Presenting the various approaches to the study of integration, a wellknown mathematics professor brings together in one
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English Pages 452 Year 1985
Table of contents :
1. The Real Numbers. Point Sets and Sequences
10 Introduction 1
11 The real line 2
12 Notation and terminology concerning sets 7
13 Construction of a complete ordered ﬁeld 12
14 Archimedean order. Countability of the rationals 16
15 Point sets. Upper and lower bounds 21
16 Real sequences 24
17 The extended real number system 29
18 Cauchy’s convergence condition 32
19 The theorem on nested intervals 34
2. Euclidean Space. Topology and Continuous Functions
20 Introduction 37
21 The space Rᵏ 38
22 Linear conﬁgurations in Rᵏ 43
23 The topology of Rᵏ 47
224 Nests, points of accumulation, and convergent sequences 56
25 Covering theorems 61
26 Compactness 65
27 Functions. Continuity 68
28 Connected sets 76
29 Relative topologies 83
210 Cantor’s ternary set 86
3. Abstract Spaces
30 Introduction 89
31 Topological spaces 90
32 Compactness and other properties 96
33 Postulates of separation 102
34 Postulated neighborhood systems 105
35 Compactiﬁcation. Local compactness 107
36 Metric spaces 111
37 Compactness in metric spaces 121
38 Completeness and completion 124
39 Category 127
310 Zorn’s lemma 131
311 Cartesian product topologies 134
312 Vector spaces 140
313 Normed linear spaces 144
314 Hilbert spaces 153
315 Spaces of continuous functions 164
4. The Theory of Measure
40 Introduction 177
41 Algebraic operations in R* 178
42 Rings and σRings 180
43 Additive set functions 185
44 Some properties of measures 188
45 Preliminary remarks about Lebesgue measure 189
46 Lebesgue outer measure in Rᵏ 191
47 The measure induced by an outer measure 196
48 Lebesgue measure in Rᵏ 203
49 A general method of constructing outer measures 215
410 Outer measures in R from monotone functions 218
411 The completion of a measure 224
5. The Lebesgue Integral
50 Introduction 226
51 Measurable functions 229
52 The integral of a bounded function 240
53 Preliminary convergence theorems 248
54 The general deﬁnition of an integral 252
55 Some basic convergence theorems 260
56 Convergence in measure 265
57 Convergence in mean 270
58 The Lᵖ spaces 274
59 Integration with respect to the completion of a measure 278
6. Integration by the Daniell Method
60 Introduction 281
61 Elementary integrals on a vector lattice of functions 282
62 Overfunctions and underfunctions 284
63 Summable functions 288
64 Sets of measure zero 292
65 Measurable functions and measurable sets 297
66 The Nnorm 300
67 Connections with Chapter 5 303
68 Induction of Lebesgue measure in Rᵏ 310
69 Arbitrary elementary integrals on C°°(Rᵏ) 313
610 Regular Borel measures in Rᵏ 319
611 The class ℒᵖ 321
7. Iterated Integrals and Fubini’s Theorem
7—0 Introduction 324
7—1 The Fubini theorem for Euclidean spaces 326
7—2 The FubiniStone theorem 329
7~3 Products of functions of one variable 334
7—4 Iterated integrals and products of Euclidean spaces 336
7—5 Abstract theory of productmeasures 339
76 The abstract Fubini theorem 345
8. The Theory of Signed Measures
8—0 Introduction 348
8—1 Signed measures 349
8—2 Absolute continuity 356
8—3 The RadonNikodym theorem 358
8—4 Continuous linear functionals on Lᵖ(μ) 361
8—5 The Lebesgue decomposition of a signed measure 364
86 Alternative approach via elementary integrals 366
87 The decomposition of linear functionals 371
9. Functions of One Real Variable
90 Introduction 379
91 Monotone functions 379
92 Vectorvalued functions of bounded variation 382
93 Rectiﬁable curves 388
94 Realvalued functions of bounded variation 390
95 Stieltjes integrals 392
96 Convergence theorems for functions of bounded variation 398
97 Differentiation of monotone functions 402
98 Absolutely continuous functions 410
BIBLIOGRAPHY 423
LIST OF SPECIAL SYMBOLS 429
INDEX 431
General Theory of Functions
and Integration
Angus
E.Taylor
Digitized by the Internet Archive in
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 0486649881
2.
515.7 (pbk.)
Calculus, Integral.
I.
Title.
8512862
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 pointset 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. Wellprepared 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 LebesgueStieltjes 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 RadonNikodym among the main objectives. The general Riesz representation theory for continuous linear functionals on C x (R k ) occurs in Section 87 (as
Theorem 87
Much
IV).
of Chapter 9 could be studied before Chapters 48. 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 pointset 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 realvalued 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 twodimensional 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 196162 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 10
Introduction
1
11
The
2
real line
12
Notation and terminology concerning
13
Construction of a complete ordered
14
Archimedean order. Countability of the
15
Point
sets.
sets
rationals
Upper and lower bounds
16
Real sequences
17
The extended
7 12
field
16 21
24
real
number system
29
18
Cauchy's convergence condition
32
1 9
The theorem on nested
34
intervals
Euclidean Space. Topology and Continuous Functions
20
Introduction
37
k
21
The space R
22
Linear configurations in
23
The topology of
24
Nests, points of accumulation, and convergent sequences
56
25
Covering theorems
61
26
Compactness
65
27
Functions. Continuity
68
28
Connected
76
29
Relative topologies
38
Rk
sets
210 Cantor's ternary
Rk
set
43 47
83
86
1
Contents
viii
3.
Abstract Spaces
30
Introduction
89
31
Topological spaces
90
32
Compactness and other properties
96
33
Postulates of separation
102
34
Postulated neighborhood systems
105
35
Compactification. Local compactness
107
36
Metric spaces
1 1
37
Compactness
in metric spaces
121
38
Completeness and completion
124
39
Category
127
310 Zorn's lemma 311 Cartesian product topologies
134
312 Vector spaces
140
Normed
144
313
4.
131
linear spaces
314 Hilbert spaces
153
3 15 Spaces of continuous functions
164
The Theory of Measure 4 0
Introduction
177
41
Algebraic operations in
42
Rings and aRings
R*
178
180
43
Additive set functions
185
44
Some
188
45
Preliminary remarks about Lebesgue measure
properties of measures
R
k
46
Lebesgue outer measure
47
The measure induced by an outer measure
48
Lebesgue measure
49
A
general
in
in
Rh
method of constructing outer measures
410 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
50
Introduction
226
51
Measurable functions
229
52
The
240
integral of a
bounded function
1
ix
Contents
6.
53
Preliminary convergence theorems
248
54
The general
252
55
Some
definition of
an integral
basic convergence theorems
260
56
Convergence
in
measure
265
57
Convergence
in
mean
270
58
The LP spaces
274
5 9
Integration with respect to the completion of a measure
278
Integration by the Daniell
60
Method
Introduction
28
61
Elementary integrals on a vector
62
Overfunctions and underfunctions
284
63
Summable
288
64
Sets of
65
Measurable functions and measurable
66
The Nnorm
67
Connections with Chapter 5
68
Induction of Lebesgue measure in
69
Arbitrary elementary integrals on C^/?*)
measure zero
6 11 The class
8.
of functions
functions
&
282
292 sets
297 300
R
610 Regular Borel measures in
7.
lattice
303
Rk
k
v
310 313 319 321
Iterated Integrals and Fubini's
Theorem
70
Introduction
71
The Fubini theorem
72
The FubiniStone theorem
73
Products of functions of one variable
74
Iterated integrals
324 for Euclidean spaces
and products of Euclidean spaces
326 329
334 336
75
Abstract theory of productmeasures
339
7 6
The
345
abstract Fubini theorem
The Theory of Signed Measures 80
Introduction
348
81
Signed measures
349
82
Absolute continuity
356
X
9.
Contents
83
The RadonNikodym theorem
358
84
Continuous linear functionals on L%u)
361
85
The Lebesgue decomposition of
364
86
Alternative approach via elementary integrals
366
87
The
371
a signed measure
decomposition of linear functionals
Functions of
One Real
Variable
90
Introduction
379
91
Monotone
379
92
Vectorvalued functions of bounded variation
382
93
Rectifiable curves
388
functions
94
Realvalued functions of bounded variation
390
95
Stieltjes integrals
392
96
Convergence theorems for functions of bounded variation
398
97
Differentiation of
98
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
10
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 realvalued 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 onedimensional 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 11). 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 14.
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 13 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 12. The ideas of pointset 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 15.
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 17.
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 11).
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 15). 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 11. set S in F with an upper bound in F has a least upper bound in F. archimedean (see Section 14). Every Cauchy sequence in Fhas a limit in F(see Section 18). F is archimedean. If {/„} is a nest in F, there is an element of F which is in every /„ (see Section 19).
F is
(b) (a)
4.
(b)
The
Properties 1
:
—*
2 —*
2 (see 1
(see
1, 2, 3,
4 are equivalent. This
Theorem 15 Problem
1
is
shown by
the following
list
of implications.
I)
in Section 15)
» 3(a) (see Theorem 14 I) 2 > 3(b) (see Theorem 18 I) 2 + 4(b) (see Theorem 19 I) 3(b) > 4(b) (see Problem 1(a) 1
4^2 Because 3(a)
Problem the same as
(see is
in Section 19)
1(b) in Section 19). 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
11
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 "ed 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 a1 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 16
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
36
(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 24
Exactly the same as the proof of
(b).
holds true in any space which
like that in the
the
satisfies
proof of Theorem 32 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 37.
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 27
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
36
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 36. 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 36 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 14 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 dotproduct of ordinary classical vector algebra
The
allimportant 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 (dotproduct) 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
21
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 CauchySchwarz inequality, the Schwarz CauchySchwarzBunyakovsky 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 22.
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 313
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)\