General theory of functions and integration 0486649881, 0486152146, 9780486152141

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General Theory of Functions

and Integration

Angus

E.Taylor

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2014

https://archive.org/details/generaltheoryoffOOtayl

General Theory of Functions

and Integration ANGUS

E.

TAYLOR

University of California, Berkeley

and Los Angeles

DOVER PUBLICATIONS, NEW YORK

INC.

a

La Girelle notre balcon sur

Copyright

©

1965, 1985 by

Leman

le

Angus E. Taylor. American and

All rights reserved under Pan

International Copyright

Conventions. Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario. Published in the United Kingdom by Constable and Company, Ltd.

This Dover edition,

first

published in 1985,

is

an unabridged republication

of the second, corrected printing (1966) of the work Blaisdell Publishing

first

published by the

Company, Waltham, Massachusetts, 1965. This edition

incorporates a few additional corrections and emendations by the author.

Manufactured

in the

Dover Publications,

United States of America. Inc., 31 East

2nd

Street,

Mineola, N. Y.

1

1501

Library of Congress Cataloging in Publication Data Taylor,

Angus

Ellis,

1911-

General theory of functions and integration. Reprint. Originally published:

Waltham, Mass.

:

Blaisdell Pub. Co.,

1966.

Bibliography:

p.

Includes index. 1.

Functional analysis.

QA320.T28 1985 ISBN 0-486-64988-1

2.

515.7 (pbk.)

Calculus, Integral.

I.

Title.

85-12862

Preface

This book had It is

the

its

beginning in a course of lectures which

I first

gave in the Fall of 1951.

intended as a basic text in certain fundamental parts of higher analysis. The scope of

book

will

be indicated in subsequent parts of this preface. The book can be used as a

text in a variety of ways, at various levels of study. It

is

kinds of honors courses and in "reading courses," for

also well adapted to use in certain

planned to be read and used by independent study, seeking to acquire a mastery of these parts of analysis. Moreover, the book is arranged so as to provide a natural transition from the classical theory of point sets in Euclidean space, and the theory of functions of one or more real variables, to the more abstract settings in which the ideas of topology, continuous functions, and integration find their most natural development and reveal their essential structures in the simplest and most transparent fashion. Chapter 1, dealing with the real number system and point sets on the real line, provides a basis for the development, in Chapter 2, of general point-set theory in the Euclidean space R k Chapter 2 provides motivation and sets the stage for Chapter 3. The introductory sections in Chapters 2 and 3 explain the carefully planned structural relationship between Chapters 2 and 3. Rather than to repeat or summarize these introductions here, I suggest that the reader examine these introductions as if they were part of this preface. Well-prepared students will not need to study Chapter 1 intensely as new material. Such students can begin directly with Chapter 2. As a teacher and counselor of students I have observed during the past twenty five years that the available expositions of the theory of integration have fallen short in various ways when measured against the desires of students to learn reasonably quickly and effectively what they need in order to qualify for more advanced graduate study and for research work. as a guide for further reading by students

who

it is

are,

.

The character of this book is to a large extent the outcome of my attempt to bring together in one volume a judicious blend of the particular and the general, of the concrete and the abstract, as

an aid to graduate students and as a guide to the further expansion of

their

mathematical horizons.

There has been a proliferation of ways to approach the theory of integration.

When

students do not carry their study of integration far enough, the existence of the various

approaches can lead to various

difficulties later iii

on when students schooled

in the various

Preface

iv

common

approaches enter a

course of study in which extensive use

theories of integration. There are students

theory (a la Riesz or Daniell),

who know

little

spaces. If they

know about find

it

may

is

made of modern

studied integration without measure

only the rudiments of general measure theory, and

or nothing about the detailed theory of Lebesgue measure in Euclidean

know

only the bare outlines of

integrals,

necessary to

it

Or,

who know

who have

it is

for

how

to

them more than a short

know about Lebesgue measure

be that a student

is

found measure theory on what they exercise to work out what they may

in its classical setting.

attracted to functional analysis, but has learned his inte-

gration by the classical route: measure theory and then integration of functions of one real variable. Perhaps he has even spent

defined on a compact interval.

what

He

most of

then finds

his it

time on bounded point sets and functions

necessary in functional analysis to cope with

and formidable extension of this theory: integration of funcon general measure spaces. Moreover, he encounters the idea that certain types of linear functionals induce measures (in some instances, even, the functional themselves are called measures), and he sees reference to the fact that the linear functionals are representable as integrals of a Lebesgue type. Such a student may feel overwhelmed in trying to decide where to begin in filling in his knowledge. This book aims to help students round out their knowledge of integration and measure theory and to understand the approach via linear functionals as well as the older, more standard approach via measure. The subjects are done fairly completely from two points of view: once with measure theory first and then integration based on measure, and once by the Daniell method of integration first, with subsequent theory of the measure induced by the integral. Then the two approaches are linked, and the ultimate complete theories are shown at first looks like a vast

tions defined

to be equivalent under rather general conditions. Chapter 5

is

devoted to the theory of

on the prior development of the theory of measure in Chapter 4. The theory in Chapter 4 includes both the abstract theory of measures and outer measures, and the concrete theory of Lebesgue measure in Euclidean space of ^-dimensions, as well as the theory of Lebesgue-Stieltjes measures on the real line. Chapter 6 begins with the general abstract theory of integration and the induced measure, by the Daniell method. A comparison is then made with the result of using this induced measure to develop a theory of integration, as in Chapter 5. The last part of Chapter 6 is concerned with the particular case of measures (among them classical Lebesgue measure) induced in Euclidean space by certain integration, as based

linear functionals.

The general theory of integrals and measures is continued in Chapters 7 and 8, with the theorems of Fubini and Radon-Nikodym among the main objectives. The general Riesz representation theory for continuous linear functionals on C x (R k ) occurs in Section 8-7 (as

Theorem 8-7

Much

IV).

of Chapter 9 could be studied before Chapters 4-8. The introduction to Chapter 9

explains this in

more

detail.

In concluding this preface,

I

wish to comment on the names used to describe the parts of

analysis dealt with in this book. Earlier generations of graduate students studied "the theory

of functions" in two distinct parts: the theory of functions of a complex variable, and the more real variables. In fact, of course, the theory of functions of

theory of functions of one or a complex variable

is

not a theory of arbitrary functions of a complex variable, but a

theory of analytic functions. Because of this stringent limitation, the theory free of pathology,

elegant.

and the subject matter

The theory of functions of

real

is

relatively

main lines is remarkably neat and variables, on the other hand, came under in its

—

Preface

v

the influence of the general investigations of point-set theory, the study of discontin-

uous functions, and the abstractionist tendencies of Frechet, Hausdorff, Hahn, and others. In the outcome, a good part of the traditional "theory of functions of real variables" was displaced by a theory of more or less arbitrary real-valued functions defined on an abstract space of some sort a metric space, a Hausdorff space, or a measure space. More recently, abstraction has gone still further, and the function values need no longer be real. Instead,

—

they

may

be in a real or complex linear space of some kind, with some sort of topological

For some years now it has been a rather form of the old theory of functions of a real variable structure.

functions."

Now,

it

is

true that the real

common

thing to refer to this evolved

as "real analysis" or "the theory of real

number system

plays a vital role in this part of

no less vital in the theory of analytic functions, for the topology of the complex plane derives from the real number system. The theory of analytic functions of a complex variable does, of course, lean heavily on a special fact about complex numbers the fact that they form a field. In this respect the complex plane must be distinguished from the two-dimensional real Euclidean space. Likewise, some forms of the theory of integration lean heavily on the special fact that the real numbers form a totally ordered system. But the true distinction between the theory of analytic functions and the kind of analysis considered in this book is not well drawn by the use of the adjectives "complex" and "real," respectively. Rather the kind of analysis here considered is characterized by its generality and by the absence of highly restrictive assumptions about the properties of the functions which are considered. It is on this account that I have named the book General Theory of Functions and Integration, rather than give it some title which invokes the word "real." analysis.

But

its

role

is

Angus Los Angeles

E.

Taylor

Acknowledgments

of the writing of this book was done during my 1961-62 sabbatical leave from the University of California, Los Angeles. I offer sincere thanks to the UCLA Administration for allowing me to interrupt my duties as Chairman of the Department of Mathematics to take this leave. Without it quite possibly the book would never have been written. During this period of leave I lived in the charming village of Lutry, on Lake Geneva close to the city of Lausanne, Switzerland. I was accorded the use of the Mathematics Library at the Ecole Polytechnique of the University of Lausanne, and I enjoyed the courtesies and kind help of various members of the Mathematics Faculty there. I am especially grateful to Professors Blanc, Methee, de Rham, and Vincent. The book has been in my thoughts since 1952, and has been much influenced by my teaching and my contacts with graduate students. For the appreciation and responsiveness

The great bulk

of

many I

students

I

am

typed manuscript with late

deeply grateful.

wish also to acknowledge

and lamented

skill

my appreciation and

and patience.

secretary, Mrs.

Finally,

who prepared

the

have been helped in many ways by

my

thanks to Elaine Barth, I

Mildred Webb, and by her successor, Mrs. Helene Gale. A. E. T.

Contents

1.

2.

The Real Numbers. Point Sets and Sequences 1-0

Introduction

1

1-1

The

2

real line

1-2

Notation and terminology concerning

1-3

Construction of a complete ordered

1-4

Archimedean order. Countability of the

1-5

Point

sets.

sets

rationals

Upper and lower bounds

1-6

Real sequences

1-7

The extended

7 12

field

16 21

24

real

number system

29

1-8

Cauchy's convergence condition

32

1- 9

The theorem on nested

34

intervals

Euclidean Space. Topology and Continuous Functions

2-0

Introduction

37

k

2-1

The space R

2-2

Linear configurations in

2-3

The topology of

2-4

Nests, points of accumulation, and convergent sequences

56

2-5

Covering theorems

61

2-6

Compactness

65

2-7

Functions. Continuity

68

2-8

Connected

76

2-9

Relative topologies

38

Rk

sets

2-10 Cantor's ternary

Rk

set

43 47

83

86

1

Contents

viii

3.

Abstract Spaces

3-0

Introduction

89

3-1

Topological spaces

90

3-2

Compactness and other properties

96

3-3

Postulates of separation

102

3-4

Postulated neighborhood systems

105

3-5

Compactification. Local compactness

107

3-6

Metric spaces

1 1

3-7

Compactness

in metric spaces

121

3-8

Completeness and completion

124

3-9

Category

127

3-10 Zorn's lemma 3-11 Cartesian product topologies

134

3-12 Vector spaces

140

Normed

144

3-13

4.

131

linear spaces

3-14 Hilbert spaces

153

3- 15 Spaces of continuous functions

164

The Theory of Measure 4- 0

Introduction

177

4-1

Algebraic operations in

4-2

Rings and a-Rings

R*

178

180

4-3

Additive set functions

185

4-4

Some

188

4-5

Preliminary remarks about Lebesgue measure

properties of measures

R

k

4-6

Lebesgue outer measure

4-7

The measure induced by an outer measure

4-8

Lebesgue measure

4-9

A

general

in

in

Rh

method of constructing outer measures

4-10 Outer measures

in

R

from monotone functions

4- 11 The completion of a measure

5.

The Lebesgue

189 191

196

203 215 218

224

Integral

5-0

Introduction

226

5-1

Measurable functions

229

5-2

The

240

integral of a

bounded function

1

ix

Contents

6.

5-3

Preliminary convergence theorems

248

5-4

The general

252

5-5

Some

definition of

an integral

basic convergence theorems

260

5-6

Convergence

in

measure

265

5-7

Convergence

in

mean

270

5-8

The LP spaces

274

5- 9

Integration with respect to the completion of a measure

278

Integration by the Daniell

6-0

Method

Introduction

28

6-1

Elementary integrals on a vector

6-2

Over-functions and under-functions

284

6-3

Summable

288

6-4

Sets of

6-5

Measurable functions and measurable

6-6

The N-norm

6-7

Connections with Chapter 5

6-8

Induction of Lebesgue measure in

6-9

Arbitrary elementary integrals on C^/?*)

measure zero

6- 11 The class

8.

of functions

functions

&

282

292 sets

297 300

R

6-10 Regular Borel measures in

7.

lattice

303

Rk

k

v

310 313 319 321

Iterated Integrals and Fubini's

Theorem

7-0

Introduction

7-1

The Fubini theorem

7-2

The Fubini-Stone theorem

7-3

Products of functions of one variable

7-4

Iterated integrals

324 for Euclidean spaces

and products of Euclidean spaces

326 329

334 336

7-5

Abstract theory of product-measures

339

7- 6

The

345

abstract Fubini theorem

The Theory of Signed Measures 8-0

Introduction

348

8-1

Signed measures

349

8-2

Absolute continuity

356

X

9.

Contents

8-3

The Radon-Nikodym theorem

358

8-4

Continuous linear functionals on L%u)

361

8-5

The Lebesgue decomposition of

364

8-6

Alternative approach via elementary integrals

366

8-7

The

371

a signed measure

decomposition of linear functionals

Functions of

One Real

Variable

9-0

Introduction

379

9-1

Monotone

379

9-2

Vector-valued functions of bounded variation

382

9-3

Rectifiable curves

388

functions

9-4

Real-valued functions of bounded variation

390

9-5

Stieltjes integrals

392

9-6

Convergence theorems for functions of bounded variation

398

9-7

Differentiation of

9-8

Absolutely continuous functions

410

BIBLIOGRAPHY

423

LIST OF SPECIAL

INDEX

monotone functions

SYMBOLS

402

429 431

CHAPTER ONE

The Real Numbers. Point Sets and Sequences

1-0

Introduction

The foundation stone of view and

in actual subject

analysis

is

the real

number system. Generalizations

in point of

matter sometimes take us far from the real numbers, or even far

from real-valued functions of a real variable. But most of the structures which are built up in some way related to the real numbers. The system of all real numbers can be looked upon geometrically by thinking of the numbers as the points on a line. This line is

analysis are in

then called the real

The

line. It is

a one-dimensional Euclidean space.

some part of it, occupies our attention when we study a function of one When we pass to functions of two or more real variables we need to study

real line, or

real variable.

compound structures formed by using the real Hence it is fundamental to begin by gaining a firm understanding of the properties of the real number system. In this chapter we describe the system of real numbers axiomatically as a complete ordered field (Section 1-1). Since the rational numbers play a very important role in the real number system, it is essential for a student to be aware of the fundamental facts about rational and irrational numbers. These higher dimensional spaces. These spaces are line or parts of

it

several times over.

are considered in Section 1-4.

and taste to decide how far a student should go, in the early toward the actual construction of a model complete ordered field. In Section 1-3 we describe the construction of such a model, using Dedekind's method of sections in the ordered field of rational numbers. This section requires the use of the most elementary notions of general set theory. These notions, and the common notations for set operations (union, intersection, difference, complement) are explained in Section 1-2. The ideas of point-set topology are not taken up in this chapter, however. Such ideas come in Chapter 2, where they are developed in a manner as free as possible of entanglement with It is

a matter of judgment

stages of his work,

dimensionality.

Our main concern with

point sets on the real

line, in

Chapter

1, is

in the dis-

cussion of least upper bounds and greatest lower bounds, in Section 1-5.

Chapter ideas

1

concludes with several sections on sequences. The aim

and theorems

in

is

a logical order without unnecessary ramification. 1

to present the basic

Much

or

all

of this

The real numbers. Point

2

may

not be

new

inferior

—

+ oo, — oo) may be new to many students.

number system (by adjunction of the symbols is

and sequences

depending on his schooling in advanced and superior limits and of the extended real

to the student at this stage

However, the discussion of

calculus.

sets

It

in Section 1-7.

DISCUSSION OF THE CONCEPT OF COMPLETENESS book we choose

In this

to define the concept of completeness of

requirement that to each section (L, R) in the the largest element of

L

field

or the smallest element in

an ordered

field

corresponds an element which

R

(see Section 1-1).

As

is

well

is

by the either

known, an

is this The field shall have the property that each nonempty set in the field which has an upper bound shall have a least upper bound (see Section 1-5). There are still other ways of characterizing the property of completeness. Let us consider the following properties which may be possessed by an ordered field F:

equivalent alternative requirement

1.

F is

2.

Every nonempty

3.

(a)

complete, as defined in Section 1-1. set S in F with an upper bound in F has a least upper bound in F. archimedean (see Section 1-4). Every Cauchy sequence in Fhas a limit in F(see Section 1-8). F is archimedean. If {/„} is a nest in F, there is an element of F which is in every /„ (see Section 1-9).

F is

(b) (a)

4.

(b)

The

Properties 1

:

—*

2 —*

2 (see 1

(see

1, 2, 3,

4 are equivalent. This

Theorem 1-5 Problem

1

is

shown by

the following

list

of implications.

I)

in Section 1-5)

-» 3(a) (see Theorem 1-4 I) 2 -> 3(b) (see Theorem 1-8 I) 2 -+ 4(b) (see Theorem 1-9 I) 3(b) -> 4(b) (see Problem 1(a) 1

4-^2 Because 3(a)

Problem the same as

(see is

in Section 1-9)

1(b) in Section 1-9). 4(a), the

foregoing implications are sufficient to show the equiv-

alence which

is

claimed for the Properties

Property

is

not as convenient in practice as Property

it is

1-1

in

1

some ways more

The

1, 2, 3, 4.

2, for the

purposes of analysis, but

intuitively natural at the outset.

real line

presumed that a person studying this book already has some knowledge of the this knowledge extends beyond what is ordinarily encompassed in a course based on a beginning text in calculus. In particular, it is presumed that readers of this book have studied continuous functions of one real variable, using the "e-d definition," and that they have become acquainted to some extent with such things as least upper bounds, greatest lower bounds, points of accumulation, and Cauchy's necessary and sufficient condition for convergence of a sequence of real numbers. A reasonable prerequisite ought to include at least the equivalent of Chapters 2, 3, and 14 in the author's Advanced Calculus (Taylor and Mann [l]t). Accordingly, the discussion here of the real number system and its geometrical representation as the real line, is briefer and less detailed It is

rudiments of analysis, and that

t

Numbers

in square brackets after

an author's name

refer to the bibliography.

The real

line

3

than such a presentation would be for complete novices. However,

in a logical

sense,

all

the

essentials are here.

The collection of all real numbers can be characterized by the statement: The real numbers form a complete orderedfield. To elaborate the meaning of this we must explain what a field is, what it means for a field to be ordered, and what it means for an ordered field to be complete.

FIELDS In

modern algebra we say

that a collection

F of elements

a, b, ... is

a

field if

each pair of

elements (distinct or not) can be combined in two ways, called addition and multiplication,

each combination yielding an element of F, these rules of combination being subjected to certain laws, as follows: (I)

The

+ b and called the sum, and the and called the product, satisfy the com-

additive combination of a, b, denoted by a

multiplicative combination of a, b, denoted by ab

mutative, associative, and distributive laws:

a

+

a

(II)

F contains

(b

+b= + c) =

+ a, (a + b) + c, a(b + c) = ab +

distinct special elements,

a

+

= ba, a{bc) = (ab)c, ab

b

0

=

and

a

ac.

denoted by

a

1

such that

0, 1,

=

a

for each a in F.

Corresponding to each a

(III)

in

a

F there

+

is

(-a)

an element

=

in

F denoted

0.

(IV) Corresponding to each a in F, with the exception of

denoted by a-1 such that 3

by —a, such that

0,

there

is

an element

in F,

,

a{a~ 1 )=\.

In terms of the notations introduced in (III) and (IV) two further rules of combination (subtraction and division) are defined as follows:

a

—

+ —

b

is

a

-

is

ab' 1 (assuming b

(

£>),

0).

b

We

on the details of deducing from the definition of a field all of the and the rules governing manipulations. The facts and rules are common knowledge, and the complete story of what can be proved about fields, as well as the development of technique of proof, are part of the study of modern abstract algebra. shall not dwell

familiar algebraic facts

ORDERED FIELDS

A field

Fis said to be ordered

if

there

is

a special class

P of elements

If a is an element of F, then either a is in P, or a one of these three situations actually occurs; if a and b are in P, then so is a + b; (2) (1)

(3)

if

a and b are

in P, then so is ab.

=

0,

or

in

—a

Fsuch is

in P,

that

and exactly

The real numbers. Point

4

It

follows from (1) and (3) that a 2

be expressed either as aa or 1

P if a ^

in

is

(— a)(— a).

and sequences

sets

and

1

—a is

For, either a or

0.

Since 0 5^

1

=

l

2 ,

in P,

and a 2 can

follows in particular that

it

in P.

is

P

The elements of

we

element,

are called the positive elements of the field.

say that a

Each nonzero element

negative.

is

is

When —a

is

a positive

either positive or negative, but

not both.

Given two elements a, b in F, we write a < b and say "a is less than b" if b — a is in the Then 0 < b means the same as "b is in P." The following assertions about the

class P.


*, y n

^

PROBLEMS 1

(a)

.

Prove that a convergent sequence

impossible

A^

Prove Theorem 1-6

3.

If

>

bounded,

is

0,

1

-* x,

such that

5.

Each

divisible

by

e if

—

\x n

is

n

>

positive integer values defined for 1

x\

N

lt


?*)

J

Therefore \l/2

From this

result

2

(6) is

1/2

2 (ft + ^)

2

+

we

once infer the truth of (6), by a simple argument which we leave to the

at

^)

student. (The case in which f4

of

\l/2

/




1.

M as

+—

Then least

m

,

S'

where is

m

and n can be

the set consisting of

upper bound, and suppose

Euclidean space. Topology and continuous functions

50 that

M

is

S such

M

member of S. Then e S'. For, e < x < M. The fact that this

not a

M—

that

>

if e is

must

there

0,

exist

>

true for each e

some point x

in

0 clearly implies that

MeS'. Suppose a

S

set

has the property that S'

S

S

e2,

< it

an open box with center

at a.

For the

simply the open interval of length 2e1 with center at a x In the

is

.

,

the interior of a rectangle, as

shown

in Figure 9,

and

in the case

k = 3 the open box under discussion is the interior of a rectangular parallelepiped bounded by the three pairs of parallel planes £ = a, ± e„ i = 1, 2, 3. = a, + e, and ^ = a, — e,- define parallel hyperplanes (parallel In the general case z

= 0). The set of points x for which — e < | < + e — a J < e,) is the set of points between the two parallel

to the coordinate hyperplane (or,

what

is

same

the

thing, |f

a.

t

f

i

£

•

i

is an open set. The open box defined by (5) is the intersection of the k formed in this way, one set between each pair of parallel hyperplanes. Hereafter, whenever we refer to open boxes we take it for granted (unless there is explicit mention to the contrary) that the boxes are formed in the manner here described, using

hyperplanes. This

open

sets

hyperplanes parallel to coordinate hyperplanes. In Chapter 4

we

shall

speak of open boxes as open

intervals,

even when k

>

1.

Euclidean space. Topology and continuous functions

54

£2

«2

+

«2

a2

1

(ai,a 2 )

— €2

«2

—

ai

£1

Figure 9

The open box defined by

On

(5) is

the other hand, the open

numbers

smallest of the

Figure It is

box

e lt

.

.

.

contained in the open sphere S(a;

e) if e is

open sphere S(a; 6) if For the case k = 2 the situation

(5) contains the ,

e fc

.

Rk

in

,

in

order that

it

be convergent

0 correspond some positive integer

N < m and N
, S, and a function / with domain and range in (perhaps, but not necessarily, equal to) S. We sometimes abbreviate this by saying "consider

2

the function

f\2^>

S."

Sometimes we indicate the definition of a function without using a symbol for the function, by the following device. Instead of writing "the function / defined by f(x) = sin x," we may write "the function ^->sin;c." The arrow here indicates the idea of the correspondence; the functional value corresponding to x is sin x. The domain of the function must of course be made clear in some way.

Example J. Let k = 2, 1 = with x = (fi, f 2 ), y =

Let 3> be

3.

(x, y)

f2 f x ,

+

all

of

fa , f*

R 2 and ,

+

f").

let

the function

In this case,

if

y

/ consist of all pairs

=

(r}

u

rj

2 , rj 3 ),

we can

write

m=

=

fi

These equations

(1)

become

function as a

into a part of

(2)

fA>

V2

(2) are what equations mapping of the plane R 2

+

h,

=

V3

ff

+

£2-

We can think

for the present case. jR

3 .

The range of

of the

the function can be

visualized as the surface of which (2) are the parametric equations (with fx £ 2 as parameters). ,

Example

2.

Let k

=

3, /

= 2.

Let

+

(3)

^ be the set of points (| II

with the exception of the point

(0, 0, 1).

(4)

x

ri

=

is,

(f,

-

-

\f

£ 2 | 3 ) for ,

which

=h

Let the function be defined by the equations

—

l -

,

ri

2

=

- f. let the function consist of all pairs (x,y), where x = 1

That

+

l5

f.

1

£ 2 |3 ) ,

and j

is

given by

(4).

/

/

/

Euclidean space. Topology and continuous functions

70

We

have here a function which maps

graphically onto the whole plane

R

2

all

except one point of the sphere (3) in

(so that the range 21 in this case

is

R

2 ).

R3

stereo-

We shall not

go

into the details about stereographic projection.

Example 3. Let k = 3, / = and let the function/ make y

3.

=

(5)

Vi

and

r

+

y\\

= ^+

2




+

% = -r

,

In this case the range of/ consists of all points y for which 0

The function

defines an inversive

1,

where

mapping of part of

R3


x and if x n 5^ x for infinitely many S which contains all the xn 's.

any

set

X, if {x n }

The propositions

is

a sequence

in S,

and

if x n

—

x, then

x£

S.

x

values of n, then

an accumulation point of

is

theorem are not true for topological spaces

in the following

in general,

not even for Hausdorff spaces. In conjunction with the foregoing propositions they give

S and

characterizations, valid in metric spaces, of the closure of

the derived set of

S by

means of convergent sequences.

theorem

3-6

(a) If x

(b) If

xn

e

Let

I.

S be a

ye

set in

a metric space X. Then:

a sequence {x n }

S, there exists

in

S

a sequence (x n )

S', there exists

such that x n

S

in

—> x. x1} x 2

such that

,

.

are all distinct,

.

and x n -> y.

5^ y,

Proof of (a). If x e S, the open sphere with center x and radius Ijn contains some point of S, say x n Then D(x n x) —* 0, and hence x n —* x. .

Proof of write

D(x n ,y)

We

,

is

(a)

III,

except that

we now

— y\\.

in place of \\x n

remark that

The argument

Theorem 2-4

Exactly the same as the proof of

(b).

holds true in any space which

like that in the

the

satisfies

proof of Theorem 3-2 V.

first

If the

axiom of countability.

space

is

also a

T

1

space,

(b) holds true.

The

role of sequences in connection with

compactness

in metric spaces

is

discussed in

Section 3-7.

DISTANCES BETWEEN SETS Let

S

be any nonempty

metric space X, and suppose x e X.

set in the

D(x, S)

(12)

=

We

define

inf D(x, y),

veS

and

between x and S. Likewise, if S1 and S 2 are nonempty sets, we S2 ) between 5 and S 2 as the infimum of the numbers D(x, y) as Sx and S 2 respectively. It is easy to see that

call this the distance

define the distance D(Si,

x and y vary over

X

,

D(S U S 2 )

(13)

We It is

can see that D(x, S)

=

0

possible to have Z)(5l5

example,

where x

in

=

R2

is

=

inf D(x,

xeS

S 2 ).

l

equivalent to x e S.

S2)

=

0 even when

5 X and S 2

are disjoint closed sets.

For

let

S1

= {x:|

2

l},

(f lt £ 2 ).

By using the notion of distance from a point is normal. See Problem 3.

that a metric space

to a set

it is

possible to prove rather easily

Y

Metric spaces

117

CONTINUITY

X

If

1

and

definition

f X :

>

X

2

are metric spaces with metrics

of continuity

X

equivalent to

is

X

continuous at a point x 0 in

2 is


f(x). See Problem

if

function

0 corresponds a

/ is

8.

ISOMETRIES If

A'and Fare metric spaces with metrics D,

^[/(*i)»/(*2)]

=

is

i

= /(x )

mapping. Observe that /(x x ) isometry

Pa

^2) f° r eacri

f *i>

Iff(X)

—

Yin

this case,

d, i

and if/:

X -*

we

/ an

n X>

implies Xj

2

obviously continuous, and so

/(X)onto X.

*2

=

ca H

x 2 so ,

that the inverse

the inverse, for

is

we say

that

Fis a mapping such that isometry, or an isometric

f~

x

exists.

An

an isometric mapping of

it is

X and Y are in isometric correspondence.

UNIFORM CONTINUITY The concept of uniform continuity metric spaces, the function /

>

e

0 corresponds some d

This concept

is

X

:

>

is

X

-*

1

A\ and X2 are on Xx if to each whenever D^x^ x 2 ) < d.

defined exactly as in Section 2-7

2 is

:

If

said to be uniformly continuous

D

0 such that

2


7and/n X We say that {/„} converges to / uniformly on X if to each e > 0 corresponds some positive integer N such that D(fn (x),f(x)) < e when n > N, for each Let

:

(n

=

1,2,...) are given.

x

in

X. Concerning this concept

we have

:

the following important theorem about the

transmission of the property of being continuous.

It is

a generalization of a familiar theorem

of advanced calculus.

theorem

3-6

Suppose X, Y,

II.

with {fn } converging to

x 0 Then f is continuous .

Proof. Suppose

fN if

is

continuous at

x G U. Then x e

D(/(x),/(x 0))




x„,

{/„},

and f are given as

in the

f uniformly on X. Suppose that each fn at

x0

is

preceding paragraph

continuous at the point

.

0. Select TV

so that, for each x,

we can choose

a neighborhood

D(fn (x), /(x))

U of x„ so that




N. Since

Z)(/V (x),/V (x 0 ))




e

that these metrics are equivalent

0 and y e X, there corresponds a 5

>

0 and an n

>

if

0 such

that

{x

:

D

x


i(x n x) -+ 0, ,

,

,

:

:

Let

S =

|J U(x),

T=

xsA 4.

Show

(J

V(x).)

xe B

that a metric for R*, equivalent to that of

1/00 ~f(y)\> where/: R* /(-«) = -1. 5.

^R

is

Prove that the metrics d and

=

defined by/(x)

D

in

C

l

[0, 1],

Example

x(l

+

2,

Ixl)"

1

can be defined by D(x, y) if

x

defined by (11) and

is

real,/(

+ oo) =

(9), respectively,

1,

=

and

are not

equivalent.

S j£ 0 show

that the

mapping x

-* D(x, S)

continuous.

6.

If

7.

Let A' be a metric space with metric D. Define a function

,

is

D

1

by

D(x, y)

Show

that Z) x

always.

Thus

is

X

is

on X, and bounded when the metric

also a metric

equivalent to D. Observe that

that

it

D

used, regardless of whether or not

x is

is

it

D

1

(x,j)


0, b > 0, and c > 0, and if a + b > c, then

+

1

Begin by proving 8. (a) If

/ X :

X

-*

1

2,

continuity at a point (b)

Show

a

+

1

b

~

+

1

c

this.

that

still

X

where

x of

X

1

1

X

and

2

are metric spaces, prove that the

e,

^-formulation of

equivalent to the definition phrased in topological terms.

is

another equivalent formulation

xn

->

the following one (the sequential formu-

is

x always

Are the sequential and topological formulations of continuity equivalent under certain conditions when X 1 and X2 lation):

/is continuous

at

x

and only

if

if

implies f(x n ) -»f(x).

are not both metrizable? 9.

This problem deals with some inequalities which are useful for the study of certain examples

of metric spaces. The important results contained here are presented in such a way that they

will

be

readily available for reference in connection with other parts of the text. (a) If

A >

B >

0,

0,

>

a

0,

>

p

and a

0,

A^BP
0.

to the theorem of Baire.

If

II.

X

a complete metric space,

is

it

is

of the second category (as a

subset of itself).

Proof. Suppose, to the contrary, that

X

nowhere dense. As a matter of convenience B(x 0

>

0 and x Q e

X

;

is

the union of

Sx S 2 ,

,

.

.

.

where each Sn

,

is

in notation let us define

r)~{x:

D(x, x 0)


j>) with respect to ^ —> z 2 we call z 2 the iterated limit of j (x, j), first with respect — to Jt There is another iterated limit z 3 in case (/(•, j),

call Zj the

z3

,

.

and (/(x,

•),

Jf) -> g(x) for each x.

necessary to prove that

if

for

some

Show

fixed x,

that (^,

^)

N and W

x

->

z.

In the

we know

that

when j£iV, then ^(x) £

Suppose that (/(x, •)> ^) converges to ^(x) uniformly on X, and that {g,^) -*-z. Then ^) -» z. Prove this. 7. Suppose (F, J() is a directed function of Cauchy type, the values of F being in the complete metric space Z. Using the following suggestions, prove that (F, converges to a limit z in Z. 6.

(/,

Choose

M

n

1

so that D[F(x x ), F(x 2 )]




ab ix(o + xor dt)

/




y wor dtj

.

dealing here with continuous functions, but Minkowski's inequality

much

for a

[b

Lebesgue

larger class of functions, with

integrals.

satisfies the

inequality:

is

actually valid

Minkowski's inequality

proved with the aid of Holder's inequality for integrals: Suppose p

>

1

and p

,

=

rb

I

\w

rb

\x(t)y(t)\dt< []\x(t)\'dt)

(7)

i

rb

y

)a

.

p

Then

is

p

~

1

\i/*>'

WOr'

dtj

.

Here also the inequality is valid with Lebesgue integrals. However, at this stage we do not presume any knowledge of the Lebesgue theory of integration. A student should be able to prove first (7) and then (6), using simple facts about integrals of continuous functions, and assuming x and y are in C[a, b]. The proofs resemble the proofs in (b) and (c) of Problem 9, Section 3-6. For a discussion of the inequalities of Holder and Minkowski in a more abstract setting, see Problem 5, at the end of this section of the text. It is

show

possible to

llxll,


0. Now, if e > 0, the fact that {x n } is a Cauchy sequence in C[a, b] implies that there is some TV such that ||x„ — x m < e if ./V < m and N < n. Consequently, for such m and n, and for each \x n (t) - x m (t)\ < e. limit of the

sequence {*„}.

have to prove,

first,

that

\\

In this inequality (11) if

N
oo. As ~ X(0\ < e

a result

we can

assert

n. Since ./V does not depend on we see that {x n (t)} converges to x(t) uniformly on Hence, by Theorem 3-6 II, x is continuous on [a, b]. The inequality (11) now shows

Normed

linear spaces

149

Figure 30 ||x„ — jc|| < e if n >_ N; that is, x n —> x. This shows that C[a, b] when the supremum norm is used. We emphasize that convergence, supremum norm, is uniform convergence on [a, b].

that

Example the

norm

4.

Let

simplicity, with a

sense of the

X be the class of functions C[a, b], made into a normed linear space with (3). Then X is not complete. We shall demonstrate this, for = 0, b = Let x„ be the element of C[0, 1] defined as follows: 1.

0}, evident that {x n }

It is readily

is

.

.

=

x2

.),

=

£

a countable basis for

I

if

1

1

In

.

fact,

—2

x as « -> oo, It

and

= I

0

16

easy to see that the expression of

it is

jc

in the

form

can be proved that any Banach space with a countable basis

is

(14)

is

unique.

separable.

It is

a famous

unsolved problem of long standing to determine whether for every separable Banach space there exists a countable basis. It can be a matter of rather intricate analysis in particular cases to determine whether a given sequence {x n } in

Show that two norms

and

on

X is

in fact a countable basis for X.

PROBLEMS X are equivalent if and only if to each a >

0 corresponds x 2 0 such that {x ||x|| 2 < b) c {x ||jf x < a}, with a corresponding situation when the roles of the norms are exchanged. Hence prove that the norms are equivalent if and only if there exist ||jc|| positive constants m, such that m ||jc|| 2 {oix) = 1 as follows

(1) if

,

;

(2)

0

if

X

,

,

2.

1 if

>

n

if

2,

Hilbert spaces

Among normed certain

linear spaces there are certain ones in

way with a function of two

variables, called

which the norm

is

associated in a

an inner product. Such spaces can occur

with either the real numbers or the complex numbers as scalars. If ^ is a complex vector space, a mapping/ X x X-+ C called an inner product in X if the following conditions are satisfied: f{x + x s3) =/(* x +f(x x 2. f(x x ) = /(x x (the bar denoting complex conjugate), 3. /(ax b x ) = af(x u x 4. f(x, x) > 0 and f(x, x) # 0 if x ^ 0. (Observe that f(x, x) must be real, by the is

:

1.

2,

x

2,

2

3)

i,

x,

2,

3 ),

2)

2 ),

2

second condition.)

X

If

is

a real vector space, a

Conditions 1-4 are 2

when

satisfied.

X is a real space. To X together, writing

complex

mapping/

The bar

X

:

for the

x

X—

/? is

called an inner product in

complex conjugate can be omitted

simplify the exposition

we

shall treat the cases

=

a

if

the imaginary part of a

In dealing with just one inner product in a given space

x 2 ) instead of f(xx x 2 ). On occasion the notation ,

Xj

•

In fact, the dot-product of ordinary classical vector algebra

The

all-important fact about an inner

CD

we

of real

X and

bars where they are needed for the complex case. They can

be ignored in the real case, since a

a norm. If

X if

Condition

in

product

define ||X||

=

y/(X, X),

is

X

we

is 0.

shall customarily write

x 2 (dot-product) is

that

is

suggestive.

an inner product. it

can

be used

to

define

Abstract spaces

154

mapping x —*

the

step in the proof

\\x\\

is

a

norm on X. The

we prove

\(x,y)\ 2

(2)

x and

for arbitrary

Lemma

of

2-1

we consider

I.

y. This inequality

obvious that

0

Now

a

let

=

not at once apparent.

is


0, with real coefficients A, B, C, is B 2 — AC < 0. For the present case this yields

2Bt

necessarily have

(2).

is variously known as the Cauchy-Schwarz inequality, the Schwarz Cauchy-Schwarz-Bunyakovsky inequality. prove that (1) defines a norm, the only serious step is the proof that the triangular satisfied. We note that, for any real or complex number A,

result in (2)

inequality, or the

Now,

to

equality

is

+

\A

Now,

A\

1

Hilbert space of dimension k the space

course use the

from

that a real Hilbert space of finite dimen-

C k will

Rk

,

while for a complex

serve as a concrete model. In

R k or C k we

of

norm nfo,

It is

show us

can be mapped isometrically and isomorphically on

this point

.

.

.

,

&)i

=

a^i

2

+

•

•

•

\m

+

of view that we can say that any ^-dimensional subspace of

regarded as a "copy" of

Rn

.

See the remarks

made about

Rk

can be

this in Section 2-2.

CONTINUOUS LINEAR FUNCTIONALS If

y

is

a fixed element of the Hilbert space X, and

evidently a linear functional.

It is

continuous, also, and

if

we

it is

define f(x)

=

easy to see that

(x,y), then ||

f\\

=

\\y\\.

/is

For,

Abstract spaces

162




Theorem 3-13

whence


d.

for elements

easy to verify that

(15)

We

=

0 in the theorem. Hence we assume

prove that there exists

x m) eP, so

yt )

\\

d= \{x n

if (x,

2

||

=

apply this with u

w

xn

,

-

\\x n

+^ + uv = x m Then

2

2

t;||

||

||

=

2(||m||

||x

m

2

+

2

|M|

).

.

xm

2 ||

=

2(||x n


oo and n —> oo we see from this that ||x n — x m —» 0. Therefore there is some y 6 X such that x n —*-y. Also, ||x n -» ||_y||, so \\y\\ = d. The set P is closed, because / is continuous. Therefore y e P. Next, we show that d = ||/||. From/(_y) = ||/|| 2 we conclude ||/|| 2 < ||/|| \\y\\ so ||/|| < ||

||

||_y||.

We

By

the definition of

can assume f(u n )

||/||

*

0.

there exists a sequence {«„} with

Let v n

=^u

n

.

Then

vn

||« n

||

=

eP. Also,

1

—>

||/||.

= JZ!1-^

||/||.

and|/(«n )|

||y„||

/("„) It

follows that

d


(a) if

-> R be continuous. Then aeA, such that sup /(/) =

/I

teT

sup

\x\ is a function on R* to R*. We simply add the definitions |+oo| = + oo, — oo| = +oo. Then, if we have / X-> R*,

f(x)

:

|

\f(x)\