality and the Fourier transform

OF

243

AND

263

ABELIAN

GROUPS

298

324

BANACH

ALGEBRAS

SEPARABILITY

CONSTRUCTIVE

TRUTH

349 352

335

CHAPTER

Z

A CONSTRUCTIVIST

1. THE

DESCRIPTIVE

BASIS

MANIFESTO

OF

MATHEMATICS

Mathematics is that portion of our intellectual activity which transcends our biology and our environment. The principles of biology as we know them may apply to life forms on other worlds, yet there is no necessity for this to be so. The principles of physics should be more universal, yet it is easy to imagine another universe governed by different physical laws. Mathematics, a creation of mind, is less arbitrary than biology or physics, creations of nature; the creatures we imagine inhabiting another world in another universe, with another biology and another physics, will develop a mathematics which in essence is the same as ours. In believing this we may be falling into a trap: Mathematics being a creation of our mind, it is, of course, difficult to imagine how mathematics could be otherwise without actually making it so, but perhaps we should not presume to predict the course of the mathematical activities of all possible types of intelligence. On the other hand, the pragmatic content of our belief in the transcendence of mathematics has nothing to do with alien forms of life. Rather it serves

2

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

to give a direction to mathematical investigation, resulting from the insistence that mathematics be born of an inner necessity. The primary concern of mathematics is number, and this means the positive integers. We feel about number the way Kant felt about space. The positive integers and their arithmetic are presupposed by the very nature of our intelligence and, we are tempted to believe, by the very nature of intelligence in general. The development of the theory of the positive integers from the primitive concept of the unit, the concept of adjoining a unit, and the process of mathematical induction carries complete conviction. In the words of Kronecker, the positive integers were created by God. Kronecker would have expressed it even better if he had said that the positive integers were created by God for the benefit of man (and other finite beings). Mathematics belongs to man, not to God. We are not interested in properties of the positive integers that have no descriptive meaning for finite man. When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself. Almost equal in importance to number are the constructions by which we ascend from number to the higher levels of mathematical existence. These constructions involve the discovery of relationships among mathematical entities already constructed, in the process of which new mathematical entities are created. The relations which form the point of departure are the order and arithmetical relations of the positive integers. From these we construct various rules for pairing integers with one another, for separating out certain integers from the rest, and for associating one integer to another. Rules of this sort give rise to the notions of sets and functions. A set is not an entity which has an ideal existence. A set exists only when it has been defined. To define a set we prescribe, at least implicitly, what we (the constructing intelligence) must do in order to construct an element of the set, and what we must do to show that two elements of the set are equal. A similar remark applies to the definition of a function: in order to define a function from a set A to a set B, we prescribe a finite routine which leads from an element of A to an element, of B, and show that equal elements of A give rise to equal elements of B. Building on the positive integers, weaving a web of ever more sets and more functions, we get the basic structures of mathematics: the rational number system, the real number system, the euclidean spaces, the complex number system, the algebraic number fields, Hilbert space, the classical groups, and so forth. Within the framework of these structures most mathematics is done. Everything attaches itself to

A

CONSTRUCTIVIST

MANIFESTO

3

number, and every mathematical statement ultimately expresses the fact that if we perform certain computations within the set of positive

integers, we shall get certain results. Mathematics takes another leap, from the entity which is constructed in fact to the entity whose construction is hypothetical. To some extent hypothetical entities are present from the start: whenever we assert that every positive integer has a certain property, in essence we are considering a positive integer whose construction is hypothetical. But now we become bolder and consider a hypothetical set, endowed with hypothetical operations subject to certain axioms. In this way we introduce such structures as topological spaces, groups, and manifolds. The motivation for doing this comes from the study of concretely constructed examples, and the justification comes from the possibility of applying the theory of the hypothetical structure to the study of more than one specific example. Recently it has become fashionable to take another leap and study, as it were, a hypothetical hypothetical structure—a hypothetical structure qua hypothetical structure. Again the motivations and justifications attach themselves to particular examples, and the examples attach themselves to numbers in the ultimate analysis. Thus even the most abstract mathematical statement has a computational basis. The transcendence of mathematics demands that it should not be confined to computations that I can perform, or you can perform, or 100 men working 100 years with 100 digital computers can perform. Any computation that can be performed by a finite intelligence—any computation that has a finite number of steps—is permissible. This does not mean that no value is to be placed on the efficiency of a computation. An applied mathematician will prize a computation for its efliciency above all else, whereas in formal mathematics much attention is paid to elegance and little to efficiency. Mathematics should and must concern itself with efficiency, perhaps to the detriment of elegance, but these matters will come to the fore only when realism has begun to prevail. Until then our first concern will be to put as much mathematics as possible on a realistic basis without close attention to questions of efficiency. 2. THE

IDEALISTIC

COMPONENT

OF

MATHEMATICS

Geometry was highly idealistic from the time of Euclid and the ancients until the time of Descartes, unfolding from axioms taken either to be self-evident or to reflect properties of the real world. Descartes reduced geometry to the theory of the real numbers, and in the nineteenth

4

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

century Dedekind, Weierstrass, and others, by the arithmetization of the real number system, brought space into the concrete realm of objects constructed by pure thought. Unfortunately the promise held out to mathematics by the arithmetization of space was not fulfilled, largely due to the intervention, around the turn of the century, of the formalist program. The successful formalization of mathematics helped keep mathematics on a wrong course. The fact that space has been arithmetized loses much of its significance if space, number, and everything else are fitted into a matrix of idealism where even the positive integers have an ambiguous computational existence. Mathematics becomes the game of sets, which is a fine game as far as it goes, with rules that are admirably precise. The game becomes its own justification, and the fact that it represents a highly idealized version of mathematical existence is universally ignored. Of course, idealistic tendencies have been present if not dominant in mathematics since the Greeks, but it took the full flowering of formalism to kill the insight into the nature of mathematics which its arithmetization could have given. To see how some of the most basic results of classical analysis lack computational meaning, take the assertion that every bounded nonvoid set A of real numbers has a least upper bound. (The real number b is the least upper bound of A if a < b for all a in A and if there exist elements of A that are arbitrarily close to b.) To avoid unnecessary complications, we actually consider the somewhat less general assertion that every bounded sequence {z;} of rational numbers has a least upper bound b (in the set of real numbers). If this assertion were constructively valid, we could compute b, in the sense of computing a rational number approximating b to within any desired accuracy;in fact we could program a digital computer to compute the approximations for us. For instance, the computer could be programmed to produce, one by one, a sequence {(bx,mx)} of ordered pairs, where each br is a rational number and each m; is a positive integer, such that (i) z; < br 4+ k~'for all positive integersj and k, and (ii) Zp, > br — k™1 for all positive integers k. Unless there exists a general method M that produces such a computer program corresponding to each bounded

constructively given sequence justified,

by

constructive

{z:} of rational numbers, we are not

standards,

in asserting

that

each

of the

sequences {x:} has a least upper bound. To see the scope such a method M would have, consider a constructively given sequence {n;} of integers, each of which is either 0 or 1. Using the method J, we compute a rational number b; and a positive integer N = m; such that (i)

A

CONSTRUCTIVIST

MANIFESTO

5

n; < bz + 2 for all positive integers 7, and (ii) ny > b3 — 1. Either ny = 0or ny = 1. If ny = 0, then (i) and (ii) imply that n %. In the first case, a > 0, and therefore the first nonzero term of the sequence {n:}, if one exists, equals 1. Similarly, in the second case, the first nonzero term, if one exists, equals —1. Thus our theorem gives a method, for each of the sequences {n;} being considered, of either (i) proving that any term that equals 1 is preceded by a term that equals —1, or (ii) proving that any term that equals —1 is preceded by a term that equals 1. Nobody believes that such a method will ever be found.

FOUNDATIONS

6

OF

CONSTRUCTIVE

ANALYSIS

Brouwer fought the advance of formalism and undertook the disengagement of mathematics from logic. He wanted to strengthen mathematics by associating to every theorem and every proof a pragmatically meaningful interpretation. His program failed to gain support. He was an indifferent expositor and an inflexible advocate, contending against the great prestige of Hilbert and the undeniable fact that idealistic mathematics produced the most general results with the least effort. More important, Brouwer’s system itself had traces of idealism and, worse, of metaphysical speculation. There was a preoccupation with the philosophical aspects of constructivism at the ex-

pense of concrete mathematical activity. A calculus of negation was developed which became a crutch to avoid the necessity of getting precise constructive results. It is not surprising that some of Brouwer’s precepts were then formalized, giving rise to so-called intuitionistic number theory, and that the formal system so obtained turned out not to be of any constructive value. In fairness to Brouwer it should be said that he did not associate himself with these efforts to formalize reality; it is the fault of the logicians that many mathematicians who think they know something of the constructive point of view have in mind a dinky formal system or, just as bad, confuse constructivism with recursive function theory. Brouwer became involved in metaphysical speculation by his desire

to improve the theory of the continuum. A bugaboo of both Brouwer and the logicians has been compulsive speculation about the nature of the continuum. In the case of the logicians this leads to contortions in which various formal systems, all detached from reality, are interpreted within one another in the hope that the nature of the continuum will somehow emerge. In Brouwer’s case there seems to have been a nagging suspicion that unless he personally intervened to prevent it the continuum would turn out to be discrete. He therefore introduced the method of free-choice sequences for constructing the continuum, as a consequence of which the continuum cannot be discrete because it is not well enough defined. This makes mathematics so bizarre it becomes unpalatable to mathematicians, and foredooms the whole of Brouwer’s program. This is a pity, because Brouwer had a remarkable insight into the defects of classical mathematics, and he made a heroic attempt to set things right. 3. THE

CONSTRUCTIVIZATION

OF

MATHEMATICS

A set is defined by describing exactly what must be done in order to construct an element of the set and what must be done in order to show

A

CONSTRUCTIVIST

MANIFESTO

7

that two elements are equal. There is no guarantee that the description will be understood; it may be that an author thinks he has described a set with sufficient clarity but a reader does not understand. As an illustration consider the set of all sequences {n;} of integers. To construct such a sequence we must give a rule which associates an integer nx to each positive integer £ in such a way that for each value of k the associated integer n; can be determined in a finite number of steps by an entirely routine process. Now this definition could perhaps be interpreted to admit sequences {n:} in which n; is constructed by a search, the proof that the search actually produces a value of n; after a finite number of steps being given in some formal system. Of course, we do not have this interpretation in mind, but it is impossible to consider every possible interpretation of our definition and say whether that is what we have in mind. There is always ambiguity, but it becomes less and less as the reader continues to read and discovers more and more of the author’s intent, modifying his interpretations if necessary to fit the intentions of the author as they continue to unfold. At any stage of the exposition the reader should be content if he can give a reasonable interpretation to account for everything the author has said. The expositor himself can never fully know all the possible ramifications of his definitions, and he is subject to the same necessity of modifying his interpretations, and sometimes his definitions as well, to conform to the dictates of experience. The constructive interpretations of the mathematical connectives and quantifiers have been established by Brouwer. " To prove the statement (P and @) we must prove the statement P and prove the statement @, just as in classical mathematics. To prove the statement (P or ) we must either prove the statement P or prove the statement @, whereas in classical mathematics it is possible to prove (P or Q) without proving either the statement P or the statement Q.

The connective “implies” is more complicated. To prove (P implies Q) we must show that P necessarily entails @, or that @ is true whenever

P is true. The validity of the computational facts implicit in the statement P must insure the validity of the computational facts implicit in the statement @, but the way this actually happens can only be seen by looking at the proof of the statement (P implies ). Statements formed with this connective, for example, statements of the type ((P implies Q) implies R), have a less immediate meaning than the statements from which they are formed, although in actual practice this does not seem to lead to difficulties in interpretation. The negation (not P) of a statement P is the statement (P implies 0 = 1). Classical mathematics makes no distinction between the con-

3

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

tent of the statements P and not (not P), whereas constructively the latter is a weaker statement. Brouwer’s system makes essential use of negation in defining, for instance, inequality and set complementation. Thus two elements of a set A are unequal according to Brouwer if the assumption of their equality somehow allows us to compute that 0 = 1.

It is natural to want to replace this negativistic definition by something more affirmative, phrased as much as possible in terms of specific computations leading to specific results. Brouwer himself does just this for the real number system, introducing an affirmative and stronger relation of inequality in addition to the negativistic relation already defined. Experience shows that it is not necessary to define inequality in terms of negation. For those cases in which an inequality relation is needed, it is better to introduce 1t affirmatively. The same remarks apply to set complementation. Van Dantzig and others have gone as far as to propose that negation could be entirely avoided in constructive mathematics. Experience bears this out. In many cases where we seem to be using negation—for instance, in the assertion that either a given integer is even or it is not— we are really asserting that one of two finitely distinguishable alternatives actually obtains. Without intending to establish a dogma, we may

continue to employ the language of negation but reserve it for situations of this sort, at least until experience changes our minds, and for counterexamples and purposes of motivation. This will have the advantage of making mathematics more immediate and in certain situations forcing us to sharpen our results. Proofs by contradiction are constructively justified in finite situations. When we have proved that one of finitely many alternatives holds at a certain stage in the proof of a theorem, to finish the proof of the theorem it is enough to show that the theorem is a consequence of each of the alternatives. Should one of the alternatives lead to a contradiction, that is, imply 0 = 1, we may either say that the alternative in question is ruled out and pass on to the consideration of the other alternatives, or we may be more meticulous and prove that the theorem is a consequence of the equality 0 = 1. A universal statement, to the effect that every element of a certain set A has a certain property P, has the same meaning in constructive as in classical mathematics. To prove such a statement we must show by some general argument that if x is any element of 4, then z has property P. Constructive existence is much more restrictive than the ideal existence of classical mathematics. The only way to show that an object exists is to give a finite routine for finding it, whereas in classical mathematics

A CONSTRUCTIVIST

MANIFESTO

9

other methods can be used. In fact the following principle is valid in classical mathematics: Evther all elements of A have property P or there exists an element of A with property not P. This principle, which we shall call the principle of omniscience, lies at the root of most of the unconstructivities of classical mathematics. This is already true of the principle of omniscience in its simplest form: if {n:} is a sequence of integers, then either n; = 0 for some &k or n; # 0 for all k. We shall call this the limited principle of omniscience. Theorem after theorem of classical mathematics depends in an essential way on the limited principle of omniscience, and is therefore not constructively wvalid. Some instances of this are the theorem that a continuous real-valued function on a closed bounded interval attains its maximum, the fixedpoint theorem for a continuous map of a closed cell into itself, the ergodic theorem, and the Hahn-Banach theorem. Nevertheless these theorems are not lost to constructive mathematics. Each of these theorems P has a constructive substitute ¢, which is a constructively valid theorem @ implying P in the classical system by a more or less simple argument based on the limited principle of omniscience. For instance, the statement that every continuous function from a closed cell in euclidean space into itself admits a fixed point finds a constructive substitute in the statement that such a function admits a point which is arbitrarily near to its image. The extent to which good constructive substitutes exist for the theorems of classical mathematics can be regarded as a demonstration that classical mathematics has a substantial underpinning of constructive truth. When a classical mathematician claims he is a construetivist, he probably means he avoids the axiom of choice. This axiom is unique in its ability to trouble the conscience of the classical mathematician, but in fact it is not a real source of the unconstructivities of classical mathematics. A choice function exists in constructive mathematics,

because a choice is implied by the very meaning of existence. Applications of the axiom of choice in classical mathematics either are irrelevant or are combined with a sweeping appeal to the principle of omniscience. The axiom of choice is used to extract elements from equivalence classes where they should never have been put in the first place. For instance, a real number should not be defined as an equivalence class of Cauchy sequences of rational numbers; there is no need to drag in the equivalence classes. The proof that the real numbers can be well ordered 1s an instance of a proof in which a sweeping use of the principle of omniscience is combined with an appeal to the axiom of choice. Such proofs offer little hope of constructivization. It is not likely that the

10

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

theorem ‘‘the real numbers can be well ordered” will be given a constructive version consonant with the intuitive interpretation of the classical result. Almost every conceivable type of resistance has been offered to a

straightforward

realistic

treatment

of mathematics,

even

by

con-

structivists. Brouwer, who has done more for constructive mathematics than anyone else, thought it necessary to introduce a revolutionary, semimystical theory of the continuum. Weyl, a great mathematician

who in practice suppressed his constructivist convictions, expressed the opinion that idealistic mathematics finds its justification in its applications to physics. Hilbert, who insisted on constructivity in metamathematics but believed the price of a constructive mathematics was too great, was willing to settle for consistency. Brouwer’s disciples

joined forces with the logicians in attempts to formalize constructive mathematics. Others seek constructive truth in the framework of recursive function theory. Still others look for a short cut to reality, a point of vantage which will suddenly reveal classical mathematics in a constructive light. None of these substitutes for a straightforward realistic approach has worked. It is no exaggeration to say that a straightforward realistic approach to mathematics has yet to be tried. It is time to make the attempt.

CHAPTER

CALCULUS

AND

THE

REAL

NUMBERS

Section 1 establishes some conventions regarding sets and functions. The next three sections are devoted to constructing the real mumbers, as certain Cauchy sequences of rational numbers, and investigating their order and arithmetic. The rest of the chapter concerns the basic ideas of the calculus of one variable. Topics covered include continuity, the convergence of sequences and series of continuous functions, differentiation, integration, Taylor’s theorem, and basic properties of the exponential and trigonometric functions and their tnverses. Most of the material 1s a routine constructwization of the corresponding part of classical mathematics. For this reason it affords a good introduction to the constructive approach. 11

12

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

The reader is assumed to be familiar with the order and the arithmetic of the integers and the rational numbers. For us a rational number will be an expression of the form p/q, where p and ¢ are integers with ¢ 0. The rational numbers p/q and p;/q: are equal if pg1 = p1q. The integer n is identified with the rational number n/1. There are geometric magnitudes that are not represented by rational numbers, that can only be described by a sequence of rational approximations. Certain such approximating sequences are called real numbers. In this chapter we construct the real numbers and study their basic properties. Then we develop the fundamental ideas of calculus.

1.

SETS

AND

FUNCTIONS

Before constructing the real numbers, we introduce some notions which are basic to much of mathematics. A sequence is a rule which associates to each positive integer n a mathematical object a,. The object a, is called the nth term of the sequence. The rule can be given explicitly or it can be left to inference by writing, for instance, the elements of the sequence in order {al,

g,

A3,

.

.

}

until the rule of formation is clear. Different notations for the sequence whose nth term is a, are n — a,, {a1, as, as, . . .}, {a.}2_;, and {a.}. Thus the sequence whose nth term is n? can be written n — n?, or {1,9,16, . . .}, or {n%}>_,, or simply {n?}. The cartesian product, or simply product, X

of sets X1, Xz,

=

Xl

X

£t

X

Xn

. . . , X, is defined to be the set of all ordered n-tuples

(1, . . . ,zx) Witha; €E X, . . . ,2, € X,. Elements (x1, . . . , Zs) and (y1, . . ., yn) of X are equal if the coordinates x; and y; are equal for each . The cartesian product Z X Z of the set Z of integers with itself can be arranged in a sequence, as follows. We order the elements (m,n) of

Z X Z, first according to the value of [m| + |n|, then according to the value of m, and finally according to the value of n. This produces the sequence (11)

{(070)7

(_1)0)7

(O;_l))

(071)1

(—2;0);

(_1)_1)7


0, p # 0, and p is relatively prime to ¢, we obtain a sequence

((i)

(1.3)

:1_1

%

. ]

which has the property that for any given rational number r there exists exactly one term a, with a, = r. The totality of all mathematical objects constructed in accord with certain requirements is called a set. The requirements of the construection, which vary with the set under consideration, determine the set. Thus the integers are a set, the rational numbers are a set, and the collection of all sequences each of whose terms is an integer is a set. Each set A will be endowed with a relation = of equality. This relation is a matter of convention, except that it must be an equivalence relation. This means that the following properties must hold for all elements a, b, and ¢ of A.

(1.4)

i) (i) (1)

a=a b=a a =c

ifa=2> ifa=>ban b d =c

The relation pg; = pig of equality for rational numbers p/q and p1/q: is an equivalence relation. In this instance there is a finite, mechanical process for deciding whether two given rational numbers are equal. Such a process may not always exist. There are instances in which the problem of whether two given elements of a given set are equal is a nontrivial mathematical problem that we are unable to solve. Such an instance, in the theory of the real numbers, will be given later. We use the standard notation a € A to say that a is an element or member of the set 4, or that the construction defining a satisfies the requirements a construction must satisfy in order to define an element of the set A. The dependence of one quantity on another is expressed in the basic

14

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

notion of an operation. An operation from a set A to a set B 1s a rule f which assigns an element f(a) of B to each element a of A. The rule must afford an explicit, finite, mechanical reduction of the procedure for construeting f(a) to the procedure for constructing a. The set A 1s called the domain of f. The most important case oceurs when f(a1)

= f(as)

whenever a; and a, are equal elements of A. Such an operation f is called a function. A function is sometimes called a map or a mapping. A function whose domain is the set Z* of positive integers is, as above, called a sequence. A function f 1s sometimes written in the form a — f(a), so that n — n? is the function f from Z* to Z* defined by f(n) = n2. The notation f: A — B indicates that f is a function from A to B. If fi:A — B and f;: A — B, then f; = f» means that fi(a) = f.(a) for all elements a of A. Functions f: A — B and ¢g: B— C can be composed, giving a function g o f: A — C defined by

(g ° f)(a) = g(f(a)) Composition is associative,

ho(gof) =(hog)of whenever the compositions are well defined. If f:A—

and

B

g:B—A

g(f(@)) = a

for all @ in A, then g is called a left inverse of f, and f is called a right inverse of g. When g is both a right and a left inverse of f, it is called an tnverse of f. A function f: A — B which has an inverse is called a oneone correspondence, or a bijection, and the sets A and B are said to be in one-one correspondence or to be equipollent. A set which is equipollent with the integers is said to be countably

infintte. As an example, let f be the sequence (1.3). Define the function g from the set @ of rational numbers to Z* by g(r) = n, where n is the unique positive integer for which f(n) = r. Then ¢ is an inverse of f.

The set Q is therefore countably infinite. A similar proof shows that Z X Z is countably infinite.

CALCULUS

AND

THE

REAL

NUMBERS

15

A function f: A — B which has a right inverse is said to map A onto B. When f satisfies the weaker condition that there exists an operation g from B to A such that f(g(b)) = b for all b in B, it is called surjective and said to be a surjection. A set A is countable if there exists a surjection f:Z+— A. Intuitively, this means that the elements of A can be arranged In a sequence with possible duplications.

For each positive integer n let Z, be theset {0,1, . . . ,n — 1}. A set A which is equipollent with Z, is said to have n elements. Such sets are called finite. Every finite set is countable. Contrary to customary usage, every finite or countable set has at least one element. If there is a surjection from Z, to a set A, we say that A has at most n elements. Such sets A are called subfinite. It is not true that every countable set is either countably infinite

or subfinite. As an example let A consist of all positive integers n such that both n and n + 2 are prime. Then A is countable, but it is neither countably infinite nor subfinite. This does not mean that sometime in the future A will not have become countably infinite or subfinite. It is possible that tomorrow someone will show that 4 is subfinite. This set A has the property that if it is subfinite, then it is finite. Not all sets have this property.

2. THE

REAL

NUMBER

SYSTEM

The following definition is basic to everything that follows. Definition 1

A sequence {z,} of rational numbers is regular if

(2.1)

Zm — xa] Km0t

(m,n &€ ZT)

A real number is a regular sequence of rational numbers. numbers ¢ = {z,} and y = {y.} are equal if

(2.2)

20 — ya| < 2070

Two

real

(n € Z7)

The set of real numbers is denoted by R.

Proposition 1

Equality of real numbers is an equivalence relation.

Proof Parts (i) and (ii) of (1.4) are obvious. Part (iii) is a consequence of the following lemma.

16

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

Lemma 1 The real numbers x = {z,} and y = {y.} are equal if and only if for each positive integer 5 there exists a positive integer N; such that

(2.3)

|Zn — ¥l max {j,N,}, then lxn

-

ynl

S

Ixn

-

N1

for some n in Z+. A real number z = r & R if

{z,} is said to be nonnegative, or

(2.9)

(n € Z7)

Tn >

—nl

The following criteria are often useful.

Lemma 2 A real number x = {x,} s positive if and only if there exists a posilwe integer N such that ' (2.10)

Tm > N1

(m > N)

CALCULUS

AND

THE

REAL

NUMBERS

19

A real number x = {x.} 18 nonnegative 1f and only if for each n tn Z+ there exists N, in Z such that

(2.11)

(m 2 Nn)

Tm 22—~

Proof Assume thatx & R*. Then z, > n~! for some n in Z*. Choose N in Z* with 2N 1 < z, — n™1

Then

Tm 2 Tn — |Tm — Tu| = 2n — m™ > Xy —n ! — N1

— 01

> N1

whenever m > N. Therefore (2.10) is valid. Conversely, if (2.10) is valid, then (2.8) Therefore z € R,

holds with n = N + 1.

Assume next that x € R%". Then for each positive integer n, Tm 2> —m

>

—n1

(m > n)

Therefore (2.11) is valid. Assume finally that (2.11) holds. Then if k, m, and n are positive integers with m > N., we have

Tk > Tm — |Tm — x| > —n7! — k71 — m™! Since m and n are arbitrary, this gives xy > —k~1. Therefore x & R*. As a corollary of Lemma 2, we see that if z and y are equal real numbers, then z is positive if and only if y is positive, and z is nonnegative if and only if y is nonnegative. It is not strictly correct to say that a real number {z,} 7s an element of R*. An element of R+ consists of a real number {z.} and a positive integer n, such that z. > n~!, because an element of R* is not presented

until both {z,} and n are given. One and the same real number {z,}can be associated with two distinet (but equal) elements of RT. Nevertheless we shall continue to refer loosely to a positive real number {z.}. On those occasions when we need to refer to an n for which z, > n™, we shall take the position that it was there implicitly all along. The proof of the following proposition is now easy, and will be left

to the reader. For convenience R* represents either R+ or R°*.

20

FOUNDATIONS

Proposition 4

(2.12)

OF

CONSTRUCTIVE

ANALYSIS

Let x and y be real numbers. Then

(@) b)

whenever x & R* and y € R* z+yER*andzy ER* whenever t & Rt and y € RO+ =+ y&eRT

(c)

|z| € RO+

(d) (¢)

max {z,y} € R* min {z,y} € R*

whenever x & R* whenever x & R* and y € R*

We now define the order relations on R.

Definition 4 (2.13)

Let xz and y be real numbers. We define x>y

(ory y

(ory < x)

if 2 —y € RO

and

(2.14)

A real number x is negative if x < 0%, thatis, if —x is positive. Consider real numbers z, 2/, y, and y’ with

(2.15)

x =2

y =1y

-y

Then

and

x>y

R =x—yec

Therefore

' >y

(2.16)

We express the fact that (2.16) holds whenever (2.15) is valid by saying that > is a relation on R. We express the fact that z > y if and only if y < z by saying that > and < are transposed relations. Similarly, and > are transposed relations.


—m™!

By Definition 3, we have n~! — |z — z,| € R, Therefore [t — z.| < n—L.

Lemma

4

If x = {z,} and vy = {y,} are real numbers, with x < y,

there exists a rational number o with x < a < y.

Proof By Definition 2, we have y — z = {y2 — T2,}2.;. Since y — x & R*, by Definition 3 there exists n in Z+ with yen — 22 > 171

24

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

Write a = 2(xm + Y2n)

Then

o — 22> a — Zon — |Z2n — 2| > (Y20 — Z2n) — (2n)"1 > 0

Also,

¥y — a2 ym —a — |[yam — Y| 2 1¥2m — 22) — (20)71 > 0

Therefore r < a < y. As a corollary, for each z in R and r in R* there exists « in @ with

|t — «| < r. Here is another corollary. Proposition 7 If x, . . . , x, are real numbers Zn > 0, then x; > 0 for some z (1 < 7 < n).

with x; +

- - - +

Proof By Lemma 4 there exists a rational number « with 0 < o < 1+ - - - 4 x.. Let a; (1

2n)~la

i

Therefore a; > (2n)~la for some 7. It follows that T > a; — |T; —ai

Corollary

>0

If z,y, and z are real numbers with y < z, then either x < 2z

orxr > Y.

Proof Since z—x+xr—y=2—y >0, x — y > 0, by Proposition 7. The

next lemma

either

gives an extremely useful method

z—2

>0

or

for proving in-

equalities of the form z < y.

Lemma 5 Let x and y be real numbers such that the assumption x > y implies that 0 = 1. Then x < y.

Proof

Without loss of generality, we take y = 0. For each n in Z¥,

either z, < n~! or z, > n~L. The case z, > n~! is ruled out, since it implies x > 0. Therefore —z, > —n~!, for all n, so that —z > 0.

CALCULUS

AND

THE

REAL

NUMBERS

25

Theorem 1 Let {a,} be a sequence of real numbers. Let xo and yo be real numbers, xo < yo. Then there exists a real number x with

(2.22)

To < T < Yo

and

(2.23) Proof

r#a,

(n€Zt)

We construct by induction sequences

{x,} and

{y.}

of rational

numbers such that

(1)

(i)

2z, > anor Y, < Gy,

yn — 20 < n?

(n >1)

(n2>1)

Assume that n > 1 and that 2o, . . . constructed. Either a, > x,_; or @, < any rational number with z,_; < 2, < rational number with z, < y» < min relevant inequalities are satisfied. In

, n_1, Yo, - - . , Ya—1 have been y»—_1. In case a, > x,—1, let z, be min {a., y»—1}, and let y, be any {@,, Yn—1, Z» + n~1}. Then the case a, < yYa—1, let y. be any

rational number with max {@,,z.—1} < y» < Y»—1, and z, any rational number with max {a., T,—1, y» — 771} < 2, < yY». Again, the relevant inequalities are satisfied. This completes the induction. From (i) and (ii1) it follows that

|Tm — Tu| = T

— X0 < Yyn — T < W71

(m > n)

Similarly |y» — ya| < n~1for m > n. Therefore x = {z.} and y = {y.} are real numbers. By (iii), they are equal. By (i), 2. < z and y» > ¥ foralln. If a, < z, thena, < z,s0a, # z.Ilf a, > ynthena, > y = z, s0 a, # x. Thus z satisfies (2.22) and (2.23). Theorem 1 is the famous theorem of Cantor, that the real numbers are uncountable. The proof is essentially Cantor’s ‘‘diagonal” proof. Both Cantor’s theorem and his method of proof are of great importance. The time has come to consider some counterexamples. Let {n.} be a sequence of integers, each of which is either 0 or 1, for which we are unable to prove either that n; = 1 for some k or that n, = 0 for all k. This corresponds to what Brouwer calls ““a fugitive property of the natural numbers.” Such a sequence can be defined, for example, as

26

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

follows. Let ni be 0 if u* + »* % w* for all integers u, v, w, { with 0 < u, v, w < kand 3 < ¢t < k. Otherwise let n; be 1. Then we are unable to prove n, = 1 for some k, because this would disprove Fermat’s last

theorem. We are unable to prove n; = 0 for all k, because this would prove Fermat’s last theorem. Define now z, = 0 if n; = 0 for all j < k, and x;, = 2™ otherwise, where m is the least positive integer such that n, = 1. Then z = {x;} is a nonnegative real number, but we are unable to prove z > 0 or z = 0. Since nothing is true unless and until it has been proved, it is untrue that £ > 0 or x = 0. Of course, if Fermat’s last theorem is proved tomorrow, we shall probably still be able to define a fugitive sequence {n:} of integers. Thus it is unlikely that there will ever exist a constructive proof that for every real number x > 0 either x > 0 or z = 0. We express this fact by saying that there exists a real number x > 0 such that it is not true that x > 0 orx = 0. In much the same way we can construct a real number x such that it is not true that x > 0 or z < 0.

3.

SEQUENCES

AND

SERIES

OF

REAL

NUMBERS

We develop methods for defining a real number in terms of approximations by other real numbers. Definition 6 A sequence {z,} of real numbers converges to a real number z, if for each k£ in Z* there exists Ny in Z+ with

(3.1)

s — 2o < k!

(n > Ny

To express that {z.} converges to xo we write (3.2)

lim . =

or (3.3)

Tn — Zo

as n —

©

A sequence of real numbers is said to converge when there exists a real number to which it converges.

It is easily seen that if {z.} converges to both z, and zj, then 2, = x5, A convergent sequence is bounded: |za| < r for all n.

there exists r in Bt

such that

CALCULUS

AND

THE

REAL

NUMBERS

27

A convergent sequence of real numbers limit zo and the sequence { N} are given, itself. Even when they are not mentioned are implicitly present. Similar comments definitions, including the following.

is not determined until the as well as the sequence {z,} explicitly, these quantities apply to many subsequent

Definition 7 A sequence {z,} of real numbers is a Cauchy sequence if for each k in Z* there exists M in Z* such that

(3.4)

[@m — 2a| < KT

(m,m > M)

Theorem 2 A sequence {x,} of real numbers converges if and only if it 1s a Cauchy sequence.

Proof Assume that {z,} converges to a real number sequence {N;} satisfy (3.1). Write My = Na. Then

|Tm — 2] < |2m — 20| + |20 — 20| < (2k)"1+ F)L

z,

Let

the

= k1

for m, n > M. Therefore {z.} is a Cauchy sequence. Assume conversely that {z.} is a Cauchy sequence. Let the sequence {M} satisfy (3.4). Write N} = max {3k,M 5}. Then lxm

-

xnl

S

(Zk)_l

(m7

n

Z

Nk)

Let yx be the (2k)th rational approximation to xzy,. For m > n,

[Ym = Yn| < |Ym — 2n,| + 2§, — 2N,| + |28, — Y2l N and compute ly — Za|


Ni)

1Zn — o] < 2mk)™Y, [y — yo| < (2mk)~? Then for n > Ny, Ixn@/n -

xoy0|


0,

Za| > |o| — |Tn — o] > L|zo| whenever n is large enough, say for n > no. Let k£ and n be positive

integers such that n > no and |z, — zo| < (2k)~!|x|%. Then

a7t = 207 = |27zl M2 — @o| < 2]ao|TH(2R) M| = £ Therefore z,” ! — zy 1 as n —

.

(f) We compute

Yo — %o = lim y, — lim z, = lim (y, — 2,) = lim |y, — .| n—

o

n—»

n—r

«©

n—

o

= lim max {y, — Tn, Tn» — Yo} = max {yo — o, To — Yo} > 0 n—r

by (a), (b), (c), and (d). For each sequence {z.} of real numbers the number

k=1

is called the nth partial sum of {x.}, and {s.} is called the sequence of partial sums of the sequence {z.}. A sum sy of {z,} is a limit of the

30

FOUNDATIONS

sequence

OF

CONSTRUCTIVE

ANALYSIS

{s,} of partial sums, and we write

to indicate that so 1s a sum of {z.}. A sequence which is meant to be summed is called a series. A series is said to converge to its sum. Thus the sequence {2-"}>_, converges to 0 as a sequence, but as a series it con-

verges to » 27 = 1. n=1

A convergent series remains convergent, but not necessarily to the same sum, after modification of finitely many of its terms. The series {x,} is often loosely referred to as the series

z

L.

n=1 0

If the series

2

z, converges, then z, - 0 asn—

«.

n=1 o0

A series

o0

z

T, is said to converge absolutely when the series

n=1

z

||

n=1

converges. In classical analysis a series of nonnegative terms converges if the partial sums are bounded. This is not true in constructive analysis. However, we have the following result. 0

Proposition 9

If z

Yn 18 G convergent series of nonnegative terms, and

n=1

if |xn| < yu for each n, then

z

Zn CONverges.

n=1 o0

Proof

Since

z

Y. 18 convergent, the sequence of partial sums is a

n=1

Cauchy sequence. Z* with

Therefore

there

exists for each k in Z+ an Ny in

>, ¥iSkt

(m=n2Ni

i=n+1

Then

l

2 I=n+1

x,-l
n > Ny)

j=n+1

Therefore the sequence of partial sums of the series

2 n=1

sequence. By Theorem 2, the series converges.

z, is a Cauchy

CALCULUS

THE

AND

31

NUMBERS

REAL

The criterion of Proposition 9 i1s known as the comparison test. 1t follows from the comparison test that every absolutely convergent series 1s convergent. The

terms

of an

absolutely

convergent

series

z

x, may

be

re-

ordered without affecting the sum s, of the series. More precisely, if

AN ZT — Z7 1s a bijection, then

z

Thny €xists and equals so. This may

n=1

not be true if the series

z

x, 1s merely convergent.

A sequence {z,} is said to diverge if there exists ¢ in R* such that for

each k in Z* there exist m and n in Z+ withm, n > kand |z, — 2. > . The motivation for this definition is, of course, that a sequence cannot be both convergent and divergent. A series is said to diverge if the sequence of 1ts partial sums diverges. The series z

x, diverges whenever there

exists 7 in R* such that |z,|> r for 1nfin1tely many values of n. The following very useful test for convergence and divergence is called the ratio test. Proposition 10

Let

z n=

a posttwe integer. Then

1

T, be a series, ¢ a positive real number, and N

z

x, converges if ¢ < 1 and

n=1

(3.6)

21|

< el

(n > N)

and diwerges if ¢ > 1 and

(3.7)

|Zat1| > clzal

(n > N)

Proof

Assume

c¥|zy|

for n > N. By the comparison test, z

that

¢ < 1 and

that

(3.6) is

valid.

X, converges.

Next, assume that ¢ > 1 and that (3.7) holds Then

|Za] 2 cm Ny and

|tnga| > clen| > 0 o0

Therefore

2 [Twi]

z n=1

z, diverges.

Then

(n>N+1)

|[z,|
1. Important real numbers represented by series are (nh)—1!

2

e =1+

n=1

4120 (—1)*(2n + 1)1

T

and

The series for e converges by the ratio test. The convergence of the series for 7 is a consequence of the general result that a series 2

(— 1)z,

n=1

converges whenever (i) z, > O for all » and (ii) the sequence {z,} converges monotonely to 0. To see this, consider positive integers m and n, with m > n. Then O

S

-

(xn

"I"

xn+1)

-

(xn+2

xn+3)

+

el

+

(_1)m+nxm

= (=1 ), (1)t k=n

=

Tn

—

(xn+l

-

xn+2)

-

"I"

(_1)m+nxm

S

In

It follows that the sequence of partial sums of the series is a Cauchy sequence. Therefore the series converges.

4. CONTINUOUS

FUNCTIONS

A property P which is applicable to the elements of a set S is defined by a statement of the requirements that an element of S must satisfy in order to have property P. To construct an element of S with property

P we must (a) construct an element of S, (b) perform certain additional constructions which depend on the property P, and (c) prove that the entities constructed satisfy certain requirements that are characteristic of the property P. Each property P applicable to elements of S determines a subset of S, which is denoted by

{x:xz € S, = has property P}

CALCULUS

AND

THE

REAL

NUMBERS

33

Properties P applicable to S and subsets A of S are essentially the same things regarded from different points of view. The intersection

{x:x € 8, x has property P} n {z:z & 8§, x has property Q} is defined to be the set

{x:x € S, x has properties P and @} For certain purposes it is convenient to extend the real numbers R by adjoining elements — «© and «. To construct an element of the set R, of extended real numbers we either construct a real number, pick the element — o, or pick the element 4 . The order relation < is extended to R, by defining — o < o, —o 0 there exists w(e) > 0 such that

[f(z) —FW)| < e

(4.3)

whenever z, y € I and |z — y| < w(e). The operation e — w(e) is called a modulus of continuity for f. A real-valued function f on an arbitrary interval J is conttnuous on J if 1t 1s continuous on every compact subinterval I of J. For example,

when a and b are finite and a < b, then f is continuous on (a,b) if and only if it is continuous on [a + 4§, b — §] for each 6 with 0 < 6
0 there exists zin 4 withc

— x < ¢

(respectively, ¢ — ¢ < e). Theorem 3 Let the subset A of R have the property that for each ¢ > 0 there exist finilely many points Y1, . . . , Y. tn A such that for each x in A at least one of the numbers |x — yi1|, . . . , |t — ya| s less than e. (Such a sel s called totally bounded.) Then Lu.b. A and g.l.b. A exist.

Proof For each positive integer k& choose points y1, . . ., ¥, in A such that for each z in A at least one of the numbers|z — 1|, . . . , |z — ¥4

CALCULUS

AND

THE

REAL

NUMBERS

35

is less than k~1. For some value of m (1 < m < n) we have

Ym = MAX {Y1, . . . , Yo} — k7 Write zx = yn. If z is any point in 4, we have |x — y;| < k~! for some value of 7 (1 < 7 < n). Hence

T— Tk =T — Y+ Thus the sequence

Y — Yu < k714 k71 = 271

{z;} of elements of A has the property that = —

rr < 2k7! for all z in A and all ¥ in Z+. In particular, |z; — 7| < 257! + 2k~ for all j and k. Therefore {z:} is a Cauchy sequence, whose limit we call zo. For all zin A, r

—

Xy

=

(x

hm k—

«

—

xk)

S

2](3—1

hm k—

=

o

Therefore o 1s an upper bound for 4, and since zo =

lim z; it is a least k—

«

upper bound. The proof that g.l.b. 4 exists is similar. Corollary If f:[a,b]@ R is a continuous interval, then the quantities

and

function

on

a compact

¢ = Lu.b. {f(z): 2 E [a,b]} d=glb. {f(x):z € a,b]}

(called, respectively, the supremum and the infimum of f on the interval [a,b]) exist.

Proof

Consider ¢ > 0. Choose real numbers a=a ¢ for all x in J and some ¢ > 0 (depending on J), then f~! 1s conttnuous on I.

Proposition 11 implies that the quotient of continuous functions is continuous, provided that the denominator is bounded away from O on every compact subinterval. It also implies that a polynomial function

r—cor” +cx"

4

- - 0+

is continuous on every interval, and that |f| is continuous on each interval where f is continuous. The composition of continuous functions is continuous, in the sense that if f: I — J and ¢g: J — R are continuous, then g o f is continuous, provided that f maps every compact subinterval of I into a compact subinterval of J. To prove this it is sufficient to consider the case in which I and J are both compact. Let w be the modulus of continuity of f and ¢ the modulus of continuity of g. Then if 2z, y € I, ¢ > 0, and |z — y| < w(a(e)), we have |f(x) — f(y)| < o(e). Therefore

g(F(2)) — 9(f(¥)]| < e It follows that g o f is continuous, with modulus of continuity w o o. Just as sequences of real numbers can converge to real numbers, sequences of continuous functions can converge to continuous functions. In fact, most of the important functions of analysis are defined in this way.

Definition 11

A sequence {f,} of continuous functions on a compact

interval I converges on I to a continuous function f if for each ¢ > 0 there exists N, in Z* such that

(4.6)

o)

—f@)| M)

0 choose N, in Z* satisfying (4.6), and write M. = N2

@)

Then, whenever m, n > M. and x & I, we have

= fa(@)| < |fal@) = f@)] + 1fula) = f@)| 0 write

w(e) = wy(e/3)

where M = M,;;. Then wheneverz,y € I and |z — y| < w(e), we have | f(z)

— fy|


0,and

of continuity w. Finally,

if

[fa(@) — f(@)| = Lm |fa(z) — fu(@)] < e m—> o

Therefore {f.} converges to f on I. Notations to express the fact that {f.} converges to f are lim f, = f, and f, > fasn— «. To each sequence {f.} of continuous functions on an interval I corresponds a sequence {g.} of partial sums, defined by n

gn = Ic;l Jx

If {g.} converges to a continuous function g on I, then g is the sum of the series

i

fn:

n=1

g =

Zlfn

and the series is said to converge to g on I. If 2

|f»| converges on I,

n=1

then

2 fx 1s said to converge absolutely on I. n=1

The comparison test and the ratio test carry over to series of functions.

CALCULUS

AND

THE

REAL

NUMBERS

39

The comparison test states that if

E

g» 18 & convergent series of

n=1

nonnegative

continuous

funections

on an interval I, then

the

series

0

z

f» of continuous functions on I converges on I whenever

|f,(x)|

n=1

< gn(z) for all n in Z+ and all z in 1. The ratio test states that if E fn 1s a series of continuous functions n=1

on an interval J such that for each compact subinterval I of J there exists a constant ¢r, 0 < ¢f < 1, and a positive integer Ny with

@& I, n2= N

[frrr@)] < erlfal@)] then

E

f» converges absolutely on J.

n=1 o0

A power series is a series of the form Z

a,(x — xo)", where a,(x — xo)”

n=0

represents the function z — a,.(r — z¢)® and where for all . The ratio test has the following corollary.

Proposition 12

a¢(z — 20)° = ao

Let the power series o0

(4.10)

Y, anlz — zo) n=0

have the property that there exists r > 0 and N in Zt such that |an1| < r~Ya,| for all n > N. (xo —

7, Ty +

Then

(4.10)

converges absolutely on the interval

r).

Proof If I is a compact subinterval of (xo — r, o + r), then there exists ro with 0 < r¢ < r such that |z — xo| < 7o for all z in I. Then

tnpr@ — 2™ < 1Mrolan(@ — 27|

(> N)

By the ratio test, (4.10) therefore converges absolutely on I.

5. DIFFERENTIATION

The rate at which a function is changing is a fundamental property of the function. Here is the precise definition of this concept.

40

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

Definition 13 Let f and g be continuous functions on a compact proper interval I such that for each ¢ > 0 there exists §(¢) > 0 with (5.1)

1f@) —f@

@ye&l

—g@)y — )| 9(x) d

if f < g. Proof

Equation (6.9) follows from the fact that

S(af + By, a, b, n) = aS(f,a,b,n) + BS(g,a,b,n) for all n, and (6.10) from the inequalities S(f,a,b,n) < S(g,a,b,n) As a corollary we have

(6.11)

b

b — a) < [f@) dz < eald — a)

whenever ¢; < f(z) < c;for all zin [a,b]. By setting —c; = ¢2 = ||| 1a,81

CALCULUS

AND

THE

REAL

NUMBERS

49

we obtain

(6.12)

[ 7@ dz| < [llen® — a)

Proposition

14

If f is a continuous function

on a compact

interval

[a,b], then

613

[f@do = [[f@do+ [(f@de

Proof

Tor arbitrary partitions P = {a, . . ., ¢} of [a,c] and @ =

{c, . . ., b}

(@ 0 and each pair of points z, y in I with |z — y| < w(e), we have

90) — 9@) — f@@ — )| = | ["O &t = [ 1) dt|

=| [ G® - @) dt| < dy — =l Therefore g is differentiable on I with derivative f and modulus of differentiability w. If also Dg, = f on I, then D(g — go) = 0 on I. By the mean-value theorem, (g — go)(a) = (¢ — ¢o)(b) whenevera, b & I and a < b. Therefore g — go 1s a constant function. Our next lemma

has an interest of its own.

Lemma 7 Let {f.} be a sequence of continuous functions converging to a function f on a nonvoid interval I. Let a be any point of I. Write

and

g@) = [(fd

@ED

@) = [(fu)d

@ED

for each positive integer n. Then g,— gon I asn—

.

Proof 1t is no loss of generality to assume that I is compact. Let r be the length of I. Then for each x in I, we have

9:@) — 9@\ = | [7 () = 5®) | 0 and z + y > 0. Then, by (7.6),

exp (z + y) exp (2) = exp (x + y + 2) = exp (2) exp (¥ + 2) = exp (z) exp (y) exp (2) which is equivalent to (7.5).

It can be shown that exp is the unique continuous function from R to R that has the value 1 at 0 and satisfies (7.5).

CALCULUS

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53

NUMBERS

Equality (7.5) implies that exp (—z) for x > 0, it follows that

(x € R)

exp () > 0

(7.7)

= (exp (x))~L. Sinceexp (z) > 1

The theory of the exponential function can also be approached as follows. To construct a function f with f(0) = 1 and f = f, notice that f’ = f implies, by the mean-value theorem, that f(x 4 6) is approximately equal to f(zx) + 6f'(z) = (1 + §)f(x) when § is small. Therefore, if ¢ is any real number and n is a large positive integer, f(f/n) is approximately 1 4 (¢/n), f(2t/n) = f(t/n + t/n) is approximately (1 4 (¢/n))?, and so forth, and thus f(f). = f(n(¢/n)) is approximately (1 + (¢/n))". This heuristic argument suggests that we define

exp (f) = lim

t»/n! as n —

4o

. This approach to the theory of

the exponential function would therefore lead to the same series representation. The logarithmic function In is the inverse function to the exponential function, but we shall not define it that way. Arguing heuristically, on the basis of the chain rule (5.6), we say that

1 =92

_ 4 (exp (n(@))) = exp (h ()2 (0 (@) = 7 2 In @

or

(7.8)

d— (In (z)) = a1_ dx

(x > 0)

We therefore define In (x) to be the integral of 21, specifically (7.9)

In (x) = /lx =1 dt

(x > 0)

By Theorem 10, In is differentiable on (0, o), and (7.8) is valid.

FOUNDATIONS

54

Let y be any positive real number. function (7.10)

z—

OF

ANALYSIS

CONSTRUCTIVE

By (5.6), the derivative of the

In (zy) — In (v)

is 271 Also, (7.10) vanishes at 1. By the last statement of Theorem 10, In (z) = In (zy) — In (y), or

(7.11)

In(@y) =In(x) +In(y)

(r,y > 0)

This is the functional equation for the logarithmic function, corresponding to the functional equation (7.5) for the exponential function. By (7.7), the composite function In o exp exists and is differentiable everywhere on R. By (5.6), we have

L mep@) =1

@ER)

Since also In (exp (0)) = In (1) = 0, we have (7.12) by

the

In (exp (x)) = = last

of

part

Theorem

10.

(x € R) Consider

z > 0

and

write

y =

exp (In (z)). Then

In (y) = In (exp (In (x))) = In (x) by (7.12), or

0 =1In(y) — In (2) =/xyt—1dt If y > z, this gives 0 > y~1(y — z), a contradiction. By Lemma 5, we have y < z. Similarly, z < y. Hence £ = y. Thus exp (In (z)) =z

forallz > 0. It follows that the functions exp: R — Rt andIn: Rt — R are inverse to each other. The trigonometric functions sin and cos also can be approached via an intuitive analysis of their rates of change. From this analysis we are led to believe that (7.13)

d g5 C08T =

: —sinz

d . 75 ST

= cos2

(x € R)

CALCULUS

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REAL

NUMBERS

55

We therefore define these functions by power series constructed in such a way that (7.13) will hold. Remembering that cos (0) = 1, sin (0) = 0, we are forced to define

(7.14)

cose = ) (=1 (;”:;, sin & =

"’:0

on+1

(—=1)~ (2nx+

(z € R)

1

Theorem 11 implies that (7.13) is valid. Proposition equations

(7.15)

16

(@) ()

The

functions

sin

and

cos

salisfy

the functional

sin (xr + y) =sinx cosy + cos z sin y cos(x+y) =coszxcosy —sinzsiny

for all z, y in R.

Proof Consider both sides of (7.15) (a) as functions of z, say f(z) and g(x). The functions f and g have Taylor's series about x = 0 which converge on R. The coeflicients of these series are expressed in terms of the various derivatives of f and g at x = 0. Therefore, to show that f = g on R, it is enough to show f®(0) = g™ (0) for all n > 0. Since f® = —fand g® = —g, it is enough to notice that

f(0) = sin y = sin 0 cos y + cos 0 sin y = ¢(0) and

f'(0) =cosy

=cos0cosy

+ (—sin 0) sin y = ¢'(0)

Therefore (7.15)(a) is valid. Equation (7.15)(b) follows from (7.15)(a) by differentiation with respect to z. Putting y =

(7.16)

—2 in (7.15)(b) gives the useful identity

coslzy + sin?2zx =1

Further study of the trigonometric functions depends on properties of the number =, which we construct as twice the first positive zero of the cosine function.

56

FOUNDATIONS

Theorem 12 the equations

OF

CONSTRUCTIVE

ANALYSIS

The sequence {x,} of real numbers defined inductively by

(7.17)

(@)

z1=1

(b)

Zp41 = 2, + cOS 2T,

(n > 1)

18 @ monotone-increasing converging sequence, whose limit w/2 satisfies the conditions

(7.18)

Proof

(@)

cos= =0

(b)

cosz >0

0 1-L50 2 =

Next assume that (7.19) is true for a particular value of n, and consider

a value of ¢ with z, < ¢t < z441. Now [sin z| < 1 for all z in R, by (7.16), and |sin z,| < 1, because cos z, > 0. Hence cos t = CoS Tp, —

t

.

-

sinz dx > cosTp — (Tny1 — )

= 0

In

Since cos is a continuous function and cos ¢ > 0 whenever 0 < ¢ < z, or T, 0 whenever 0 < ¢ < z,41. By induction,

(7.19) holds for all n. Since cos z, > 0 for all n, the sequence monotone-increasing. From the mean-value theorem and (7.19) we obtain sinz, >sin0

{z,} is

=0

for all n, and sin ¢ > sin 1 whenever 1 < ¢t < z, for some n. Also by the mean-value theorem, for each n > 2 there exists ¢ in [2,_1,Z.] such that xn+l

—

Tn

X, + cos x, — (Xn_1 + cOS Tp_1) Tpn — Tn—1 + COS T, — COS Tp_1

CALCULUS

AND

REAL

THE

57

NUMBERS

is arbitrarily near to

Tn — Tu—1 — (SIN 1) (Xn — Tno1) < (Tn — Toa—1)(1 — sin 1) Therefore

Tnp1 — Tn < (1 — sin 1) (2, — Zu—y) This gives —

Tnyl

T

(1


0 and points z! and 2 in X with p(21,22) < lw(e). Let {ya'} and {y.?} be sequences of points

METRIC

87

SPACES

of Y converging respectively to z! and z2 Then p(y.!,y»?) < w(e) for all sufficiently large n, and thus p(f(y.!),f(y»?)) < e. Therefore

lim p(f(y=)),f(yn?)) < e p(f(x),f(x?)) = n—> The next result is called the Baire category theorem. Theorem 4 Let {U,} be a sequence of dense open sets in a complete metric space X. Then the intersection UEF\U,; n=1

18 also dense in X.

Proof Let S(zo,ro) be any open sphere in X. Since U, is dense, there exists a closed sphere Sc(zy,r1) with r; < 1, such that

Sc(z1,r1) C S(xo,ro) N Ui

Continuing by induction, we construct a sequence {Sc(z.,r)} of closed spheres such that r, < n~! and

Sc(Xn,mn) C S(xn—1,rn-1) N Uy

for all n. Then z. € S(z.,r.) whenever m > n, so that p(Tm,x.) < r. < n~L Therefore {z,} is a Cauchy sequence. Since all except finitely many terms of this sequence lie in any given closed sphere Sc(xn,rs), the limit z lies in each of these spheres. Therefore r & S(xo,r0) N ;\1 Un It follows that U is dense in X.

Theorem 4 is one of the most useful versions of Cantor’s diagonal technique (which was used in the proof of Theorem 1 of Chap. 2). The reader should show that the latter theorem is a corollary of Theorem 4. A located subset Y of a metric space X is nowhere dense in X if the metric complement —Y of Y is dense. With this definition we reformulate Theorem 4.

388

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

Corollary Let {Y.} be a sequence of nowhere-dense subsets of a metric space X. Then every open sphere S(x,r) in X contains a point y whose distance to each Y, s positive. Proof

By Theorem 4 there exists a point y in S(x,r) n M

— Y,. The

n=1

distance of y to Y, is positive because y & —Y,. A more significant concept for metrie spaces than boundedness is the concept of total boundedness, defined as follows.

Definition 12 A metric space X is fotally bounded if for each ¢ > 0 there exists a finite subset {z1, . . . , .} (where n is a positive integer depending on €) of X, called an e approximation to X, such that for each z in X at least one of the numbers p(x,21), . . . , p(z,x,) is less than e. The reader can show as an exercise that X is totally bounded if for each ¢ > 0 there exists a subfinite e approximation {z;, . . . , z.} to X. The property of total boundedness is preserved under passage to an equivalent metric. The product of a sequence of totally bounded metric spaces is totally bounded. Call a metric space X separable if it has a countable dense subset. The product of a sequence of separable metric spaces is separable.

Every totally bounded metric space is separable.

4. COMPACTNESS

Of special importance are those metric spaces which are both complete and totally bounded. Definition 13 A compact metric space, or simply a compact space, 1s a metric space which is complete and totally bounded. The compact intervals of real numbers are compact in the sense of Definition 13. Total boundedness is the more important of the two properties which occur in the definition of compactness, since a metric space which 1is totally bounded but not necessarily complete can always be compactified by passing to its completion. The following propositions show that compactness is preserved by many of the standard operations on metric spaces. Proposition 6

The product of a sequence of compact spaces is compact.

METRIC

SPACES

89

Proof This is an immediate consequence of the earlier remarks that such a product is (i) complete and (ii) totally bounded. Proposition

7

A

closed located subset

Y of a compact

space X

1s

compact.

Proof Thesubset Y is complete by the remark Toshow that Y is totally bounded, considere > be a 1e approximation to X. For each ¢ choose p(x:i,Y) + Le. Consider an arbitrary y in Y. Then p(y,z:;)
0 and let z = {z.} and y = {y.} be points of X with p(z,y) > e. Choose N in Z+ with

n=N+1

27" < ¢/2. By Definition 4, we have

l\Dlm

2 270, (Tn,n) > p(x,y) —

l\3|m

N

Therefore pn(Z.,y.) > ¢/2 for some n with n < N. Write g = f, o 74, where f,: X, — R is the function =

fu(22)

X,)

(2. €

)) 100 (Tny25) (pa(Zn)Yn

If z = {z,} € X and p(z,2) < €22~¥—1 then Ig(z)l

Ifn(zn)l

< 267 1pa(2n)2)

< 22p(z,2)

< 267127p(2,2)

< e

Similarly if p(y,2) < €22=¥-1, then |g(z)

—

ll =

Ifn(zn)

"‘fn(?/n)l

< Ze—llpn(xmzn)

—

Pn(xn;yn)l

< 261 pn(Yny2n) < 26712Vp(y,2) < € Therefore @ is separating. Corollary 2 Let X be a compact space, and let G consist of all functions r — p(x,z0), with o E X. Then A(QR) is dense in C(X).

Proof Consider ¢ > 0 and z, y in X function ¢ in A(G@) by

with

) 2) " p(z) g(2) = p(,y

p(x,y) 2 e. Define

the

100

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

Then if p(z,2) < €

9(2)] < elp(z,2) < e and if p(y,2) < €,

9(z) — 1| = p(z,y)7Yp(z,2) — p(z,y)| < e'p(y,2) < e Therefore ( is separating. Corollary 3 FEvery continuous function f on a compact set X C R" can be arbitrarily closely approximated on X by polynomzal functions p: R* > R. Proof

Tirst consider the case in which n =

1. By Lemma 5, the func-

tion x — |x — x| can be arbitrarily closely approximated on X by polynomials. The theorem then follows from Corollary 2. Next consider the case X = [a,b]” for some compact interval [a,b]. The result then follows from Corollary 1. Finally, consider the general case. Since X is bounded, there exists a compact interval [a,b] with X C [a,b]*. By the case already considered, each of the functions z — p(x,z0) (xo E [a,b]?) can be arbitrarily closely approximated by polynomials on [a,b]?. The result then follows by Corollary 2. The case n =1 and X = [—1,1] of Corollary 3 is the famous W everstrass approximation theorem. We turn now to the study of sufficient conditions for a subset of a compact space to be compact, first proving that every compact space is a union of finitely many compact sets of arbitrarily small diameters. Proposition 11 For every compact space X and every ¢ > 0 there exist finitely many compact subsets X, . . . , X, of X of diameters at most e whose union 1s X.

By the total boundedness of X, there exist subsets X!, . . . , Proof X! of X, each consisting of one point, such that for each z in X at least one of the numbers p(z,X;!) (1 2p(z,y0). Then if y

closed by Proposiclosed. Choose y, and ¢ as is any point of ¥

with p(z,y) < p(z,y0), we have p(¥,Y0) < p(x,y) + p(z,50) < 2p(x,y0) < and thus y € Y.. Therefore p(z,Y) exists and equals p(z,Y.). Thus Y is located. Assume next that Y is a closed located subset of a locally compact space X. Let yo be any point of Y. For each o in R write

X, = {z € X:p(x,y0)

< a}

Consider any ¢ > 0 for which X4 is compact. Choose r in (¢,2¢) so that the set

V={z&X:r

r, then

p(z,V) = 0. The closure U of U is therefore a compact subset of Xy,

by Proposition 7, and thus there exists « in (c,r) such that U n X, is compact. But

UnX.={y€Y:pyy) < af Since ¢ can be arbitrarily large, Y is locally compact.

The problem of extending a continuous function to a larger space arises often and in many forms. The following solution to a special case of this problem is the famous Tetze extension theorem, or rather as much of it as is constructively valid. Theorem 10 LetY be a locally compact subset of a metric space X, and I C R a compact proper interval. Let

f:Y—1I be continuous. Then there exists a function h: X — I which is uniformly continuous on bounded subsets of X such that h(y) = f(y) for all y in Y. Proof There is no loss of generality in assuming I = [—1,1]. By adjoining two additional points to X, we may assume also that there

exist points z*+ and z~ in Y with f(z*) = 1 and f(z~) = —1. (This will serve to make certain sets which will be defined below nonvoid.)

Let

g be a compactifier for Y, with g(z*) = g(z=) = 0, so that there exists a sequence {a,} of nonnegative constants with e, — © asn — o« such

that the set {y € Y:9(y) < a.} is compact for each n in Z*. By Theorem 8 there exists an @ in (1,4) such that for each n in Z+ the sets

and

A, B,

lveY: fly) £ —a,g9@¥) < an} lveY: fly) 2 a, 9@ < on}

are compact. Therefore the sets

and

A={yeY:f(y) B={ycY:f@y)

< —a} = aj

are locally compact. We define the function f; on X by

1@) = (o(x,A) + p(z,B))ap(z,A) — ap(x,B))

108

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

for x in X. Since (p(z,A) + p(z,B))~! is bounded away from 0 on bounded subsets of X, we see that f; 1s uniformly continuous on bounded subsets of X. Clearly |fi(x)] < a < 1 forall zin X. For all y in 4 we have

f@) —

)] = @) +al r with Sc(zo,t) CC U. Let v be the circular path of radius ¢ about

132

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

zo. By Theorem 4,

f@) = @riy™ [ JOE — 207 df whenever |z — 29| < ¢. By Proposition 4, we see that f’ is differentiable on Sc(zo,r). By Lemma 6, we see that f’ is differentiable on U. The formula (4.4) now follows from Theorem 4 and Proposition 4. Corollary in U.

1

A function f analytic tn an open set U is differentiable

Proof When U is simply connected, by Theorem 3 there exists a differentiable function g in U with ¢’ = f, and thus f is differentiable in U by Theorem 5. The general case follows from this special case by means of Lemma 6.

Corollary 2 Let {f.} be a sequence of differentiable functions on an open set U, converging uniformly on every compact set K CC U lo a functron f on U. Then f 1s differentiable on U. Proof

1If v is any triangular path with span

vy CC

U, then

0 = e d ) G u f [ m li = [ @ dz n—r

0

Therefore f is analytic, and consequently differentiable, on U. Corollary 3

If f is differentiable on Sc(zo,70), then

[f™(20)] < nlre™ sup {[f(2)]: |z — 2ol = o} Proof

(0 20)

For 0 < r < rylet ¥ be the circular path about 2z, of radius r.

Then

|f®(0)| = @m)~nl| [ f@)( — 2™ ds| < @r)~mllylrt sup {[FG)]: |z — 20| = 7} = nlr— sup

{|f(2)|: |z — 20| = 7}

Letting r — r, gives the result. Corollary

4

If f is differentiable on an open set U, and if zo ts an

interior point of U, then the function z — (f(z) — f(20))(z — 20)~" on U — {20} extends to a differentiable function on U.

COMPLEX

Proof

133

ANALYSIS

The result follows from Lemma 5 and Corollary 1 above.

Power series are uniquely suited to the representation of differentiable functions in the complex domain. The complex variable version of Taylor’s theorem reads as follows. Theorem 6 the series

Let f be differentiable in an open sphere S = S(zo,7). Then

(4.5)

2

an(2 — 2o)"

n=0

with coefficients a, given by

(4.6)

an = (n!)71f™ (20)

converges uniformly to f on every compact set K C C S. Conversely, if the series (4.5) converges uniformly to f on every compact set K CC S, then a. s given by (4.6).

Proof

Consider a compact set K CC

sphere, with 0 < ro < rand K C

S. Let T = Sc(z0,70) be a closed

T. Write t = L(r + ro). Let v be the

circular path of radius ¢ about zo. Then for all zin T,

f@) = @)™ [ O

— )7 de

= @e)7 [ JO& — 2071 = @ — 2§ — 2077

€ = 20t — 207} o

- @y [ 10 n=0

= @iyt Y {[ SO =z

i}

= 20

n=0

By Theorem 5 this means that (4.5), with a, given by (4.6), converges to f uniformly on 7', and therefore on K. If, conversely, (4.5) converges uniformly to f on T, we get

(W)@ (o) = @mi)t [ @& — 20t b o0

= ) a@m) [ ¢ = 2@ — 2yt df = a k=0

and thus (4.6) is valid.

134

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

Proposition 10 of Chap. 2 is valid in the complex domain. Hence a o0

power

series

2

a.(z — zo)® converges

absolutely

and

uniformly

on

n=0

every compact set K C C S(zo,70) 1f there exists N in Z+ such that

|Gni1| < r07Ya.| whenever n > N. 5.

ESTIMATES

SIZE,

OF

AND

THE

LOCATION

OF

ZEROS

It is a famous principle of classical analysis, called the maximum principle, that an analytic function attains its maximum on the boundary. Here is the constructive version of this result. Definition 12

1f K and B are compact sets, with B C K, such that

Sc(z,p(2,B)) CK

(2 € K)

then B is called a border for K.

Proposition 5

If B is a border for K, then

1fllx = ||fllz for all differentiable functions f on K.

Proof

Let ¢ be any positive number. Choose § > 0 so that K n B; is

compact and [f(2)| < ||fllz + ¢ whenever 2z € K n B;. Consider 2z, in K, with p(z0,B) > 6/2. Let v be the circle of radius r about zo, where r 18 chosen

so

that

p(z¢,B) — 6/2 0, we are through. Otherwise we continue. Either the process proceeds for N stages or it terminates at some stage n < N. Consider the former case. By the choice of N, there exist m of the points 2z;, . . . , 2y whose mutual distances are less than 4. Call these points w;, . . . , Wn. Then (5.7)

f(2) = (z —w1)

* - - (2 — wu)h(2)

(z € K)

for some differentiable function A on K. Since f(w:) = 0 and |f(2)| > M for all z in B3;, we have

ro = p(wy,B) > 36 Since B is a border for K, (5.8)

Sc(wy,ry) C K

By Lemma 9, there exists z in Bj; with

(5.9)

|z — wi| < 7o — 56

Since |w; — wi] < 6 (1 0 with

i—1

|@n—ilri > Y |@n—m|r™ + |p(0)] m=0

Then either a,—n # 0 for some m > j or else

(5.10)

lawilr?> Y 0 lanmlr™4+

3

lanmlr + |p(0)] ™

In the latter case

inf {[p(2)]: || = r} > |p(0)] and there exists a complex number 2z; with p(z;) = 0 by Lemma 11. In the former case we replace j with m and perform the same construction. Eventually this leads to a value of 5 for which (5.10) is true, and therefore to a complex number z; with p(z;) = 0. Using the process of polynomial division, we find a polynomial q1(z) = bz

1+

- - -

4+ b,_; and a constant ¢ in C with

p(z) = (2 — 21)qa(2) + ¢ Since p(z1) = 0, we trarily small if |bo|, bn_1—; #Z 0 for some Replacing p with ¢;

(z € C)

have ¢ = 0. Now |ao|, . . . , |an—s| will be arbi. . . , |bus| are all arbitrarily small. Therefore j > k — 1. Thus ¢, has degree at least ¥ — 1. and repeating the construction, we find

p(2) = (z — 21)(z2 — 22)q2(2)

COMPLEX

ANALYSIS

‘

141

where ¢; has degree at least £ — 2. By induction we finally construet the desired numbers z;, . . . , 2z and the desired polynomial ¢ = ¢,.

6. THE

RIEMANN

MAPPING

THEOREM

In this section we prove the beautiful theorem of Riemann that under very general conditions any two simply connected open sets in C have the same analytic structure. The first step is to derive some integral formulas which are important in their own right.

Proposition 6 eachz

=zo+

Let the function f be differentiable in Sc(zo,R). Then for re® (0 ER,0

< r < R) we have

(R* — r?) do

©1) §@) = @07 [ feo + Re) g

i

Proof It is enough to consider the case zo = 0 and R = 1. Let v be the circle of radius 1 about the point 0. By Cauchy’s integral formula,

(6.2)

f@) = @riyt [ f§)¢ — 27t dg = @m)t [ f@)E6 — 7M@) de

and by Cauchy’s integral theorem,

(6.3)

0 = (2ri)~t [ f@)2*(1 — 2%)7 dt = @m) [ f@)2*(* — 2T @) T i

Addition of (6.2) and (6.3) gives

(1 — z2*) f(z) = (2mr)1 /7 f(¢) {W

} (i) de

Writing ¢ = e* and z = re®, we get

(o1 [ 4

_ f(z) =

(2r)

/0

1 —r)de f(e®) (1 — 2rcos (0 — ¢) + r?

which is just (6.1) for the case zo = 0 and R = 1. A consequence of Proposition 6 is a bound for an analytic function inside a circle.

142

FOUNDATIONS

Proposition 7

OF

CONSTRUCTIVE

ANALYSIS

If f is differentiable in S = Sc(z0,R), and if |2 — 20| =

r < R, then

(6.4)

5@ < 176 3R-I- +

R

2r

”f”s

Proof There is no loss of generality in taking Write z in the form z = re¥, and abbreviate L

1 — r?

2z = 0 and R = 1.

1=

A=1—-2rcos(0-—

Un

Uy be a compact set. By the Corollary to Proposition 7, there

exists ¢ < 1 such that |po"(2)| < ¢ for all n in Z+ and all z in L. Using the Corollary to Proposition 8, from the fact that d,— we see that, for m < n,

1 as n—

loa"(2) — 0™ (2)| = |en™(00™(2)) — eo™(2)] will be arbitrarily small for all z in L if m and n are sufficiently large. It follows that {¢o"} converges uniformly on L, to a continuous function from L to Sc(0,c). Since L is an arbitrary compact set with L CC U, the sequence {po*} converges on U, to a differentiable

funection ¢: Uy — 8(0,1), such that (L) CC S(0,1) for each compact set

L

CC

Uo.

To construct the inverse ¢,: S(0,1) = U, of ¢, consider compact sets K and L with K CC int L and L CC 8(0,1). For each m < n, let o™ U,— U, be the inverse to ¢n." Since d, — 1 as n — «, the functions ¢,° are defined on L for all n sufficiently large. By the Corollary to Proposition 8, ¢," converges uniformly on K as m — « and n — o to the identity function z — 2. Therefore we may choose the positive integer N so large that if m, n > N then ¢,° and ¢,° are defined on L and ¢,"(K) C L. For all zin K and all m, n > N we have lom®(2)

—

on’(2)|

=

lfiono(fiomn(z))

—

n(2)|

Hence the sequence {¢,°} converges uniformly on K. Since ¢x° is an equivalence, ox°(L) CC U,. Therefore

on’(K) = on%(en¥(K)) C on®(L) CC

Uy

(n > N)

COMPLEX

ANALYSIS

149

Thus there exists a differentiable map ¢.: S(0,1) — U, such that for each compact set K CC S(0,1) the sequence {¢,°} is defined on K whenever n is sufficiently large and converges uniformly on K to ¢,,. Moreover ¢, (K) CC U.,. To show that ¢: Uy — S(0,1) is an analytic equivalence, it remains to show that ¢,0 ¢: Uy— Uy and ¢o ¢.:8(0,1) — S(0,1) are the identity maps. For each zin U, the points ¢o"(2) lie inside some compact set M CC S(0,1) [since the sequence {¢¢"(2)} converges to a point in

S(0,1)], and thus for each ¢ > 0 we can choose N in Z+ with

l0:% (00" (2)) — ¢ule(@))| < e

(k,n &€ Z* k > N)

Taking n = k gives

2 — 0u(p®(2))| < e Letting k —

(k> N)

o« gives

2 — ¢u(e(2))] < e Since e is arbitrary, ¢, o ¢ 1s the identity map. Similarly, ¢ o ¢, is the identity map.

Corollary

Every mappable open set U s equivalent to S(0,1).

Proof The corollary is immediate because every mappable equivalent to a sequestered set containing the point 0.

set is

PROBLEMS

1. Is homotopy an equivalence relation on the set of all closed paths lying in a given compact set K? 2. Show that a closed path in an open set U i1s homotopic to a closed differentiable path in U. 3.

Show that the convex set spanned ., 2, need not be closed.

by

finitely many

points

z;,

4. Prove that homotopic closed paths in a compact set K have the same winding number with respect to each point of —K. 5. such

Let f and g be complex-valued functions on an open set U C C, that

for every

compact

such that (2.1) holds. Show therefore f' = g).

set K CC

U

there

exists

that f and g are continuous

§: Rt—

R+

(and that

150

6.

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

Let @ be the set of all analytic functions in U = {z: |2| < 1} which

are bounded by 1, and let z;, . . . , 2, be points in U. Show that the set A = {(f(z1), . . ., f(zn):f € G} C Cnis totally bounded. 7. Let0 < r; 2n2,

Assume that

2 fi(x) > g(x) — € for some positive integer m. Then i=1

Y fdn > (g — €\, for all sufficiently large n. This gives j=1 0

/(g~6)>\ndu—

Z /ffi\nduSO =1

162

FOUNDATIONS

This contradiction shows that

OF

CONSTRUCTIVE

ANALYSIS

2 filx) < g(x) — € for all m in Z+, as i=1

was to be proved.

Our next theorem is the final step to the goal of assigning a measure to every integrable set.

Theorem 1 Let the test functions f and g approximate the complemented set A lo within €; and e, respectwely. Then JIf —glde

(2.7)

Proof

< &1+

e

Let {f;} and {g;} be error sequences for f and g, respectively.

Assume that

i {/fjdu-l-/gjdu} 0 and z in X such that

5, 5@ + 0] +25 < 1f0) - 9(a)

(2.9) 1-—f@ + 9@ + J

13

for all m in Z+. Now either |f(zx) — g(x)| < f(x) — g(x) + & or |f(x) — g(x)| < g(x) — f(x) + 6. Without loss of generality, we may assume that the first alternative holds. Then (2.8) gives

(i) +gi@)} 0}, {z: p(z,K) < 0}) = (—K,K) It will be clear from the context whether K is being thought of as a compact set or as a Borel set. We next show that for a positive measure on R there are many integrable compact sets.

3. MEASURES

ON

R

The structure of a positive measure x on R is revealed by the following result.

Theorem 3 Let u be a positive measure on R. Then all except countably many points x of R are steady, in the sense that for each ¢ > 0 there exist f in C(R) and 6 > 0 such that 0 < f u. Then S is totally bounded. To see this, consider any ¢ > 0. Let uo = a0 < u1 < ++ + < un = b be points U — U1 0, the sets Xa

{x:g(x) > a}

and

X. =

{z:6(x) < a}

are locally compact and the sets X® and — X, are integrable and have the same measure. M oreover,

(3.1)

Jgdu > au(X®)

and

(3.2)

Jg du < lgllu(X®) + [go du

where go 18 any nonnegative test function with g(x) < go(x) for all x in — Xe. Proof Consider any ¢ > 0. Let h: X — R be a proper map, with h(xz) = g(x) whenever g(x) > ¢, and h(x) < ¢ whenever g(z) < ¢. Then for all @ > ¢ we have Xa

{z:h(x) 2 a}

It follows from Theorem 4 that for all except countably many a > ¢

MEASURE

171

(and hence for all except countably the set X, is locally compact, and for each admissible a and for each any fin C(X) with0 < f 0) the set X2 is compact, the set X¢ is integrable. Moreover, ¢ > 0 there exists § > 0 such that = 1forg(z) > a-+ 6, andf(zx) =0 to within ¢, for each admissible b

C X° with [a — b| < 8. If a < b < a + 6§, we have X* C —X,

Any

function f that satisfies the above conditions therefore approximates — X, to within . Hence — X, is integrable and has the same measure as Xe.

To prove (3.1), note that for each ¢ > 0 there exists f as described, with g > af.

To prove (3.2), note that ¢ — go < 0 on —X° on X

and g — go < ||g]|

For each ¢ > 0 choose a function f as described, with f(z) = 1

for all z in X2 Then g — go < ||g||f. This gives (3.2). Another consequence of Theorem 4 is that every positive measure on R comes from a monotone function. Theorem 5 Let i be a posttive measure on R. Then there exists a monotone-nondecreasing functton o: S — R defined on the set S of steady points of u such that u = p,.

Proof Let xobe any steady point of u. Let x be any steady point with write alr) = u([zo,z]), and if z < xy, write x # xo. If x>z, a(x) = —u([z,x0]). Then « is monotone-nondecreasing, and a(z) will be arbitrarily near to 0 if « is sufficiently near to xo. Thus a can be uniquely extended to a monotone-nondecreasing function on S, with < y we have a(xy) = 0. For arbitrary points x and y in S with a(y) — a(x) = u(z,y]). We must show that u = u,, or that

[fduw = [f(z) da(2) for all test functions f. Let I = [a,b] be a proper compact interval which supports f, such thata € 8,b € §, and f(x) vanishes for all sufficiently near to a. Let w be the modulus of continuity of f on I. Let € be any positive constant. Choose a partition P = {a¢ = a, a1, . . . , an = b} of I, composed of points of S, of mesh less than w(e), such that f(a;) = 0 and

(3.3)

with

|[f(@) da(z) — S(f,P)| < e

SU,P) = 3 fla) (ala) — a(ay) 1=1

172

Choose

FOUNDATIONS

test

functions

fi, . . .., f»

OF

CONSTRUCTIVE

that

such

(@)

ANALYSIS

0 < f; a;| — |¢ if 0 = fulz) (¢) I, Y fi@) =1 for all z in

i=1

i (k+ 1n4 and |f(x) — f(y)| < (2n)¢z — y| for all x and y. Each of the sets S»* is a compact subset of C(I), by Theorem 6 of Chap. 4. Therefore the supremum

a.* = sup {[fdu:f € Si*} of the continuous function f — [f du on S,* exists. For each n partition

OF

FOUNDATIONS

174

CONSTRUCTIVE

ANALYSIS

the set {k: 0 < k < n* — 1} into disjoint subsets U, and V,, such that a,* > n—? whenever k € U, and a,* < 2n—2 whenever £k & V,. For each k£ in U, choose f,* in S,* with ffnk

Define

d}i

fa=

>

n—2

Y fa ke Un

Then ||f.]] £ 1, and 1 2> [fady > apn™? where o, is the cardinality of U,. Therefore a, < n?. Let A, be the union of the intervals [kn—% (k + 1)n—¢] for k in U,, and the intervals Then the Lebesgue for 0 N, we

I = [k(um)n* + n7°, (k(u,n) + Dn™* — n~°] for which a,*®™ < 2n=2 Consider points v and v in I — A, max {N(u),N(v)} let g. be the continuous 1 at each point of [u,], which vanishes n~8 ulv + n=8 o), and which is linear

with u < v. For each n > function which has the value at each point of (— o, u — on the intervals [u — n~8, u]

and [v, v + n~¢]. Clearly |g.(y) — g.(2)] < nbly — 2| for all y and 2. Therefore ¢, — gn41 i1s the sum of functions ¢’ in S,*®™ and ¢"” In Sk

Therefore lf(gn

—

gn+1)

dul

S

ank(“,n)

+

ank(v,n)


0. Thereforef = f’ a.e. Corresponding to each notion of Cauchyness.

of the above

notions

of convergence

is a

Definition 8 Let {f,} be a sequence of measurable functions. Define the classes I'(K,e) as in Definition 7, but without reference to the function f. The sequence {f.} is Cauchy almost uniformly if to each integrable set K and each ¢ > 0 there exists 4 in I'(K,e) such that for each § > 0 there exists N in Z+ such that |fn(x) — f.(z)| < é for all zin A and m, n > N. The sequence {f,} is Cauchy almost everywhere if to each integrable set K and each e > 0 there exists A in I'(K,e) and N in Z* such

that |fm(x) — fau(x)| < eforall xin A and all m, n > N. The sequence {f.} is Cauchy in measure if to each integrable set K and each ¢ > 0 there exists N in Z* such that for all m, n > N there exists A in I'(K ,e) with [fm(z) — fa(x)| < eforall z in A. It is obvious that a sequence {f.} of measurable functions which converges in one of the three possible senses is Cauchy in the corresponding sense. It is also obvious that a sequence which is Cauchy almost uniformly 1s Cauchy almost everywhere, and that a sequence which is Cauchy almost everywhere is Cauchy in measure. To see that a sequence which is Cauchy in any of the three senses converges in the corresponding sense, we need a lemma. Lemma 4 Let the sequence { A} of measurable sets have the property that w(K) = lim u(K N A,) for every integrable set K. Let the sequence {fn} n—

0

of measurable functions converge uniformly to a function f: U

A,—

R

n=1

on each An. Then

Proof

{f.} converges almost uniformly to f.

Consider any integrable set K and any e¢ > 0. There exists a

value of n for which u(K — K n A,) < ¢ by hypothesis. Since {fx} converges uniformly to f on K N A,, it follows that {f.} converges almost uniformly to f.

Theorem

4

A sequence

{f,} which is Cauchy almost everywhere con-

verges almost uniformly to some measurable function f. In addition {f.(x)}

converges to f(x) at each point x of some full set B. A sequence {f.} which 1s Cauchy tn measure converges in measure to some measurable function [, and some subsequence of {f.} converges almost uniformly to f.

INTEGRATION

197

Proof Let the sequence {f.} of measurable functions be Cauchy almost everywhere. Let the sequence {S.} have the property (a) of Definition 1. For arbitrary positive integers & and n choose B,* in '(S:,27%n~1) and N,* in Z* such that |fi(x) — fi(x)| < 2% whenever

u(Sn

r & B,*

— 4,)


N,*.

n7,

and

{f.}

Then

converges

A, =

M B,* is integrable, k=1 unifor—mly on each of the

sets 4, to a function f: U

A, — R. By Lemma 4, we see that {f.}

converges almost umformly

to f. It also converges pointwise to f on

the full set B = U

A..

Consider next a sequence {f} of measurable functions which is Cauchy in measure. Let {S.} be the sequence of (a) of Definition 1.

We define (by induction on m) for each m in Z+ a subsequence {g,™}*_, of {f.} such that {g,m*t'}=_, is a subsequence of {g.™} and such that for each m there exists an integrable set A,, C S,, with u(S, — An) < m~!, such that {g.m}2_, converges uniformly on A,. We prepare the 1nduct10n by taking g,” = f, for m = 0. Assume then that {g,”1}>_, has been defined for a given value of m > 1. For each k in Z* choose N(k) in Z* so that for each 7, j > N(k) there exists Bi(z,7) in I'(Sm,27*m~1) such that |g:i""1(x) — g, (x)| < 27* whenever x & Bi(¢,7).

We may assume that N(1) < N(2) < - - - . Then 4, = r"\ Bu(N (), N(k + 1)) is integrable, u(S,, — A») < m~1, and the subsequence {g.m}

= {gnsl}2, of {g.™1} converges uniformly on A,. This completes the 1nduct10n The sequence {g,"}>_, is, except for finitely many terms, a subsequence of each of the sequences {g.”}2_,. It therefore converges uniformly

on each of the sets 4,

to a function f: \U 4,,— R. By m=1

Lemma 4, we see that {g,”} converges almost uniformly to f. Since {g."} 1s a subsequence of {f.}, and the latter sequence is Cauchy in measure,

{fn] converges to f in measure. As a corollary to Theorem 4, we see that almost-uniform convergence and convergence almost everywhere are equivalent. We come now to some results concerning interchange of integrals and limits. The first is Lebesgue’s famous monotone-convergence theorem. Theorem

5

Let fy < fo
0. Choose the positive integer N so large that

L— [frdp 7 > N. Since f; — f; 1s measurable, there exists an integrable set A C K with u(K — A) < ¢/2 and a simple function x such that

0 < x < xalfi — 17 SxA 0 there exists

6 > 0 such that l /A f du‘ < € whenever A 1s integrable and u(4)

< é.

Proof It is enough to consider the case f > 0. Choose n in Z* so that [(f — f*) du < ¢/2, where f* = min {f,n}. Write § = (2n)le. Then if A is integrable and u(4) < 4, we have

c = s n + + 5 < u u d r f d [ m = G [ s n a J [ The next two results shed additional light on the interchange of integrals and limits. The first gives a good condition for the convergence of an integral to 0. The second is Lebesgue’s famous dominatedconvergence theorem.

Proposition 9

Let the sequence {f.} of nonnegative integrable functions

converge wn measure to 0. Then

lim [f, du = 0 if and only if for each n—r

0

€ > 0 there exists an integrable set K and N tn Z* such that /A fadu < €

Jor all n > N and all measurable sets A with u(A n K) < N~

Proof

The condition is clearly necessary. To show that it is sufficient,

for each € > 0 choose K and N as described. Take no > N in Z* so that

for each n > n, there exists an integrable set B C K with u(K — B) < N—!and |f.| < eg(K)! a.e. on B. Then

/fndu=/_3fndu+/3fnduSe+e whenever n > n,. Therefore [f, du— 0 asn—

= 2¢

.

Theorem 6 Let the sequence {f,} of integrable funclions converge in measure to the measurable function f. Let there exist an integrable function

f is integrable, and [ fn du — g such that |f.| < g a.e. for all n in Z+. Then [fduasn— Proof

.

Since |f| < g a.e., we see that f is integrable. To show that

Jfandp— [fdu as n—

oo, it is enough to show that [|fn — f|du— 0

INTEGRATION

as n—

.

201

By Proposition 9, it is enough

to find for each ¢ > 0 an

integrable set K and N in Z* such that /A |f — fa| du < € whenever

n > N N =

and

u(4dnK)

< N-L

Since

|f — f.] < 2¢ a.e., we

1 and conclude the existence of K from Lemma

may

set

5.

Our next lemma prepares the way for a study of the sets on which the value of a measurable function is less than a given real number. Lemma 6 If f: X — R s measurable, and tf ¢: R — R is continuous, then ¢ o f 1s measurable.

Proof

Consider

any

integrable

set K

and

any

e¢ > 0.

Choose

an

integrable set B C K with u(K — B) < ¢/2 so that |f| is bounded on B by some constant ¢ > 0. Let w be the modulus of continuity of ¢ on the interval [ —c,c]. Choose an integrable set A C B with u(B — A) < ¢/2 and a simple function x with |x| < ¢ a.e. and |f — x| < w(e) a.e. on A. Then u(K — A) < e and x4(¢ © x) is a simple function, with loof — xaleox)|

< ea.e.on A

Thus ¢ o f 1s measurable.

It 1s very convenient to be able to speak of the set A on which the value of a measurable function is less than a given real number, and to know that A is measurable. The theorem which asserts that this is actually the case bears a strong resemblance to Theorem 4 of Chap. 6. Theorem 7 Let f: X — R be measurable. Then for all except countably many o 1n R there exists a measurable set A such that f < « a.e. on A and f 2 aae on —A. Proof It 1s enough to prove the assertion for all except countably many real numbers « in each interval of the form I = [—a,a], with

a > 0. For this purpose we may replace f with

f* = max {min {f,a}, —a} so that |f|] < a a.e. Assume first that X, € 9, fore integrable. Choose a sequence

and that f is there-

.anz...ahe.

of simple functions converging to f almost uniformly. Write x* in the

202

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

form xn

where

x1,

—

Clnxln

_l_

o

o

e

_l_

CNann

. . . , xn™ are the characteristic functions

of disjoint in-

tegrable sets A1*, . . . , Ax™, respectively, whose union is a full set. (Of course, N depends on n.) Let » be the positive measure on I defined

by (h& C())

= [hofdu Jhdv

All except countably many « in I are steady relative to the measure v, and satisfy the inequalities a 52 ¢,* for all n and 7. Consider such a

value of a. Since o #

¢

for all 7, there exists an integrable set A,

such that x» < a a.e. on A, and x* > « a.e. on —A4,. Since x* > x"*!

a.e., An — Anq11s a null set for each n. We shall show that

(3.1)

im u(4.) = v(la) Nn—r

where I, = {t € steady, for each h in C(I), with Thdy — v(la)
0 there exists a monotone-nonincreasing function 0 < h [hdy — e = [(ho f)du —e> [(hox") du — ¢ (n € Z%) S fxa, du — e = u(d,) — ¢

Therefore

(n in Z+)

w(d,) < v(la)

(3.2)

Notice next that for each ¢ > 0 there exists hin C(I) with0 < h «, so that v(I,) — [hdv < e. Now {ho x"} converges to h o f almost uniformly. By Theorem 5, we have [(h o x*) du —

J(hof)du

as n—

«.

Therefore

we

can

choose

N

so large

that

Jhox™) du > [(hof)du — efor alln > N. Then (3.3)

du +)2e v(Io) L fhdv+e=f(hof)du+ e < [(hox™ (n 2 N) < [xa,du + 2¢ = p(A,) + 2¢

Inequalities (3.2) and (3.3) imply (3.1). Theset A = U

A, isintegrable

by (3.1) and the fact that A, — A,y is a null set for each n. Clearly f aa.e on —A.

INTEGRATION

203

Next consider the general case. Let the sequence {S.} of integrable sets have property (a) of Definition 1. Write B; = S; and BnESn—'Sl—S2_

e

—Sn—l

for n > 2. By the case just considered, for each n there exists an integrable subset A, of B, such that f < « a.e. on 4, and f > « a.e. on

B, — A.. Therefore the measurable set A =

\U A, has the property n=1

that f < ¢ a.e.on A andf > a a.e. on —A.

Corollary

A

mnonnegalive

integrable function f whose

integral

is 0

vanishes almost everywhere.

Proof There exist arbitrarily small positive constants a such that a measurable set A exists with f < a a.e.on 4 and f > « a.e. on —A4. Thus f > a x_4 a.e. It follows that —A is a null set, and thusf < « a.e. Therefore f = 0 a.e.

4. THE L, SPACES Our next goal is to introduce certain basic metric spaces of measurable functions. This requires some preliminary inequalities.

Lemma a+

7

B =1.

Lel a, b, a, 8 be real numbers with a, b > 0; a«, 8 > 0; and Then

(4.1)

aa + b

Proof Consider aa + b of b, @, and 8. Then

— a*b? > 0

— a2b? as a function f of a for fixed values

fl(@) = a — aa® b

= a(l — (ba—1)f)

if a % 0. Thus f'(a) > 0if a > b, and f'(a) f(b) = 0, it follows that f(a) > 0 whenever ¢ whenever a > 0. By continuity, (4.1) is valid Successive applications of Lemma 7 show

< 0if 0 < a < b. Since > 0. Thus (4.1) is valid for a > 0. that if oy, . . . , an are

positive constants whose sum is 1 and if @y, . . . , @, > 0, then a1a1+

et

+anan'—'a1al'

’

'ana">0

204

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

Theorem 8 Let fi, . . ., f. be nonnegatie integrable functions and ai, . . . , oy positive constants whose sum 18 1. Then fi* - « -+ f%n 18 integrable, and

(4.2)

[fro - o fammdp < (ffrdw) - - - ([fadp)e

Proof Let ¢, . .., 1t < n). By (4.1), f(cl_lfl)al

R

¢, be positive numbers with ¢; > [fidu

(Cn—lfn)a"

d/‘

el

+

S

f(alcl—lfl

=

alcl—lffl du +

-« + -

Sart - ton=1 fflal

or

e

fna,, d”’

S

C1™

*

*

+

¢

ancn—lfn)

+

(1
0, Consider any and f; — f2 = f. Write

1(f) = 2(f1) — 2(f2) The number 4(f) does not depend on the choice of f; and f, since for a

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choice f; and f; we

THEORY

217

have fi — f;, = f = fi — f2,

and

thus

f1 + f: = f1 + f3, and consequently 9(f]) + n(f2) = n(f1) + 2(f3). Thus 7 is a linear functional from C(X) to R, with (f)

> 0 whenever

f 2> 0. In other words, 7 is a positive measure u; on X. Write us = u; — p. Then us is a measure, since u and u; are measures. If f € C(X) and f > 0, then ffdflz=ffdfll—ffdfl=fl(f)"‘ffd#?_o

Thus u. is also a positive measure. It remains to show that u; and u, are mutually singular. Consider any °

f 2 0in C(X), and any € > 0. Choose g in C(X) with 0 < ¢g < f and

Jfduw =9(f) < [gdu + e We have

9) dm d—u + [(f ,dpr = fgdus — [g u—9) [(f Jg+d [gdur< e = [f—dp Therefore g1 and w2 are mutually singular.

The problem solved by Theorem 1 is related to the problem of representing a measure as an integral with respect to another measure. Theorem 2 Let u and v be measures on a locally compact space X, with u positive. Then there exists a measurable function h: X — R such

that (1) fh &€ Li(n) for all f in C(X)

and (1) v = hu, in the sense that

[fdv = [fhdu for allf in C(X), if and only if (a) There exists a dense subset S of R such that for each t in S the measure v — tu s the difference of two mutually singular positive measures, and (b) For each f > 01in C(X) and each ¢ > 0 there exists ¢ > 0 such that

fgdv| < e+ clgdn whenever ¢ & C(X) and 0 < g < f. Let f be any nonProof Assume first that the function h exists. negative test function and e any positive constant. Choose ¢ > 0 so that h = hy + hs, where h;, hy are measurable functions with |h;| < ¢ and

[flhs| dp < e. Then whenever ¢ € C(X) and 0 < g < f we have

€ + u gd cf < du s| lh [g + du i| lh [g fgdv| = |fghdul < Thus condition (b) 1s satisfied..

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To see that condition (a) is satisfied, consider any ¢ in R. Then v —tu = (h — t)u. Let f be a test function, with 0 < f < 1. Let the compact u-integrable set K be a support for f. By Theorem 7 of Chap. 7, for each ¢ > 0 there exists an integrable subset A of K such that h—t>0onAandh

—¢

< eon K

— A. Choo a test function seg to

approximate A so closely that

[ It =0 du < [, Joh — ) du + ¢ and

|[

Joth =D du|
¢ and 8 < —ec. Our first task will be to approximate

» by a measure » with Bu < »' < au. Since a € §,

we have y —

oM

=

V] —

Ve

where »; and v, are mutually singular positive measures. For each e with 0 < ¢ < k=1 there exists g in C(X) with 0 < ¢ « and ¢;

(2.4)

The class ¢ is said to satisfy the property that for each « exists a finite subset ¢; of ¢ C 8, such that

u(\J I) > pu(\J I) Iet

IEso

The most important case in which condition (a) is satisfied is obtained by taking X = [0,1], letting u be Lebesgue measure, and taking ¢ to consist of all proper subintervals, or of all proper subintervals whose end points lie in a given dense set S C [0,1], with 0 & Sand 1 & S. In fact, condition (a) is satisfied for all p < 1. To prove this, it is enough to show that corresponding to every finite subset ¢y of ¢ there exists a disjoint subset {; of ¢ satisfying (2.4). It is sufficient to consider the case in which all the end points of the intervals in ¢, are distinct. Because of this condition, there exists a minimal collection I,, . .., I, of intervals in {, whose union equals almost everywhere the union of ¢. We may assume these intervals to be ordered according to the values of their left end points. Let ¢; be the set of those I, (1 < k < n) with

odd index k, and ¢;’ the set of those with even index. Each of these sets 1s disjoint. Therefore one of them will serve for ¢;. This example generalizes to higher dimensions. Take X = [0,1]" and

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227

let u be Lebesgue measure. Let ¢ consist of all “rectangles’

I = [aybi] X

+ ¢ - X [@n,ba]

where 0 < a; < b; < 1 for each 7, where we restrict the a; and the b; to some dense S C [0,1], with 0 € S and 1 & S, and where !

min {by — @1, . . . ,b—p @}

> 7m{by ax —ay,

. .. ,b—,an}

for some fixed 7 > 0. Then ¢ can be shown to satisfy condition (a), with some constant p > 0 depending on 7 and n. Suppose v: ¢ — R 1s finitely additive. This means that whenever Iy, . . . ,I,aredisjoint elements of { such that I, v - - - U I, is equal almost everywhere to some element I of ¢, then

v(I) = Zv(l)) For each « in ¢ we define the integrable function f, on X by

fa(®) = v(Dud)™

(€€

)

It is clear that f, < fs whenever a < B. We are interested in convergence properties of the family {f,}. Here 1s the basic estimate. Proposition 1

Suppose that ¢ satisfies condition (a), and that the map v:

¢ — R 1s findtely additive. For each o in T and each € > 0 let {.(€) consist

of all I in ¢, with |[J(I,a),I]| > €, where

[/ (1,a),1]

v(J(L,a))u(J (L,2))™ — v(Du()™?

Let K and € be positive constants. Then there exists a positive constant o, depending only on K, €, and p, such that for each « in T having the property

that ||fslx < K and ||fsllx — ||fallh < 6 whenever 8 ET and a < B, we have u(A) < e for all finite untons A of elements of ¢a(€). Proof We imitate the proof of Theorem 3. Take the positive constant r so large that KA(r)~! < 1pe. Write 6 = 1e30(r)p. Consider an integrable set A of the form described. By condition (a), there exists B in T, with & < 8, and an integrable set B C {z: |fe(x) — fs(x)| > €} with u(B) > pu(A4). There is no loss of generality in taking the sets B, =-{z:|fa(x)| > 7} and B, = {z:|fs(x)| > r} to be integrable. Since

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[fellx £ K and [|fslls < K, we get u(Bi) < 4pe and p(B2) < 3pe, as before. We compute

pu(d) Su®B) 0, let 6 be given by Proposition 1, and choose « as described in Proposition 1. Each finite union A4 of elements of {.(¢) has measure at most e. By omniscience again, it follows that there exists an integrable set B, with u(B) < ¢ such that I C B a.e. for all I in {.(e). This implies that f has a derivative almost everywhere. Constructively, things are not so simple. In fact there are two difficulties with the above proof. First, the least upper bound K may not exist. In general there is no possibility of proving constructively that K exists. We therefore assume that it exists. Second, the integrable set B may not exist. In case f is monotone-nondecreasing (or equivalently, v 1s nonnegative), this second difficulty can be overcome, by means

of the following lemma. Lemma 5 Let f be a monotone-nondecreasing function defined on a dense set S C [0,1], with 0 &€ S and 1 & 8. Let ¢ consist of all proper intervals with end points in S. Then for given constants K > 0 and e > 0 there exists 6 > 0, depending only on K and €, such that for all a in T

having the property that ||fs|h < K and ||fsllh — [|fallh < 8 whenever B 2 a,and all v > o, there exists an integrable set B, with u(B) < €, such that for each I in ¢q(€) either I C B or I € ¢,(e/4).

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229

Proof By a slight modification of the proof of Proposition 1, we can find 6 > 0 so small that for all « as described and all finite subsets ¢ of ta(e/5) we have u(\J I') < ¢/2, where I’ is any interval containing Iet,

I whose length is less than twice the length of /. Consider « as described, and consider any ¥ > «. Let By be a finite union of open intervals that contains the end points of each of the intervals in v, with u(B,) < ¢/2. Choose ¢t > 0 so small that (i) u(I) 2> t/2 for all I in v and (ii) for each Iin ¢, with u(J) < ¢, either I C Boor I € ¢,. Take finitely many intervals I, . . . , I,, I3, . . ., I} from ¢, with I C I, and u(I}) < 2u(l;) for 1 < k < n, such that for each I in {a, with u(I) > t/2, there exists £k (1 < k < n) for which

— p(I)1}
€/2 or (i) [J(I,a),I] < —e/2. If (1) holds, then

WL By

@) = sTJuD) 2 )11 >

(2.5), it follows that [J(I,a),I:] > ¢/4. Hence I] € B;. Similarly,

if (ii) holds, then I;, € B;. Thus I C I, C B; C B whenever I & {.(e/2)

and u(I) > t/2. Now consider either ] C BorI the above proof struction of By, alternative I &

any I in {,(¢). To prove Lemma 5, we must show € ¢.,(e/4). Either u(I) > t/2 or u(I) < t. If u(I) = gives I C B. Consider the case u(l) < t. By the either I C By or I € ¢,. We need only consider ¢,. Since

that t/2, conthe

e < |[J(1,a),I]| £ [[JU,a),JTN]| + [[JU7),I]

either

7 (I,v),1]| > %

or

|[J,),JUI] > %

In the former case, I € ¢,(¢/4). In the latter case, J(I,y) € {a(e/2).

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Recall that u(J) = t/2 for all J in v, by the choice of {. In particular, p(J(I,v)) = t/2. By the above, it follows that J(I,y) C B. Hence I C B. Here is our substitute for Lebesgue’s result that a monotone function has a derivative almost everywhere. Theorem 4 Take X = [0,1], let u be Lebesgue measure, let S be dense wn [0,1], with 0 € S and 1 & S, and let ¢ be all proper intervals with end points in S. Let v be a nonnegative finitely additive function on ¢. Let the

set {||felr: @« E T} be bounded.

Let oy < az
0 there exists n in Z*

such that ||fallh — ||falln < € for all « > an. Then for each € > 0 there exists n 1n Z+ and an integrable set B, with u(B) < ¢, such that I C B for all I in q,(e). Proof Using Lemma 5, we construct by induction a subsequence - -0of ag «, in the sense of the following definition, at most once.

Definition 4 A (finite or infinite) sequence a;, az, . . . of real numbers upcrosses from a real number « to a real number 8 > « at most n times if N < n whenever N is a positive integer such that there exists a subsequence by, by, . . . , byy such that b, < aforet=1, 3, ..., 2N — land b; > Bforz =2,4, ... ,2N. The least such n (alternatively, the greatest such V), if it exists, is called the number of upcrossings of the sequence from « to 8. It is clear that a sequence {a,} of real numbers converges classically if and only if it is bounded, and for all real numbers « and g with a < B3, upcrosses from « to 8 at most a certain finite number N («,8) of times. Constructively, however, convergence is much the stronger property. Thus we have found a possible constructive substitute for the convergence of a sequence of real numbers. In the light of these remarks, the following developments will be seen to constitute a constructive substitute for the Chacon-Ornstein ergodic theorem. We first prove a combinatorial lemma.

Lemma6 Let a(—1), a(0), . .., a(n) and b(—1), b(0), . . . . b(n) be real numbers. Let N be any posiiive integer such that there exist integers

(3.1)

—1fum 0 with afjv|| < 1 we have av € S, and therefore alA(v)| = [N(av)| < c. It follows that [\(v)| < c||v||. Putting v = v, — v, gives

A1)

— N@2)| = A1

— v2)| < cfjor — ve

for all vectors v; and v,. Hence X is uniformly continuous. We have thus proved the following proposition.

Proposition 2 The following are equivalent functional N\ on a normed linear space V : (@) X\ s continuous (b) X\ s uniformly continuous

conditions for a linear

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X\ s bounded on S N ts bounded on each bounded subset of V There exists ¢ > 0 such that

Aol 0 such that (i) |lu — w|| > ¢ for all w in L, and (ii) every vector v can be written in the formv = au + w, witha € Fand w & L. We have the following result.

Proposition 3 If \ 1s a nonzero bounded linear functional on a normed linear space V, then N (\) is a hyperplane. Conversely, if L ts a hyperplane with assoctated vector u, there exists a unique bounded linear functional X with N(\) = L and \N(u) = 1. Proof Let N\ be a nonzero bounded linear functional, and u any vector with A(u) = 1. By Proposition 2, there exists ¢ > 0 such that

cA@)| < ||v]| for all vectors v. If w is any vector in N(A), this gives

le — wl 2 efMu —w)| = ¢ For each vector » we have v = au + w, where a = A\(v) and w =v — Av)u; clearly w &€ N(N). Consider, conversely, a hyperplane L with associated vector u satisfying (i) and (ii). The representation » = au + w is unique, since if v = a;u + w; is another representation of the same vector v, we have (@ — a1)u = w; — w, and thus a = a; and w = w; by (i). Define the linear functional A on V by A(v») = a. Clearly AM(v) = 1 and L = N()). For each vector v for which A(») = a > 0 we have, by (i),

loll = lal lu — (—aw)|| = lalc = A@)|e or |A(v)| < ¢7!|v||. Thus \ is bounded. The most useful normed linear spaces are those which are both separable and complete. Such a space is called a Banach space.

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Recall that a metric space is separable if it has a countable dense set. There are no important constructively defined normed linear spaces that are not separable. Thus, as far as existing mathematics is concerned, there is no loss of generality in postulating separability, and there is a great gain in power and convenience. The fact that every metric space can be completed leads us to hope that the same is true of every normed linear space V. This turns out to be the case, and the completion of V as a normed linear space can be identified with its completion as a metric space. Here is an outline of the construction. We take V to consist of all Cauchy sequences from V, with termwise addition and termwise scalar multiplication. Two elements {z,} and {y.} of V are equal if lim ||z, — .|| = 0. Clearly ¥ is

a linear space. Under the norm

I {za}ll = tim o] it is a complete normed linear space, called the completion of V. The inclusion map ¢: V. — V defined by ¢(v) = {v.}, where v, = v for each n, preserves norms and realizes V as a dense linear subset of V. We have thus proved the following result. Proposition 4 To each normed linear space V is associated a complete normed linear space V, and a norm-preserving linear inclusion map % from V into V. Moreover, V is dense in V.

Remarks (a) In case V is separable, V is also separable and therefore a Banach space. (b) The bounded linear functionals on V are in one-one correspondence with the bounded linear functionals on V: If A is any bounded

linear functional on V and {z.} any pointin ¥, then {A(z,)} is a Cauchy sequence in F whose limit A\({z.}) = lim \(z.) defines an extension of Ato V.

nw

The simplest instances of Banach F*, with norm

spaces

[(ay,

- . . ,a)]

=

(21

are the euclidean spaces

Iai|2)1/2

These spaces are finite-dimensional, in the following sense.

Definition 4 A normed linear space V is finite-dimensional if it has a finite basis, that is, if there exist nonzero vectors vy, . . . , v and

248

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bounded linear functionals Ay, . . . , A\, such that

@ v= YN0

@EV)

1=1

and

0)

N()

=0

1 m~! and n < N, h(n) = —1 whenever a, < —m™ and n < N, and h(n) = 0 whenever n > N. Then

N

AR)| =

z

a.h(n) ‘ 0 in L,(u) and each ¢ > 0 there exists an integrable set A and a positive integer N, with |f| < N a.e. on A, such that lf — xafll» < e. Moreover,

g < xaf < g+

there exists a simple function g with 0
0 there exists n in Z+ with

flxa — xaldp = u(4d — 4;) + p(4n — 4) < We then have the following result.

Proposition 6 If (X,F,5,9,u) is a separable measure space, then L,(u) 18 a Banach space for each p > 1.

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Proof We must show that L,(u) is separable. It is enough to consider the case F = R. The set T' of all linear combinations with rational coefficients of characteristic functions of the sets 4, is countable. Every simple function is in the closure of T' in L,(ux). Since the simple functions are dense, I' is dense. Of particular interest is a measure defined on a locally compact space. The following lemma will be useful later. Lemma 3 If uis a positive measure on a locally compact space X, then the set C(X,F) of all f in C (X, F) with compact support is dense in Ly(u).

Proof 1t is enough to consider the case F = R. Since the simple functions are dense in L,(u), it is enough to show that if A is any

integrable set, there are test functions f such that ||[x4 — f||» is arbitrarily small. This is done by means of an estimate which is interesting in its own right. Consider any ¢ > 0, and let the test function f approximate A to within ¢, with error sequence { f;}. By Theorem 5 of Chap. 7, the series 2 fi converges at every point of some full set B to an integrai=1

ble function A (Av —A)nB. 1 — h(x) for all lows that [x4(z)

with [hdu < e. Consider any z in the full set C = If f(z) + h(zx) z in A. Similarly, f(z) < h(x) for all x in —A. It fol— f(z)| < h(z). Thus we have

e < du [h < de fl — xa fl < p 2d fl — xa fl Ixa — fll2*» = Proposition 7 If u is a positive measure on a locally compact space X, then (X,F,5,9,u) 7s separable. Proof Again, we may restrict our attention to the case F = R. As shown above, there exists a dense sequence {f.} in C(X). IFor each n choose an, with 1 < a, < 2, so that

An = {2 fu(2) 2 on) is compact and integrable. We shall show that the sequence {A.} satisfies the requirements of Definition 6. To this end consider an integrable set A and a constant ¢ with 0 < ¢ < 1. Choose a compact integrable set K C A with u(4)

— u(K)

< e. Choose

a test function

f, 0 < f 0 a.e. Write ¢ = p/(p — 1). By

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Lemma 4,

S+

g@rdu = [f(f + 9>~ du + [g(f + g)* du > ([frawt*(J(f + g)? dw)*—DI7

+ (fg? dw)1n(f(f + g)» du) 1o

This is equivalent to (2.3). Lemma

6

Letzx, y, p, and q be real numbers, with p~! + ¢

= 1. Then

@ + yl7 + o — yle > 2(|z]? + |y|7)e?

(2:4) for p > 2, and

(2.5) forl Proof

[z +yl2+ |z — yle < 2(el? + [y[7) < p < 2. Assume first thatp

> 2, so that 1 < ¢ < 2. By continuity and

symmetry we may take |z| > |y| > 0. Dividing by |z|? and setting ¢ = |yz~!| (so that 0 < ¢ < 1), we reduce (2.4) to

(2.6)

L+ 0r+ (1 — 1 —2(1 + )1 >0

Expanding the left side of (2.6) in powers of ¢, we get

@7

2 Y (@) lglq —1) - - - (g — 2k + 1)e k=1 o0

-2 kz (CR)N)~g — 1) - - - (g — 2k)c*» —

9

i

(Qk

—

DN g —1)

- - - (¢ —

2k +

1)ct@*—Drp

k=1

(@— 2k +1)

=2 Y (@)D Mg —1) o0

k=1

X

{qczk

—

(q

—

2k)02kp

—

2k0(2k-l)p}

BEach of the products (¢ — 1) - - -+ (¢ — 2k 4 1) is nonnegative, because 1 < ¢ < 2. Consider k in Z*, and write o = 2k(p — 1). Then

gc* — (g — 2k)c?r — 2kc2k—Dr = 2kpc*Pla~lc™® — (! — p7Y) — p~ic7} = 2kpc*?la~(ce — 1) —p~Hc?—1)} =20 for all kin Z+, since t~1(¢c~¢ — 1) is an increasing function of ¢ for ¢ > 0,

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and a=2k(p—1)22(p

—1)

=p+p — 2 > p. Thus

each term

in the expansion (2.7) is nonnegative. Hence (2.4) is vald. Assume next that 1 < p < 2. Then 2 < ¢ < =, and thus, by the case already considered, we have

z + yl* + | — yl* = 2(|2]? + [y[97! Substituting (z + y)/2 for x and (z — y)/2 for y gives

9r yl — |z + gle + (z 0 2 > * lol o + =

21—P(|x

—+

qu

+

|x

—

y|Q)P—1

which is equivalent to (2.5). We now prove Clarkson’s basic inequalities for the L, norms. Theorem 1 we have

Take F to be R. Letl f and g be elements of L,. For p > 2

(@ and

2(Ifz+ lglnyt < If +gls + 1If — gli2

® 17 +gllz + I — gllz < 200171 + gl

while for 1 < p < 2 the reverse inequalities hold.

Proof The inequality (f + 9)/2 and ¢ with (f Consider first the case inequality and Lemma 6

(b) — p we

arises from (a) when we replace f with g)/2. It is therefore enough to prove (a). > 2. Write s = p/q. From Minkowski’s get

17+ ollg + I = glig = 1| |7 + glelle + 1 1f — glells = | [f +gle+ |f — gl 2 2[|(fl” + gyl

as desired. In case 1 < p < 2, using first Lemma 5 with s = p/q and then Lemma 6, we get

If+gllg+I1f—gllg = I If + glells + 1| 1 — glelle < f+gle+1f — gl < 2[[(|fl> + lgl)e. = 2(|lfllz + llgl|lz)=? as desired.

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Corollary Take F to be R. Then for each p > 1 the space L, is uniformly convex, in the sense that to each ¢ > 0 corresponds a positive constant r < 1 such that |1(f + 9)|l» < r whenever ||f]|, = ||gll, = 1 and ||f — gll» 2 e

Proof

For p > 2 we have, by (b) of Theorem 1,

147+ D)l = 31F + gll» < 3@ — @)1r = (1 — Gor)1o For 1 < p < 2 we have, by Theorem 1,

IG5+ Do = 317 + gll, < 327 — e)Ve = (1 — ()9 Thus the desired inequality holds for all p > 1, with

r = max {(1 — GOM, (1 — (399 We

can use Theorem

1 to determine all normable linear functionals

on L,, for p > 1.

Theorem 2 Take F to be R. Then, for each p > 1, and each ¢ in L, the linear functional N\, on L, defined by

(2.8)

A(f) = [fg du

ts normable, and |[\;|| = ||g|lq. Conversely, if N\ is any normable linear functional on L,, then N\ = \, for some g in L,.

Proof

By Holder’s inequality,

(2.9)

NN

There exists a unique |fol? = |g|? a.e. Then

(2.10)

< llgllall Al

element fo of L, such

that fog > 0 a.e. and

Ao(fo) = Jfog du = [lg|"* @ du = [lg|2dp = |gllllSoll»

Together with (2.9), this implies that A, is normable Consider, conversely, a normable linear functional any constant, 0 < ¢ < 1. Choose f in L, with ||f]|, Al — €. Let g be the unique element of L, with fg > a.e. Then |lgl[, = 1. Let h be an arbitrary vector in and A;(h) = 0. Then M(f+

€h)

= N(f — €b)

= N(f)

= [fgdu

with norm ||g|}q. A on L,. Let € be = 1 and \(f) > 0 and |f|? = |g|¢ L, with ||A]] = 1

= [|flrdp

= 1

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Therefore ||f + €h||, > 1 and ||f — €h||, = 1. In casep > 2, Theorem 1 gives

cr e? + 1 < 1 — ) P + t 2( 1 < el If + |z

where ¢; is some positive constant. Therefore If + ehll, < 1+

coe?

where ¢; is some positive constant. Similarly for 1 < p < 2, Theorem 1 gives

I + ehllz < 21 + )t — 1

and thus

If 4+ ehllp < 1+ cze? Thusin case 1 < p < 2 or2 < p we have If + ehllp < 1 + cue

(2.11)

where ¢ = min {p,q}. Hence (2.11) holds for all p > 1. Therefore

AT+ eh)] < (1 + cae) M| and

AR < (NS + er)| — M) < A + cae)[IM] — M+

e

< et

€2)

It follows that for all A in L, with A;(h) = 0 we have

(2.12)

NR)| < (cae™ MM + )[|R]]

For any h in L ,, A(h

N(R)f)

=

ka(h)(l

=

—

Aa(f))

=0

If we set ¢ = A(f), inequality (2.12) therefore gives |>‘‘ 0. For each k in Z+ let zx be a vector with |[zx]| = 1 and A(zx) > |[M| — kL. For each z in B and yin N(\) we have

lz — yll = [A7MA@)] On the other hand, the vector 2z = x — A(x)A(zx)"'xx belongs to N(N), and :

lz — 2l = A@)A@D)™ < M@

— )~

Thus p(z,N(\)) exists and equals [|[A]||7YA(z)]. Assume, conversely, that p(x,N(\)) exists for all z in V. Since X is nonzero, there exists a vector xo with A(xy) = 1. Hence

inf {[|z]|:Mx) = 1} = inf {[[z0 — y[l:y € NN}

= s, N(N))

exists. Therefore ||A| exists and equals p(xo,N (N\))~L Lemma 7 Let K be a bounded located convex subset of a normed linear space B, whose distance to 0 is positive. Then the cone

¢(K) = {tx:t > 0,2 € K} generated by K is located.

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259

For eacht > 0 write

tK = {tx:x € K} Then {K is located. Since p(0,K) > 0, for each 2 in B we have lim p(x,tK) = . Therefore p(z,c(K)) exists and is the infimum of the {—

continuous function ¢t — p(x,tK) of i. The fundamental geometric fact about normed linear spaces states that under certain conditions two convex sets can be separated by a normable linear functional. This result, which we now prove, is called the separation theorem.

Theorem 3 LetF and @G be bounded convex subsets of a separable normed linear space V, whose algebraic difference

ly —z:z€F,y € G} 18 located, and whose mutual distance

d=infl{lly —zll:z€F,y € Gl 18 positive. Then for each ¢ > 0 there exists a normable linear functional A on V of norm 1 such that

)— € My) > ME+d

(3.1)

for all x in F and y in G, where N\, s the real part of \. Proof Consider first the case F = R. We By Lemma 1, the bounded open convex set

may

assume that ¢ < d.

K={ly—z—2z22@,E|z €F O, xEK}

mfa+n|t t 4+ 1 2>

P(OyK)

= é’e

t+1

(=)

:t>0,x€K}

260

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Let {z.} be a sequence of vectors which is dense in V. We construct inductively

a

sequence

K = K, C K; C

- - - of

bounded

located

open convex sets such that the following properties hold for all n > 1.

(@) (¢)

p(xo,c(Ka)) > (1 — 27")p(x0,¢(Kn-1)) Either p(z.,c(K,)) < n~! or p(—2Z,,c(K,)) < n!

Assume

that

Ko,

...,

K,_;

have

been

constructed.

If

either

< n~!, wemay take K, = K, _;.

p(Tnyc(Kn—1)) < nlorp(—zn,c(Ka._1))

Assume therefore that both these distances are at least (2n)~!. Write

and

Kt={tz, + 1 —)k:0 = (¢ — Px,x — Px)

or

a’||z||* + a(z, 2 — Px) + a*(x — Px,2) > 0

This implies that (x — Pz, 2) = 0. Conversely, if y is any vector in M with (x — y, 2) = 0 for all z in M, then since y — Px & M, we have (y — Pr,y

— Px)

= (x — Px,y

— Px)

— (x —y,y

and thus y = Pz. Hence Px is the unique (x — Px,2) =0 forall zin M. Since

(ax + by — (aPx + bPy),2)

vector

— Px)

in M

=0

such

= a(x — Px,2) + b(y — Py,2)

that

=0

whenever a, b € F; z, y € H; and z €& M, we see that P(ax + by) = aPx + bPy

Thus P is linear. Since Pxr & M (for each z in H), it follows that P2r = Px. Also, |z||? = (z,x) = (x — Px + Pz, x — Px + Pz)

= ||z — Pz|? + ||Pz|?

since (x — Pz, Px) = 0. Therefore ||Pz|| < ||z||. Hence 1 is a bound for P. On the other hand, ||Py|| = |ly|] > 0 if y is any nonzero element of M, and thus P is normable, and ||P| = 1. Vectors x and y in an inner product space B are orthogonal, written x 1 y,if (x,y) = 0. For orthogonal vectors x and y we have the important equality

lz + y||2 = (z,2) + (=,9) + (W,2) + @y

= =]+ ||y

The orthogonal complement K+ of a set K C B consists of all zin B such that ¢ 1 y for all y in K. It is a closed linear subset of B.

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Consider vector x in sum of an vector y in Therefore

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269

a subspace M of a Hilbert space H has a unique representation x element Px of M and an element M+ the vector z = (x — Px) — y

H. By Theorem 6, every = Px 4 (x — Pzx) as the x — Px of M+. For each &€ M+, and thusz | Pz.

lz — yl|* = [Pz + 2|I* = [[Pz[* + [l2]|* = [[Pz[|* = [z — (z — Pz)|? and thus x — Px 1s the closest vector in M+

to . Thus M+

i1s a sub-

space of H, and the projection of x on M+ is x — Px.

Definition 10 A sequence {z,} of vectors of a Hilbert space H is orthonormal if z,, 1 x, whenever m # n and if for each n either ||z.|| = 1 or ||z.|]] = 0. Such a sequence is called an orthonormal basis if each vector z can be written uniquely in the form o0

r

=

Z

Anln

n=1

where {a,} is a sequence of scalars, called the coordinates of x with respect to the orthonormal basis {z.}, such that a, = 0 whenever x, = 0. An orthonormal basis can always be obtained by the so-called GrammSchmidt orthogonalization process. Theorem 7 Every Hilbert space has an orthonormal basis {x,}. {a.} are the coordinates of x, and {b,} are the coordinates of y, then An

=

(x:xn)

bn

=

If

(y;xn)

@) = % abt lalt = Y ol 0

n=1

Proof Let {y.} be a countable dense subset of H. We define inductively an orthonormal sequence {z,} such that for each n there exists a linear combination y’ of 21, . . . , x, with

(4.8)

lyn — 4]l < m1

To start the induction, note that either |ly:]| < 1 or ||y1]| > 0. In the first case write z; = 0, and in the second case x; = |[y1l|~! v1. Assume that 1, . . . , z, have been defined. Let M, be the subset of H consisting of all linear combinations of x;, . . . , 2,. Then M, is a

270

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ANALYSIS

finite-dimensional Banach space, and so is locally compact. It is therefore located in H. Hence M, is a subspace of H. Let P, be the projection

on M,. Either ||[yos1 — Potyna]] < (0 4+ 1)~ or ||yns1 — PaYnsall > 0. In the first case set .41 = 0. The above condition holds with y, ., =

P.yu+1. In the second case set Tnil

=

—

Pnyn-l-l“—l(yn-i-l

”yn-i-l —

Prynt1)

Condition (4.8) is satisfied with [|[Yns1 — PrlYnsa|l Togr = Yapa

Yny1 = Pryns1 +

The vector z,41 1s orthogonal to z;, . . . , x, because Yyp41 — Prlyns1 & M.*. Thus the sequence {z,} is orthonormal. To finish the proof, consider arbitrary vectors « and y in H. FFor each n we have unique representations n

n

P,.x =

P,y =

Z a.x; i=1

Z b.x; i=1

with a;, = b; = 0 whenever z; = 0. Taking inner products with xx, we get

ar = (Puz,xr) = (x,2r) + (Pox — x, )

= (x,2x)

and similarly bx = (y,zx), so that a; and b; are independent of n. We have n

(an,Pny)

=

z

a,-b;-"

t=1

Now the distances of and y to M, approach 0 asn — and P,y — y as n — . Therefore

«, or P,z — x

0

(z,y)

=

lim n—

(P.x,P,y) o

=

z

a:b¥

;

Setting y = x gives

An n-dimensional Hilberl space admils a norm-preserving Corollary linear map onto F*. An infinite-dimensional Hilbert space admats a normpreserving linear map onto l,. (The space H is infinite-dimensional if for every n > 0 there exists an n-dimensional subspace of H.)

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271

Proof Assume that H is n-dimensional, and let 4, . . . , y. be a basis for H. The orthogonalization process of Theorem 7 produces an orthonormal basis z1,

o: H—

. . . , z, for H, with l|z;ll = 1 (1 + [Py — z|* > |ly — Pyl|? since Py — x & M and y — Py 1 M. Hence Py is the closest element in M to y. It follows that M is located, and P = Py.

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ANALYSIS

We now prove some lemmas which will help disclose the structure of an arbitrary hermitian operator. Lemma

8

Let A be a hermitian operator on a Hulbert space H. For

simplicity assume that ||Az| < ||z|| for all z in H. Let x be any vector of norm 1. Write

y = Az — (Az,x)x Then y 1 x (and thus y 1s the projection of Ax on the orthogonal complement of x). Write

o = |z + 1l + 1)

Then

and if B 1s any hermitian operator which commutes with A, then

(4.10) Proof

|Bzal| < 2| Bz]| Set N\ = (Az,x) and compute

(Al + 1y),z + 3y) = Oz A+ N+ =X+

+ y + 3(yy) Ly (yy)

LAy, + Ly) + 3(Ay,x) + 1(Ay,y) + 3y, Ax) + 1(Ay,y) +3(Ay,y) 2 N + 2(y,y)

Therefore

(Aza,za) — (Az,x) 2 (1 4 2(y,y)(N + 2(y,1)) — A = (1 + +yy)'GW,y) — A\ wyy) = = 3(y,y) = :l|Ar — (Az2)z|*

& W,y)

Since z + 1y = 2 — 1(Az,x)z + 1Az, we also have

[Bza|l < |B(z + 3u)|l = |[Bx — 3(Az,z)Bx + }ABz| < {1 + 3|(Az,2)| + 3}||Be| < 2| Bzl Lemma 9 Let Ay, . . ., A, be commuling hermitian operators on a H7lbert space H, let x be a vector of norm 1, and let € be a positive constant. Then there exists a vector y of norm 1 with

(4.11)

(Aw,y) > (Aiz,x) — ¢

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and

(1

2

3 — € € T N + ) U € , U € 1 ~ ‘ ‘ * ( 2

Z

This contradicts the fact that 1 is a bound for A;. Thus (4.13) holds for

some k. Set y = ;. Then (4.12) is valid, and (41y,y) = (Aiz,x). Thus the lemma is true when n = 1. Assume therefore that n > 1, and that the lemma is true for all smaller values of n. Choose the positive integer N so that Ne? > 32. By the inductive hypothesis, there exists a unit vector u with €

(Alu,u)

and

A

— (Au,u)u||

>

(A]IE,IE)

< -V

—_

2

1 52

@ (A un—1,un—1)

>

-

+ 2

16

2> (Auu+)N

62

16

> —-14+2=1 This contradiction shows that (4.14) is valid for some value of k. Write Yy = ux. Then (4.14) gives (4.12) forz = n. If 1 0 there exist commuting projections Py, . . . , P,, with P;P; = 0 for 1 0, o(f) < [|fll, o(f + 9) < o(f) + o(9),

o(cf) = lelo(f) and o(fg) < o(f)a(g) for allf and g in C(X,F) and all c in F. Then there exists a compact set K C X such that o(f) = ||f||x for all f in C(X,F). Proof Let K consist of all z in X such that for each € > 0 there exists hein C(X,F), with a(he) > 0, for which h(y) = 0 whenever p(y,z) > e. Consider any z in K, any cin F, and any g in C(X,F) such that g(y) = ¢ at all points y belonging to some neighborhood of z. Then gh. = ch. if e 1s sufficiently small. Hence

lclo(he) = o(che) = a(ghd) < a(g)a(he) or a(g) > |c|. Now, if f is any element of C'(X,F), and k is any positive integer, we can find a function ¢ in C(X,F) such that ||f — g|| < k!

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279

SPACES

and g(y) = f(x) for all ¥ in some neighborhood of x. Hence

o(f) 2 0(g) —o(f —g) 2 |f@)]| — Kk Letting k — o gives o(f) > |f(x)|, for allf in C(X,F) and z in K. We next show that if fo is any element of C(X,F) with ¢(fo) > 0, K is nonvoid. To this K, is any compact support for fo, then Ko we define inductively a sequence fo, f1, . . . of elements of C(X,F) a sequence Ko D K; D - - - of compact sets, so that ¢(f,) > 0, supports f», and diam K, < n~!for all » > 1. The function f, and set K, have

already

defined.

been

Assume

therefore

and end and K. the

that n > 1 and

that fo, . . . , faz1, Ko, . . . , K._; have already been defined. Choose elements g1, . . . , gy of C(X,F) whose sum is f,—1, each of which is supported by some compact subset of K,_; of diameter at most n~1. Since

a(gr) +

-+

+algy)

= a(far) >0

a(g:) > 0 for some 7. Set f, = g: and take K, to be a compact support for g; with K, C K,_; and diam K, < n~!. This completes the induction. The sets K, have a unique point z in common. Clearly + € Kon K, K is nonvoid. and thus Ky To see that K is totally bounded, fix ¢ > 0 and choose functions fi, - . ., foin C(X,F), with fy + - - - + f, = 1, so that f; 1s supported by some compact set K; of diameter less than e. The set of positive integers ¢ with 1 < 7 < n is the union of finite subsets S and

T,

with ¢(f;)) < @2n)~!for 7 € §, and o(f;) > 0 for 7 & T. By the above, for each 7 in T there exists a point z; in K; K. We shall show that these points z; are an e approximation to K. To this end, consider z in K. Then |f:(x)| > (2n)~! for some 7. Therefore o(f;) > (2n)~1, and thus 1 € T.Sincezx € K;and z; € K; and diam K; < ¢, we have p(z,z;) < e. Thus the points x; are an ¢ approximation to K. Hence K is compact. Now consider any f in C(X,F). For each ¢ > 0 there exist ¢ and

h = f — gin C(X,F) such that ||g|| < ||fl|lx + € and & is supported by a compact set L C —K. If ¢(h) > 0, then, by the above, nonvoid. This contradiction shows that (k) = 0. Hence

a(f) < a(g) +oh)

LN K

is

=a(g) < |gll < |Ifllx + e

Thus o(f) < ||f||x- Since we have already shown that o(f) > |f(z)| for all z in K, it follows that o(f) = ||f||x. We now state and prove our supplement to the spectral theorem.

280

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

Supplement to Theorem 8 In case H is nontrivial, in the sense that 1t has a nonzero vector, and in case the operators in U are all normable, there exists a compact set K C X such that u(X — K) = 0 and

(4.18)

IF

= lIf]lx

for allf in C(X).

Proof Since |f(A)|| < ||f|| for all f in 2, the operators f(4) are normable for all f in C(X), and ||[f(4)]] < ||fll. The hypotheses of Lemma 11 are satisfied by the function o(f) = ||f(4)||. Hence there exists a compact set K C X satisfying (4.18). If f is any function in C(X) vanishing on K, we have f(A) = 0, by (4.18), and thus [fdu = 0 by the definition of u. Hence u(X — K)

= 0.

In order to reformulate this supplement, we remark another consequence of Lemma 11, which has considerable interest of its own.

Proposition 10 Let K be a compact space and T' the set of all nonzero multiplicative bounded linear functionals v on C(K,F). For each x in K define the element u, of T by u.(f) = f(x). Then the map \: K — T defined by N(x) = u, ts a metric equivalence [where T has the metric induced by the double norm on C(K,F)*].

Proof 1If |u(f)| > 1, and ||f|| < 1 for some u in T' andf in C(K,F), then ||f*|| = 0 as n — o, and |u(f*)| = |u(f)|*> « as n— o, contradicting the boundedness of u. Therefore |u(f)| < ||f]| for all w in T and f in C(K,F). By Lemma 11, for each u in T there exists a compact set L C K such that |u(f)| = ||f||z for allf in C(K,F). Since u is multiplicative, L must consist of a single point z, so that |u(f)| = |f(z)|. Thus u(f) = 0 whenever f(z) = 0. Hence, for eachf in C(K,F),

u(f) = u(f = J(@)) +f@u)

= f(z) = u(f)

and thus ¥ = wu,. Thus the map \ is onto. Since each f in C(K,F) is uniformly continuous, the formly continuous, by the definition of the double norm To show that A~! is also uniformly continuous, consider Ty, . . . ,%, be an e approximation to K. By Proposition

map \ is union C(K,F)*. ¢ > 0 and let 9, there exists & > 0 such that |p(x,z;) — p(y,z:)| < € 1 < 7 < m) whenever z and y are points of K with |[||A(z) — A()||| < é. If we choo¢se with p(z,z;) < e, 1t follows that p(y,z;) < 2¢, so p(x,y) < 3e. Thus A\~!is uniformly continuous. Hence X\ is a metric equivalence.

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281

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We now give three corollaries, which are actually reformulations of the Supplement to Theorem 8. To do this, we need to define the uniform neighborhood structure on Hom (H,H). For each A in Hom (H,H) and each ¢ > 0 the uniform neighborhood U(4,e) consists of all B in Hom (H,H) such that A — B is bounded by some constant ¢ < e. These neighborhoods define a neighborhood structure. Such concepts as uniformly closed and uniformly separable pertain to this neighborhood structure. The set of normable operators is uniformly closed, and on any linear subset of the set of normable operators the uniform neighbor-

hood structure is induced by the metric p(4,B) = |A — Bj|. Corollary

1

Let B

be a commutative

algebra of normable

hermitian

operators on a nontrivial Hilbert space H, with I & B, which 18 uniformly separable and uniformly closed. Let the spectrum Z of B consist of all nonzero bounded multiplicative linear functionals on B. Then each u in Z

s normable, with ||ull = 1, and Z is compact in the metric induced by the double norm. M oreover, if for each operator U in B we let v(U) be the element of C'(Z) defined by

(v(U))(w) = u(U) then v 18 a norm-preserving algebra C(Z).

isomorphism from

the algebra B

onto the

Proof Let A = {A,} be a sequence of elements of B, with common bound 1, whose linear combinations are uniformly dense in 8. Let X be defined as in Theorem 8, and let K be the compact set defined in the Supplement to Theorem 8. Since u(X — K) = 0, the element f(A4) of B 1s well defined for all f in C(K). By Theorem 10 of Chap. 4, each element of C(K) extends to an element of C(X). By (4.18), it follows that the map f — f(4) from C(K) to Hom (H,H) is a norm-preserving isomorphism F of C(K) with 8. Since £ C B™* stands in the same relation to B as the space T' C C(K)* of Proposition 10 does to C(K), we see from Proposition 10 that 2 is compact and that each u in Z is normable with |u| = 1. Moreover, the map y: 8 — C(Z) stands in the same relation to B and 2 as the map w: C(K) — C(I'), defined by w(f) = fo X1 stands to C(K) preserving isomorphism.

and

TI'. It follows

that

v is a norm-

In case H is a complex Hilbert space, this reformulation admits a modification.

Corollary 2

Let B’ be a complex commutative algebra of normable linear

operators on a nontrivial complex Hilbert space H, with I € B', which s

282

FOUNDATIONS

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ANALYSIS

uniformly separable and uniformly closed, and s selfadjoint in the sense

that each operator B in B’ has an adjoint B* that also lies in B'. Let the spectrum 2’ of B’ consist of all nonzero multiplicative bounded linear functionals w' on B’. Then each v’ in Z' is normable, with ||u'|| = 1, £’ s compact, and the map v': B — C(Z',C), defined by

YO)W)

=4w(U)

@WezZ,Ue®)

18 a norm-preserving tsomorphism of B’ onto C(Z’,C), with

(4.19)

Y (U*) =~ (U)*

(U

Proof Each Uin %’ canbewritten uniquelyintheform U = U, + 1U,, where U, = L(U + U*) and U, = 1/2¢(U — U*) are hermitian. Let B be the set of all hermitian operators in B’ and 2 the set of all nonzero multiplicative bounded linear functionals v on 8. By the above, the map v: B — C(Z) is a norm-preserving isomorphism. Let 4’ be any element of £’ and U any operator in 8. If 4'(U) is not real, then there exists V in 8 with «/(V) = 1

for all v in =. We have 0 = «/ (V2 4+ Du'(W) = w'(I) = 1. This contradiction shows that «’(U) is real for all 4’ in Z’ and U in 8B. Therefore the restriction u of 4’ to B belongs to Z. Moreover U’(U*)

=

u’(U1

-

’LUz)

=

’U/,(Ul)

-

Zu,(Uz)

=

u’(U)*

(U


0 there exists §,

LOCALLY

COMPACT

301

GROUPS

ABELIAN

with 0 < § < 1lr, such that |f(z) — f(y)| < e whenever z, y € K, and p(x,y) < 8. Consider any points z and y of G with p(z,y) < 4. Either

p(x,K) < r — & or p(x,K) > 4. In the first case p(y,K) < r. Therefore z, y € K,, and thus |f(x) — f(y)| < e. In the second case p(y,K) > 0, and thus |f(z) — f(y)| = |0 — 0] = 0. Thus f is uniformly continuous. For each f: @ — R and each s in G we define the left translate T'(s) f of f by s as follows:

(T(s)f) () = f(sz) The right translate is defined similarly. A measure u on G is called leftinvariant, or invariant under left translations, if

JT@S)f du = [fdu for all test functions f and all s in G. Right-invariant measures are defined similarly. Our immediate goal is to construct a positive left-invariant measure. We begin by defining certain quantities which approximate to a leftinvariant measure.

Definition 2 Let Ct = C(@)* consist of all nonnegative elements of C = C(G), and @ consist of all nonzero elements of C+. For each fin C+ and each ¢ in ® we consider the set A of all finite linear combinations

(1.1)

S = Ze;T(s:)

where the c¢; are nonnegative constants and the s; are points of G, for which f < 8. We write

(f: o) =inf {Z¢;:S € A} whenever the infimum exists.

Lemma 2 To prove the existence of (f: ¢) only elements s; in a certain compact set Y C G need be considered.

Proof Let K and L be compact Choose € > 0 so that the set

supports for f and ¢, respectively.

Y = {z: p(x, LK)

< €}

is compact. For each element S = Z¢,T(s;)¢ of A, partition the set of

302

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

indices ¢ into free subsets M and N, so that s; & Y for all 7 in M and

> 0 for all 7in N. For each z in K and each 7 in N, we have

p(s;, LK)

= 0. Hence

p(siz,L) > 0; therefore ¢(s;x)

f
0 for some ¢ in G. Choose ¥ > 0 so that ¢(tx) > v whenever p(x) < v. Choose t1, . . . , tm in K71 so that for each in K we have p(t;x) < v for some j, with 1 < j < m. Then o(ttix) > +v. Hence

1 0 there exists a subfinite set B C A such that for each S = Z¢;T(s;))¢ in A, there exists S' =

2ciT(sl)e in B with

Zci < et e

(1.3)

To this end, write § = (2 + v~2||f||m? + v 'm)~'e. By Lemma 1, we see that ¢ is uniformly continuous. By Proposition 1, there exist x, . . . ,zy1n Y such that for each sin Y thereexists kt (1 < k < N) with

J in Z+, with 6~ 'Nv~!||f|lm < o(sy) < o(xry) + 6 for all y in K. Choose J, and let @ consist of all N-tuples M = (M4, . . ., My) of nonnegative integers such that 0 < M; < J for all k. For each M in Q we write

(14)

8" =6N-1Y

N

k=1

(M +2)T@)e + oy (v

m

flm + 1) 3 T()e 1=1

Partition Q into finite subsets ©; and Q, so that (i) f < 8’ for all M in Q, and (ii) for each M in Q, there exists y in K with f(y) > S'(y) — é.

LOCALLY

COMPACT

ABELIAN

GROUPS

303

The set B

=

{S,Z

M

E

91}

is a subfinite subset of A. Let S = Z¢;T'(s;)¢ be any element of Ao. By the choiceof z;,

. . . | 2n

there exist nonnegative constants a1, . . . , ay with 2Zar

(15)

=

2C;

and

(y € K)

Sy) < Zarp(zry) + 62¢;

(1.6)

By (1.2) and the inequality Zc¢; < y7!||f||m, inequality (1.6) gives

(L.7)

fly) + 8 < S) + 9

< Zae(@y) + oy (v Iflm + 1) Y o(ity) i=1

for all y in K. Since ar < Z¢; < v7Y|fllm < 8N-J for all k, there exists M

in

Q@

such

that

N-M;

< ar < N7 (M

+ 2)

for

all

k. Let

S = ZciT(s})e be given by (1.4). By (1.7), we see that f(y) + § < S'(y) for all y in K. Therefore M & Q,, so that " & B. Also,

¢t = SN

Z(My + 2) + v (v Y|f|lm + 1)m

< Zar + 62 + v flm? + v~'m) By (1.5) and the definition of §, inequality (1.3) is valid. For later use we note some simple properties of the function (f: ¢).

¢ € @, then g € @, and If f € @, then (f:¢) > 0. If fE Ct,

(o) S (F19)(g:@) Hence, if f & @, then

(1.8)

G: NS

(o)™

< (i g)

As a function of f, (f: ¢) 1s homogeneous,

Af:e) =NJ19)

(A ER™)

and subadditive,

(fi+feio)

£ (fiio) + (foi )

304

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

It is also left-invariant, in the sense that

(T)f:0) = (fio)

(s€EG)

Now fix a particular function f, in @, for the rest of this section, for

purposes of normalization. I'or each f in C* and each ¢ in ® we write

1.(f) = (f: ) (fo: 9)* In casef € @,

(fo: /)71 < L(f) < (f:10)

(1.9)

by (1.8). A function ¢ & @ is small of order ¢ (where c is a positive constant) if p(x) = 0 whenever p(x) > c. The following lemma shows that I, is approximately linear if ¢ is sufficiently small. Lemma 4 Let e and M be positive constants and f1, . . . , f. elements of C*. Then there exists ¢ > 0 such that, whenever \i, . . . , A\, are real numbers with 0
0 so that

ZNIL (i) < (T(2Nf5) + (1 + ¢ whenever ¢ 1s small of order c. Let g be any element of @ such that g(x) = 1 for all zin some compact set K which supports each of the functions f;. Write d = e(g: fo)™! By (1.9), 61,(g)


0 with (119)

f <
1, ¢ > 1,r > 1, and p!

+

gl=1+

r—1

Then

(1.23)

1f = gll- < [I£]l2llgllq

Proof It is sufficient to consider the case f > 0and ¢ > 0. The numbers a =771, 3 =p ! — 71 and y = ¢! — r~! are positive and sum

to 1. Holder’s inequality gives

(f* @) (@) = [{f@Wrgly—x)}of (y)*Pg(y~'z) ™ du(y) < (Jf@)rg(y=x)e duy)) =] fl|2%]| g2

312

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

Therefore

H )) lx du ) ly J du ( e a ) l —x (y l Pg g ) l (W l [ 8 l l f 1S * gll- < [I = [IfIzllglgrCllANzllglD = 1 flllgllq Proposition 2 extends to more general functions.

Proposition 3 Let p, q, and r be constants with p > 1,q > 1, r > 1, p~ '+ g ' =14 rt Let f and g be functions in L, and L4, respectively, on the unimodular group G. Then the integral (1.22) exists for almost all x

i G, the function f = g so defined belongs to L., and (1.23) s valid.

Proof

It is enough

to consider the case f > 0 a.e. and g > 0 a.e.

In case both fand g are test functions, the result is already established.

Next consider the case in which fis a test function and g 1s arbitrary. Let {g.} be a sequence of test functions converging to g in L,. By (1.23), we see that {f * ¢.} is a Cauchy sequence in L,, whose limit we call 4. By Theorem 4 of Chap. 7, we may assume that {f * g.} converges to A almost everywhere. Since for each z in G the linear functional

F— [f(y)F (y~'z) du(y) on L, is bounded, we have

. a.e z) h( = x) )( ga * (f m li = x) )( (f*g Also,

If*glls

Next consider the the complement of uniformly bounded and converging to f (f * g)(x) for all z, L., and &g f*x

=

k]l

= 7}_1_{2 | f * gnll+

< 151l lim ligalla = 171l

case in which f1s a simple function that vanishes on a compact set and ¢ is arbitrary. Let {f.} be a sequence of test functions having a common support almost everywhere. Then {(f. * g)(x)} converges to and {f. * g} is a Cauchy sequence in L,. Hence

IF gl < lim [[fa = gll- < [[f]l5llglle n—r 0

Next consider the general case. Let {f.} be a nondecreasing sequence of nonnegative simple functions converging to f in L,. By the case just considered, .{f, * g} is a Cauchy sequence in L., whose limit we denote

LOCALLY

COMPACT

ABELIAN

GROUPS

313

by h. We may assume that {(f, * g)(x)} converges to h(z) for almost all z. For such an x we see by Theorem 5 of Chap. 7 that (f * g) (x) = h(x). Thus f * g is defined almost everywhere, f x ¢ & L., and

If= gll: = [|All, = lim [|fa = gll- < |[fll2llglls n—r

The reader may check that in case f & L; and ¢ & L, the proof of Proposition 3 holds for an arbitrary locally compact group (not necessarily unimodular).

2. CONVOLUTION

OPERATORS

In the sequel ¢ will be a locally compact abelian group. We have seen

that if f € Li(@) and g € Ly(G), then f*g € Ly(G) and ||f * g
0 there exists

U, in ¥ such

that

U —

U, is

bounded by e. The operators in % are normable, and ¥ is commutative and selfadjoint, as well as uniformly closed. Let Z be the spectrum of . By Corollary 3 of Theorem 8 of Chap. 9 (page 283), Z is locally com-

pact, and the map v, from U to C(Z,C) defined by vo(U) (u) = w(U) is & normpreserving isomorphism. For each f in L;(@) and each « in G* we write

(T(f)) = a(f) By

(3.4), we see that

|a(T())| < ||[T(f)| for all T(f)

in A. Thus

a(T(f)) defines a bounded homomorphism of A onto C, and so extends

to a bounded homomorphism

U — «(U) of A onto C, that is, to an

element of Z. Conversely, if « & homomorphism of L; onto C, and Thus we may identify G* with 2, G* into the element U — «(U) of metric of 2, so that G* is a locally

Z, then f— u(T(f)) 1s a bounded so comes from an element « of G*. via the map which takes each « in Z. This permits us to give G* the compact metric space. To write the

metric explicitly, consider a dense sequence { f.} in L1(G), so that { T'(f») } is dense in . By the definition of the double norm on Z, we have

p*(a,8) = [lla = Bl[| + [([lladlI7* = [[I8lII71)] for all « and 8 in G*, where

llelll = 3, 2-latICt + ITG)D for all @ in G*, and |||« — B]|| is defined similarly. Combining the norm-preserving isomorphism v, from ¥ to C,(Z,C) with the map which identifies G* with 2, we obtain a norm-preserving

LOCALLY

COMPACT

ABELIAN

321

GROUPS

isomorphism v of A with C(G*C) given by v(U)(a)

(3.5)

= a(U)

(U & A

o € G%)

In particular, for each f in L,(G),

Y(T(N)(e) = «(T(f)) = a(f)

(3.6)

The function f = v(T(f)) is called the Fourier transform of f. For notational convenience, we also write f = F(f). The Fourier transform F 1s an isomorphism of the algebra L;(G) into C,(G*C), which is norm-decreasing:

1Tl

= V(TN

= ITADON < 1112

It is given by the formula,

B.7)

fla) = a(f) = [f@a(@) du(x)

(f € Li(B), a € G¥)

For later use, we shall wish to express the Fourier transform of a translate T'(z~!)f in terms of the Fourier transform of f. We have

F(T(xN () = [fx"y)aly) du(y) = [f@x 'y)a(@y)a(x) du(y) a(x)(F(f)) () or

F(T(x)f) = £5(f)

(3.8)

where £ is the function &« — a(x) on G*. The metric p* on G* is not easy to work with. It is helpful torelate the metric p* to the behavior of the characters a on certain subsets of G. First we give a criterion for boundedness.

Lemma

11

For each ¢ > 0 and each subset K of G we define

N*(K,e) = {a € G*: |a(x) — 1| < € for all z 1n K} Then N*(K,e) 1s a bounded subset of G*, for each real number e, with 0 < € < 1, and each neighborhood K of e in G. Conversely, if M* 1s any bounded subset of G* and e any positive constant, there exists a neighborhood K of e in G such that M* C N*(Ke).

Proof

OF

FOUNDATIONS

322

First, consider any real number

CONSTRUCTIVE

ANALYSIS

¢ with 0 < ¢ < 1, and any

neighborhood K of ein G. Then Re a(x) > 6 > 0, forall zin K and a in N*(K,e), where & is some absolute constant. Let f be any nonnegative test function supported by K, with [fdu = 1. Then

Re a(f) = [f(z) Re a(z) du(z) 2 6

(e € N*(K,e))

Hence |||«||| is bounded away from 0 on N*(K,e). Therefore N*(K,e) is bounded. Conversely, consider any bounded set M*

C G* and any ¢ > 0. Then

l|||l| is bounded away from 0 on M*. Let {f.} be the sequence from L,(G) we used in defining the double norm ||| ||| on Z. Then |a(f1)| + + « + 4 |a(fs)| is bounded away from 0 on M*, for some n in Z*. Thus to show the neighborhood K of 0 exists, it is sufficient to take M* to be the set

{a € G*:|a(f)] 2 7}

M*

where 7 is a positive constant and f is an element of L;(G). By Lemma 9, for each 6 > O there exists a neighborhood K of e in G such that

IT(x)f — fll+ £ 6 forall zin K. For each « in M* and z in K we have

() — 1

a(@)* — 1| < 7 Ha(@)* — 1] |a(f)] ra(T@)f — ] 0 there exists 6 > 0 and a compact set K C G such that p*(a,8) < ¢ whenever

o, B E M* and ||a — B||x < 8. Conversely, for each § > 0 and each compact set K C G there exists ¢ > 0 such that || — B|lx < & whenever

a, 3 € M* and p*(a,8) < e Consider any ¢ > 0. By the definition of the metric p*, to make Proof p*(a,8) < e (for arbitrary elements « and 8 of M*) it is enough to make the quantities [(a¢ — B)(f1)|, . - . , |(a — B)(f»)| each less than some sufficiently small »r > 0, where f1, . . . , f» are certain test functions

on G. To do this it is enough to make ||a — B||x < &, where K is a common constant.

support

for fi, . . . , f» and

6 is a small

enough

positive

LOCALLY

COMPACT

ABELIAN

323

GROUPS

Assume, conversely, that K and é are given. Since M* is a bounded subset of G*, there exists a positive constant » and a finite subset F of the unit sphere of L;(G) such that for each « in M* there exists f in F

with |a(f)| > 2r. By Lemma 11, there exists a neighborhood V of e in G with |a(y) A be a finite so small that F, and o and with p*(a,8)

— B8(y)| < 8/2 for all @ and Bin M* and all y in subset of G such that e € A and K C AV. Choose |a(T(z)f) — B(T(x)f)| < min {r,r25/2} forallzin 8 in M* with p*(a,8) < e. Consider any « and 8 < e. Choosef in F with |a(f)| = 2r. Then |8(f)| >

V. Let ¢ > 0 A, fin in M* 2r —

la(f) — B(f)| > r. Hence for all z in A and y in V we have

|a(zy) — By)| < la@)] |lay) — B®)] + [BW)] |a(z) — B(=)]

< 2 + la(T@Na() - BTENBU < 5 + laDIa(T@ — B(T@))] + BT @] |l = BU)Y < 2+ @)1 5 + [flilaDI SN al) — B 0 there exists ¢ > 0 such that P*(afi,aofio)

< e

whenever a, 3, ao, Bo € M*; p*(a,a0) < €'; and p*(8,80) < €. Note first that (M *)?is a bounded subset of G*, by Lemma 11. Therefore to prove

324

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

the existence of ¢, by Lemma 12 it is enough to show that for each compact set K C G and each § > 0 there exists a compact set K’ C G and a constant & > 0 such that | — aoBol|x < 6 whenever |a —

aollxr < & and ||B — Bollxr < &'. This is straightforward: Take K’ = K and take &’ to be very small.

4. DUALITY

AND

THE

FOURIER

TRANSFORM

The full duality between G and its character group G* is established by

inverting the Fourier transform §. The possibility of doing this is already contained in the fact, derived from the spectral theorem, that the map v of A onto C(G*C) given by (3.5)is a norm-preserving isomorphism.The inverse y~! of v is also a norm-preserving isomorphism.

For each f in Li(@) we have v~ 1(f) = T(f), by the definition of f. In other words,

v 1(f)g=F*g

(9 € L(Q))

This identity forms the point of departure for inverting the Fourier transform §. To each ¢ in C(G*,C) we shall associate a function @ on G, such that

(4.1)

v Ho)g=2*g9

(9 € L))

If ¢ is of the form £, for some f in L;(®), then @ will turn out to be equal to f. The map ¢ — @ is called the tnverse Fourier transform F*. Proposition 6 For each ¢ in C(G*C) there exists a unique function ? = F*(p) in C(G,C) N Ly(@) satisfying (4.1).

Proof

Notice first that if g and & are in L.(G) the convolution

(9 *h)(x) = Jgh(yz) du(y) is well defined, and belongs to C.(G,C). The integral defining (g * A) (x) exists for all z, because the product of two functions in L, is integrable. If f and ¢ are test functions, then f *x g 1s a test function, and thus f*g € Co(G,C). By an approximation argument, f * ¢ € C,(G,C) for all f and g in L (G).

Let K* C G* be a compact support for ¢. Choose h in C(G,C) with

k()| = |[h(z)a(@) du(@)| > 1

(« € K¥)

LOCALLY

COMPACT

ABELIAN

This can be done by taking that «(z) is uniformly very taking h to be a nonnegative There exists a unique ¢ &

325

GROUPS

a compact neighborhood V of ¢ in G near to 1 for all z in V and « in K*, function supported by V, with [h du C(G*,C) supported by K* with ¢ =

such and = 2. h¥}.

Clearly ||[¢¥|le < ||¢]lw. [We write ||¢|l. and ||¢||. rather than ||¢l and |¢|| to emphasize that norms in the space C,,(G*,C) are what is meant.]

Y~ He)g

I

For each g in C'(G,C) we have

v 1(R (@)) v~y (h)g

I

)T Th) v(h)g hx (v 1) T(9h) = hx (T(@v1(¥))h =h*xg*xy Yh = (h*v 1 Y)k) * g @=hxy1(Y)h

Write

Then (4.1) holds for all ¢ in C(G,C). Since h € C(G,C) and v~ 1(y)h E Ly(®), it follows that € C(G,C) n Lo(G). Finally, @ is unique because 0 is the only element of L.(G) whose convolution with an arbitrary element of L,(G) vanishes.

For later use we estimate |3/, and ||3||2:

and

|2lle < (22l @Rl < (v @] [124E = W]lo]Rll} < Cllell ()] @R[l < [IAflllA]l2llv [|gllz < [[All" lly = [[A[l:]A]l=¥]le < C"llell

where C and C’ are constants depending only on K*. If ¢ and ¢ are in C(G*,C), then for all g in L,(G) we have

ebrg=vUe)g = v

T IWg =p+x{F*g) = (p*V) *g

and thus

(4.2)

o=y

Similarly, ¢* * g = v (¢*)g = v (¢)*g = & * g, and thus ——

(4.3)

o* =

The next two lemmas

Lemma

(4.4)

13

~

contain formulas which will be needed later.

For each ¢ in C(G*,C) and each x in G we have

F*(&e) = T (@ 1)F*(e)

326

FOUNDATIONS

Proof

Choose a sequence

OF

CONSTRUCTIVE

ANALYSIS

{f.} of functions in C(@,C) so that ||¢ —

falle — 0asn— «.Todothisitis enough to have [[y~1(¢) — T(f.)|| — 0 as m —

o, since vy 1S a norm-preserving isomorphism and

Y(v" o) — T(fa)) = ¢ — fa

This can be done because U is dense in A. Since ||¢£n — fafnllo — O as n —

o, for each g in L,(G) we have

F*(2p)

g = v U £p)g = n—limoo v~1(f,)g

= lim v X(5(T(@)fo))g = lim T(T (@) fa)g

= 7}3{10 (T

)fn) g

= T(@™) lim fo xg = T(c™) lim v~1(fx)g =T@ )7y ey =Ta)(@*xg) = (TlxHe) g Therefore $*(£0)

Lemma

(4.5) Proof

14

= T(z7)e.

For each a in G* and each ¢ in C(G*,C) we have

F*(T(a)e) = aF*(¢) For each fin Ly(@) and each 8 in G* we have

F(af)(B) = [f (z)a(z)B(z) du(z) = f(ap) or

(4.6)

Flaf) = T()F()

For each positive integer n choose f» in Li(G) with |l — fall. < n~L Then for each g in C(G,C) we have, by (4.6),

4.7)

§*(T(a)e) xg = v (T(@)e)g = lim v (T(2)F(f2))g = lim v~1(F(ofz))g = lim (afz) * g n—r o

n—r

Now

o0

n—>r oo

for each z in G we have

((afz) *9) () = [a(y) fr(¥)g(yz) du(y) " x)* du(y) = a(2) [fn(Y)g(yx)a(y = a(x)(fr * a™g) (z) or (afn) * g = a(fn * a™g). Similarly, (ap) *g = a(@ * a*g). Combined

LOCALLY

COMPACT

ABELIAN

327

GROUPS

with (4.7), this gives

lim a(fn * a¥g) = a(p * a™g) = (ap) *g F*(T(a)e) *g = n— This is equivalent to (4.5). The inverse Fourier transform F* gives an explicit formula for Haar measure on G*.

Proposition 7

For each ¢ in C(G* C)

write

Jo du* = p(e)

(4.8)

Then u* s a Haar measure on G*.

Proof

(4.9)

Clearly u*isa linear functional on C(G*,C). For each ¢ we have

flel? du* = [op* du* = (2 * @)(e) = [|a|2du > 0

Therefore p* is a positive measure on G*. For each « in G* we have, by (4.5),

[T(a)e du* = F*(T(a)e)(e) = (ap)(e) = p(e) = [ du* Thus x* 1s a Haar measure on G*.

The formula (4.8) for u* leads to a formula for the inverse Fourier transform, analogous to the formula (3.7) for the Fourier transform:

2(x) = (T@)@)(e) = T*(£*)(e) = [E*e du*

(¢ € C(G*,0))

We extend the map ¢ — @ to all of the set L;(G*) by defining

px) = [$*odp*

(¢ € Li(GY),z € G)

Then ¢ € C,(G,C) forall ¢in L;(G*), by continuity (since g & C,(G,C) whenever ¢ & C(G* C)). The proof, already given, that

F(f*g) = 5()F(g) shows that F*(¢ * y¥) = By (4.9),

F*(o)F*(¥) for all ¢ and ¢ in L;(G*).

lell: = liall

328

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

for all ¢ in C(G* C). By approximation, this equality extends to all ¢ In Li(G*) n Ly(G*). Thus §* is linear on L:1(G*) n Ly(G*), and preserves the norm | [[2. It therefore extends to a norm-preserving linear map, which we also call §*, from L.(G*) into L.(G). Thus §* is defined on L;(G*) and on L2(G*), and on the intersection the definitions agree.

Theorem 5

F* maps L.(G*) onto L:(G).

Proof 1t is enough to show that the image of L.(G*) is dense in L:(G@). To prepare the way, consider ¢ in C(G* C) and f in C(G,C). Then for each g in L,(G) we get

Frg=veh)g=ve)*g=(p*f)*g or of = p *f. Now consider any & in C(G,C), and any ¢ > 0. Choose f in C(G,C)

with ||f * h — hl]2 < e Since b € C,(G*,C), there exists ¢ in C(G*,C)

with [[h — ¢|le < €l f]s"%. Then

b= offls = llh —Fxplle < |Ih —F*hls+ [If* — o)l < e+ [lvih 0 there exists & in Le(®)

with ||g — h?||i < e. Choosf ein S with ||f — Al < e. Then

lg — 7l < llg — B2l + [|f* — A% < e+ ([fll + Rl — &[2) < e+ @[hlls + e)e Since e is arbitrary, ¢ is in the closure of S2. The set

18

Lemma

S ={fre:f € Li(G), ¢ € C(G*C), 2 € Li(G)} 1s dense in Li(G) N Ly(G)

[in the sense that for each g in Li(G) n Ly(@)

and each ¢ > 0 there exists h in S with ||g — k|1 < eand llg — hll; < €. Proof that

Clearly {¢:¢¥ € C(G*,C)}isdenseinL:(G). By Lemma 17, we see {¢ =

¢:¢ € C(G*,C)}

Li(G) n Ly(@)

there

is dense in L;(G). Moreover,

exists g in Ly(G@)

such

that

for each f in

||[f — f =g|;

and

|f — f * g||2 are arbitrarily small. There also exists ¢ = ¢ x ¢ in C(G*,C) such that @ is arbitrarily near to ¢ in L,(G). Hence ||f — f * ||; and

|f — f * @||. are arbitrarily small. We now prove the inversion theorem.

Theorem 6 For cach hin Li(Q) n L:(G) we have h € Ly(G*) and b = h. Proof

TFirst consider the case where 2~ = f * @ belongs to the dense

set S of Lemma

18. By Lemma

16, we see that A € Lo(G*) and h = &.

FOUNDATIONS

330

OF

CONSTRUCTIVE

ANALYSIS

Next consider an arbitrary & in L;(G) n L2(G). FFor each positive integer

n there exists h, in S with ||A — &.]|1 < n7 ! and ||h — k.|l < 771, by Lemma 18. Thus {h.} converges pointwise to k. Since

lhn = hnlle = [0 — Rallz = [lha — Bnl2 for all m and n, we see that {h.} is a Cauchy sequence in L.(G*), which

therefore converges to h in Ly(G*). Moreover,

in the metric of Lo(G).

Our final task is to prove the famous Pontryagin duality theorem, which states that the natural map z — £ of G onto G** is both a homeomorphism and an isomorphism. To do this we must first prove two lemmas, analogs of Lemmas 11 and 12, that characterize the metric p on G in terms of the behavior of the elements z of ¢ on certain subsets

of G*. The first is a criterion for boundedness. Lemma

19

For each ¢ > 0 and each K* C G* we define

N(K*e)

= {z € G: |a(zx) — 1| < efor all « in K*}

Then N(K*e) is bounded, for each real number ¢, with 0 < e < 1, and each neighborhood K* of the identity element 1 of G*. Conversely, if M is any bounded subset of G and € any positive constant, there exists a neighborhood K* of 1 in G* such that M C N(K*e).

Proof Consider a real hood K* of 1 in G*. Let By Lemma 11, N*(K,1) exists a compact set K

a € N*(K,,1)

number ¢ with 0 < K, be any compact is a bounded subset of C G and 6 > 0 such

and o € N*(K,s).

We

may

¢ < 1, and a neighborhood G*. By Lemma that « & K*

assume

neighborof ¢ in G. 12, there whenever

that 6§ < 1 and

K; C K. Then N*(K,s) C K*. By Lemma 8, there exist compact integrable neighborhoods V and W of e in @G such that KV C W and u(W — V) < (6/4)u(W). Write f = xw. Consider y in K. Then T(y)f = xy-w. Since V C y~'W, this gives

If = TWflly = (W — y='W) + ply='W — W)

= 2u(W — y=1W) < 2u(W = V) < 2 u(W) = 3 |1l

LOCALLY

COMPACT

ABELIAN

for all ¥y in K. Since

TN

= Ifllo = [Iflls

f

GROUPS

331

e < |Iflls and f(1) = [fdu = ||fll;, we have

Take g in C(G) with ||g|ls = 1 and ||f = g||2 > 272||T(f)||. Let L, be support

compact

any

WW-1L,L,—'.

for g. Let

L be any

We shall show that N(K*,¢)

any x in N(K*e).

Assume that x &

—L.

set containing

compact

C L. To this end, consider

If y is any point of G with

(f *xg)(y) # 0, then y € WL, If also (f * g)(xy) # 0, thenxy & WL,, and thus z € WW-1L,L;=* C L. This contradiction shows that if

(f xg)(y) # 0, then (f *g)(xy) = 0. Therefore f x g and orthogonal elements of Ls(G). Hence

|1T(f — T@NIE > IIf g — T(@)f =g} = 2|f= gllz > 2@ T(HD?2

T'(x)f = g are

= TN

There therefore exists « in G* with (4.10)

la(f) = a@)*a(f)| = la(f = T@N| > [TKIl

and thus 411)

1 —a@| =01 = a@)* > [«

ITO] 21

By (4.10),

la(N] = $la(f) — a@)*a(f)] > L[ TH Hence, for all y in K,

JNTDI = 17 = Tl 2 lalf = T@H] = le(] 1 = o) > 3HTHI 1 = a®)! and thus |1 — a(y)| < 6. It follows that « € N*(K,5) C K*. Therefore (4.11) contradicts the fact that + € N(K*,e¢). This contradiction arises from the assumption © & — L. Therefore x+ & L. Hence N(K*,e) C L, and thus N(K*e) is bounded. Consider, conversely, any bounded subset M of G and any ¢ > 0. Let K be any compact subset of ¢, with M C K, and M™* any compact neighborhood of 1 in G*. By Lemma 12, there exists § > 0 such that

if « € M*

and p*(e,1) < 5, then

||a — 1||x < e Thus

K* = {a €

M*: p*(a,1) < 8} is a neighborhood of 1 in G*, and M C N(K*,e). Our next lemma characterizes the metric p on bounded subsets of G.

332

Lemma

FOUNDATIONS

20

Let M

OF

CONSTRUCTIVE

ANALYSIS

be any bounded subset of G. For each ¢ > 0 there

exists 6 > 0 and a compact set K* C G* such that p(x,y)

< e whenever

z,y E M and ||£ — §||gx < 8. Conversely, for each § > 0 and each compact set K* C G* there exvsts e > 0 such that ||£ — §||x+ < 8 whenever z,

y € M and p(z,y) < e Proof

Consider

¢ > 0.

By

Proposition

1, there

exists

a compact

neighborhood L of ¢ in G such that p(z,y) < e whenever z, y & M and xy~! € L. Let Lo be a compact neighborhood of e in G with L¢2L,2 C L.

By the proof of Theorem 2, there exist test functions f and ¢ on G, sup-

ported by L,, such that ||g|l = 1 and ||f * g|]. > 2-V2||T(f)||. Choose

a neighborhood K of ¢ in @ so that ||f — T'(y)f|l: < LI|T()|| for all y

in K. Write 6 = 1. By the proof of Lemma

19, for each x in — L there

exists @ in G* with |1 — a(z)] > 1 and |1 — a(y)| < 1 for all y in K. Hence o« €& K*, where K* is any compact set containing the bounded subset N*(K,}) of G*. It follows that N(K*,6) C L.

Consider elements z and y of M, with ||£ — §||x+« < 8. Then ||£)~! — l||gx < 6, or zy~! € N(K*,5). Hence zy~! & L. By the choice of L, we have p(z,y) < e Conversely, consider 6 > 0 and a compact set K* C G*. By Lemma 11, there exists a neighborhood K of ¢ in G such that K*C N*(K,s). Choose ¢ > 0 so small that zy~! &€ K wheneverz, y € M and p(z,y) < e. Then for all x and y in M with p(x,y) < e and all « in K* we have

(@) — ()| = |a(@y™) — 1] 0. Then there exists » > 0 in A(u) > 0, and a compact neighborhood K** of & in G** & —i(G) whenever A(x) > 0 and y € K**. Let ¢ be any test function on G**, with g(¢) > 0, that is supported by * h)(u) > 0and g * h vanisheson 7(G). Let ¢ be the inverse

Fourier transform of g * &, so that ¢ = jh € Li(G*) n Ly(G*). Also, ol = |lg * || > 0. For all x in G we have

p@) = [£*0 du* = 6(¢*) = (g% H)(@*) = 0 since £* € 72(G). Hence 0 = ||@||s = ||¢||2 > 0. This contradiction proves that p**(u,7(@)) = 0. It remains to show that 7 is an isomorphism. Since ¢ is bijective, it is

enough to show that it is a homomorphism. identity zZy= £7.

This follows from the

PROBLEMS

1. Let G and H be locally compact groups. Show that G X H is a locally compact group, and construct Haar measure on G X H in terms of the Haar measures on ¢ and H. 2. Show that any left-invariant measure on a locally compact group (G is a multiple of Haar measure (not necessarily positive).

3. Let A be an integrable set of positive measure in a locally compact group (. Show that AA~! contains a neighborhood of e. 4.

Exhibit Haar measure on the locally compact group of all 2 X 2 a O real matrices < >, with ac # 0. Show that this group is not b

¢

unimodular.

5. Prove that Proposition 3, for the case p = 1, holds for an arbitrary G (not necessarily unimodular). 6. Does Lemma 8 hold for an arbitrary (not necessarily commutative) locally compact group?

7.

Show

that R can be identified with its own character group and

{z:]2| = 1} with the character group of Z. 8. Find the character group of the additive group D of all rational numbers z such that 27z is an integer if n is sufficiently large (D has the discrete metric p defined by p(z,y) = 1if z = y).

334

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

9. Let u and » be finite measures on G (by which we mean that |ff du| + |ff dv| is bounded as f runs over the set of test functions with

|f]l < 1). Define the convolution u * v of p and v by

Jfd(u*v) = [[f(xy) du(z) dv(y) Show that u * » is a finite measure. Show that convolution of finite measures is associative, and commutative if G i1s commutative.

Prove that there exists an identity element for convolution in 10. L,(@) if and only if ¢ is discrete (which means that for each x in G the

set {x} is open). 11.

Prove that G* is compact if and only if G is discrete.

12.

A continuous function ¢: G — C is positive definite if

Jfe(

) f(x) f(y)* du(x) du(y) = 0

for all f in C(G,C). If h is any element of C(G,C), show that A x A is positive definite. 13.

Let

¢ be any

positive

definite function

on G. Prove

Bochner’s

theorem, that there exists a positive finite measure » on G* such that

o(x) = [a(zr) dv(a) for all z in G. 14. A continuous real-valued function f on G is almost periodic if for each ¢ > 0 there exists a compact subset K of G such that for each x in G there exists y in K with |f(x2) — f(y2)| < efor all zin G. Show that an almost-periodic function f is uniformly continuous on all of @, in the sense that for each ¢ > 0 there exists § > 0 such that |f(x) —

f(y)| < e whenever p(y~'x) < 8. 15. Show that f is almost periodic if and only if for each ¢ > 0 there exist points 1, . . . , ¥, in G such that for each z in G there exists

G. n zi all r efo < | ;2) f(z — ) xzz |f( th wi n) < 0 such that if 4; 1s an analytic function

onU; (1 ||z||z, 2D. Equivalently, (¢ — ¢x)~!1is analytic for is bounded by Do = 2D(||z||z,, + €/2). For have the expansion

where D is the + ¢/2, and is |¢| < (||z|s, + all sufficiently

norm of B. bounded by ¢/2)7, and small ¢ we

€ —goyt = 3 By (a) (page 339), we see that

n=0

Jo) < Do (el +5 )

for all n in Z+. Hence ||z*||Y* < |z||s,, + € for all sufficiently large . PROBLEMS

1. A subset S of a Banach algebra U generates A if ¥ is the smallest closed subalgebra of % containing S U {e}. Let ¥ be a Banach algebra with a single generator xo. Show that the map ¢ — o(z) is a metric equivalence of 2 with {¢(z¢): 0 € =} C C. 2. Call the spectrum X of a commutative Banach algebra ¥ firm (a) if 2 is compact, and (b) if p(Z,,2) — 0 as n — «. Show that if % has a single generator and 2 is compact, then ¥ is firm.

348

3.

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

Let 9 be a commutative Banach algebra whose spectrum Z is firm.

Let 23,

. . . , 2, be elements of Y and € a positive constant such that

(1.13) holds for all fying (1.14).

in Z. Show that there exist 1,

. . . , y» in I satis-

4. Let A be the Banach algebra of all continuous functions on 2| < 1} that are analytic on {z: |¢| < 1}. Show that 2 is firm.

{z:

5. Let G be a discrete locally compact abelian group. Let f be any element of Li(@) such that |f(a)| > ¢ > 0 for all « in G*. Show that there exists ¢ in L;(G) such that (g = f)(e) = 1 and (g = f)(x) = 0 for all z # e.

NOTES Certain classical Banach algebras are not Banach algebras in our sense, because the norm is not computable. Again, L, is an example. It would be interesting to extend the theory to cover such algebras. The fact that we work with partial ideals rather than ideals causes ugly com-

plications. Lemma 1 would not be necessary if we could work with ideals. These complications seem somehow extraneous. It may be that the theory has not found its proper form. Another possibility is that the complications are not extraneous, but that they could be avoided by means of a metamathematical result, to the effect that under appropriate conditions there exists a routine way for constructivizing a given mathematical theory. In certain circumstances we might be content with the knowledge that the construectivization exists. Chapter 11 contains the only material in this book in which it is at all plausible that such a metatheorem could be successfully applied.

APPENDIX A

METRIZABILITY

AND

SEPARABILITY

How much have we lost by restricting our attention almost exclusively to metric spaces and by our liberal use of separability hypotheses? There are really two questions, depending on whether we take the constructive or the classical point of view. From the constructive point of view, we ask whether there is significant mathematics that cannot be done, or perhaps cannot be done in a good way, within the framework we have developed. From the classical point of view, the problem is to see how much of the nonmetric or nonseparable theory can be recovered from our results, and how much cannot. In this appendix, we take the constructive point of view. The situation is easily summarized: Nonmetric spaces and nonseparable metric spaces play no significant role in those parts of analysis with which this book is concerned. To illustrate this point, consider the concept of a uniform space, as developed in Probs. 17 to 21 of Chap. 4. A uniform space at first sight appears to be a natural and fruitful concept for constructive mathematics, a promising substitute for the concept of a topological space. In fact, this is not the case. For instance, just to construct a compact uniform space X, such that the assumption that X is metrizable

349

350

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

leads to a contradiction, seems to be a hard problem. Here is an attempt that fails. Let I be a compact proper interval and S a set such that the assumption that S is countable leads to a econtradiction. Let X consist of all functions from S into /. For all sin S and all f and ¢ in X write p,(f,g) = |f(s) — g(s)|. There is no known choice of S which makes X totally bounded (in the sense of Prob. 18 of Chap. 4) relative to the uniform structure defined by the pseudometrics p;, because the known sets S all have the property that there exist points s and ¢ in S for which we are unable either to construet f in X with f(s) £ f(¢) or to prove f(s) = f(?) for all fin X. This means we are unable to show that S is totally bounded in the pseudometric p, + p;, and thus we are unable to show that S is

totally bounded relative to the given uniform structure. Of course, important constructively defined uniform spaces that are not necessarily metrizable exist: every locally convex space has a natural uniform structure. At first glance, the concept of a locally convex space would appear to be important for constructive mathematics, since examples exist in profusion. However, in most cases of interest it seems to be unnecessary to make use of any deep facts from the general theory of locally convex spaces. For example, the dual B* of a separable Banach space B is most conveniently studied in terms of the double norm, rather than the locally convex structure. As another example, consider the theory of distributions. Let © denote all infinitely differentiable functions ¢: R — R with compact support. I'or each sequence N = {N;} of positive integers and each ¢ in D, write

lelly = i N sup { f o™ (@)|: |z] = & — 1} k=1

n=0

where, of course, the sum is actually finite. These seminorms

|

||»

generate a locally convex structure on ©. With this strueture, D is called the space of test functions. Classically D is complete (in the sense of Prob. 19 of Chap. 4), but constructively the completion of D consists of all infinitely differentiable functions ¢: R — R such that the norm

redeby overcome be could difficulty This N. each for l|¢||» exists fining O, although such a procedure seems slightly artificial. The same difficulty arises, in more acute form, with the dual space D* (the space of distributions). Here again the constructive completion of D* cannot be identified with ©*, and in this case it seems definitely artificial to enlarge D* to make it complete. Although detailed studies are needed, tentatively we conclude that the constructive theory of distributions should rely heavily on the concept of sequential convergence, defined

METRIZABILITY

AND

SEPARABILITY

351

ad hoc both for sequences in D and for sequences in D*, rather than on general theorems about locally convex spaces. A final remark: From the constructive point of view, there is no loss of generality in taking all metric spaces to be separable. Not only are the spaces that naturally arise separable spaces, but the author knows of no constructively defined nonvoid metric space that can be proved to be nonseparable.

APPENDIX B

ASPECTS

OF

CONSTRUCTIVE

TRUTH

The role of contradiction in constructive mathematics deserves further comment. As remarked in Chap. 1, there are two possible points of view. Suppose that in the course of proving a certain proposition P we have constructed an integer n, which we know must have one of finitely many values 1,2, . . . , N. Which of these N values n actually assumes will of course depend on the particular numerieal data for the proposition P. For given data, n is finitely computable. We may prove P, using an argument by cases: there are N cases to consider (call them cases Cy, C., . . ., Cy, depending on the value of n), and we are at liberty to give a separate argument in each case. Suppose that one of the cases C» leads to a contradiction—that is, the assumption n = m allows us to deduce the equality 0 = 1. Then either we may rule this case out on the ground that it cannot occur for any permissible choice of the numerical data, or we may be more meticulous and prove that in this case too the proposition P is valid. Let us examine the first possibility. To rule out the case C,. because 1t leads to a contradiction seems to require a belief in the consistency of constructive mathematics, and so raises the question of the constructi-

352

ASPECTS

OF

CONSTRUCTIVE

TRUTH

353

vist attitude to questions of consistency. For the construetivist, consistency is not a hobgoblin. It has no independent value; it is merely a consequence of correct thought. The consistency of some particular formal system, even if it were constructively proved, would not be a very interesting result for the constructive mathematician. Since consistency 1s a consequence of correct thought, an inconsistency should be regarded as a consequence of incorrect thought. The construetive mathematician, being human, will not be surprised if one of his results leads to a contradiction. In this sense, he does not believe in the consistency of constructive mathematics. An inconsistency will

cause him to examine his thinking to find out where he went wrong. On the other hand, a constructive proof which leads to a contradiction s wrong, perhaps by definition. In this sense, he does believe in the consisteney of construetive mathematics, and therefore has no hesitation in ruling out a case C,, that leads to a contradiction. The practice of ruling out a case C, that does not occur involves no loss in computational meaning, because the character of the proof for the cases that do oceur is not affected. In spite of this, certain skeptics seem to think that this practice compromises the constructivist position. These skeptics are entitled to prove every result in all cases, if they wish, including those cases that lead to a contradiction. We have not chosen this course, because to us it seems silly to insist on finishing a proof that already contains a mistake. Nevertheless, it would be easy to rewrite this book along such lines. In some instances, definitions would have to be modified. Some mathematicians would still be dissatisfied, if the book were rewritten as described, because to them it also compromises the constructivist position to prove a theorem as a consequence of a contradiction. The book could be rewritten again, with these mathematicians in mind. Each occurrence of a contradiction would be cleverly disguised; all cases involving a contradiction would be made to appear affirmative. (The book as it stands undoubtedly contains instances of arguments in which certain cases that are not treated as contradietory do in fact lead to contradictions.)

Contrary to the opinion expressed above, Hilbert thought that a constructive proof of the consistency of a sufficiently powerful formal system F would be of great value, even to the constructive mathematician, because it would guarantee the constructive validity of every theorem T proved in F whose statement (standing by itself) makes

constructive sense. For instance, a constructive proof of the consistency of such a formal system F together with a proof in F of the theorem 7', that every even integer n > 4 is the sum of two primes could auto-

354

FOUNDATIONS

OF

CONSTRUCTIVE

ANALYSIS

matically be transformed into a constructive proof of 7';. On the other hand, a constructive proof of the consistency of F, together with a proof in F of the theorem T'; that there exist 100 consecutive digits equal to 7 in the decimal expansion of =, would not necessarily give rise to a constructive proof of T's, because the statement 7', (standing by itself) does not make constructive sense. (In order for T'» to make constructive sense, an upper bound for the location of the 100 digits in question would have to be given. In other words, computing one digit after another in the decimal expansion of = until we either find 100 consecutive sevens or get tired and quit is not a finitely performable computation.) Hilbert’s implied belief that there are a significant number of interesting theorems whose statements (standing alone) are constructive but whose proofs are not constructive (or cannot easily be made constructive) has not been justified. In fact we do not know of even one such theorem. It is worthwhile to probe the dictum that all mathematics should have computational meaning. It would be a mistake to attempt to formalize the notion of computational meaning. Rather let us ask how the computational content of a given mathematical result is to be realized. From one point of view, a constructively valid proof already constitutes a realization of the computational content of the result being proved. In other words, a proof