Mathematical Analysis. A Special Course

484 Pages·1965·17.973 MB·English

by  G. Ye. Shilov and D. A. R. Wallace (Auth.)

G.YE. SHILOV
MATHEMATICAL
ANALYSIS
A Special Course
TRANSLATED BY
J.D.DAVIS
ENGLISH TRANSLATION EDITED
BY
D. A. R.WALLACE
Department of Mathematics
The University of Glasgow
PERGAMON PRESS
OXFORD • LONDON • EDINBURGH • NEW YORK
PARIS •FRANKFURT
Pergamon Press Ltd., Headington Hill Hall, Oxford
4 & 5 Fitzroy Square, London W.l
Pergamon Press (Scotland) Ltd., 2 & 3 Teviot Place, Edinburgh 1
Pergamon Press Inc., 122 East 55th St., New York 22, N.Y.
Pergamon Press GmbH., Kaiserstrasse 75, Frankfurt-am-Main
Copyright © 1965
Pergamon Press Ltd.
First edition 1965
Library of Congress Catalog Card No. 65-18619
A translation of the original volume
MaTeMaTEraecKHft aHajra3, cnerjuajifeHBiii Kypc
(Matematicheskii Analiz, SpetsiaVnyi Kurs)
Fizmatgiz, Moscow 1961
PREFATORY NOTE
THE book was written as a textbook for University Faculties of
Mathematics, in a special course of Mathematical analysis. Prob-
lems in the theory of functions of a real variable, the calculus of
variations and integral equations are treated here from the single
viewpoint of the theory of linear spaces. The reader requires a
grasp of a general course of mathematical analysis as covered in
a degree course.
V
FOREWORD
THIS book has been written as a textbook for a special course of
mathematical analysis (in brief, "Analysis III"). "Analysis III" is
delivered as the third course of the mechanics-mathematics faculty
of the Moscow State University, and dates from 1949; the intro-
duction of such a course was initiated by academician A. N. Kolmo-
gorov, who was the first lecturer." Analysis III" is based on material
from earlier separate courses on the theory of functions of a real
variable, the calculus of variations and integral equations, and
treats all this material from a single viewpoint which has its source
in the theory of linear spaces.
The layout of the book is according to the following scheme. The
first chapter gives a straightforward minimal account of set
theory. The second chapter contains elements of the theory of
metric and normed linear spaces. In the third chapter the calculus
of variations is developed; it is presented here as a theory of
differentiable functionals in normed linear spaces. The fourth
chapter is devoted to the theory of the Lebesgue integral; the
scheme given by F. Riesz is thought to be a better starting-point
for the account than that of Lebesgue, inasmuch as it is more
economical and leads more rapidly to the heart of the subject.
The fifth chapter, "The Geometry of Hubert Space" contains the
theory of orthogonal resolutions and a geometrical treatment of
integral equations. In the sixth chapter the relation between inte-
gration and differentiation is explained and the Stieltjes integral is
constructed. In the seventh and last chapter an account is given
of the theory of the Fourier Transform; we depart from tradition
here in including some material, which, in view of its special im-
portance in mathematical physics, should long ago have had a
place in a course of mathematical analysis. To facilitate modi-
fications in a course of lectures, the material of the last three
chapters is widely diversified.
The logical dependence between chapters is given schematically
as follows:
la* XI
Xll FOREWORD
Sets
Metric spaces
Lebesgue integral Calculus of variations
Hilbert space Integration and differentiation
Fourier transform
It must be observed that the general viewpoint of functional
analysis developed in this course does not constitute an end in
itself, but only a means; the chief aim is an introduction to the
field of classical mathematical analysis.
For the sake of brevity we limit ourselves within each theme to
a discussion of only the most important questions, fully recognising
that the reader may well be dissatisfied in some instances. The
choice of material generally, and in particular for the last chapters,
presented the author with great difficulties. Some interesting but
somewhat digressional questions have been set as problems; they
can be used as material for seminars.
The reader needs to be acquainted with a general course of
mathematical analysis, such as "A short course" by A. Ya.Klin-
chin for example. The book can then be used for an independent
study of the subject. Towards the end of the book the elementary
properties of analytic functions are also assumed to be familiar.
The author is particularly grateful to M. G. Krain, 0. A. Olyanik,
and D. A. Rykov, who have read through the manuscript and with
their criticisms have contributed greatly to its improvement. In
the second edition the text has been revised, supplemented, and
improved in places. Some new problems have also been added. The
author is indebted to his many correspondents in Leningrad,
Kazan, Baku, and other cities of the U.S.S.R. for their valuable
criticisms of the first edition.
CHAPTER I
SETS
1. SETS, SUBSETS, INCLUSIONS
When we consider a number of objects ("elements"), we use
such terms as "totality", "aggregate", and "set". For example,
one can speak of the set of students in an audience, the set of
grains of sand on a beach, the set of vertices of a polygon, or the
set of its sides. The examples specified have the property that in
each of them the corresponding set is composed of a definite
number of elements (which may, in some cases, be difficult to
determine). We shall call such sets finite.
In mathematics it is often necessary to deal with sets which are
not composed of a finite number of objects; the simplest examples
are provided by the set of all natural numbers 1, 2, 3, ... and the
set of all points of an interval. We shall call such sets infinite. We
also add to the totality of sets the empty set—the set which con-
tains no elements at all.
So, for example, as can be seen from Fig. 1, the set of real roots
of the equation sin xjx = b is infinite when b = 0 (in this case it
consists in all the values x = ±7t, ±2n ...), finite but non-empty
when 0 < |6| ^ 1 (the precise number of roots can be calculated
for each 6), and empty when |6| > 1 (no value of the function
sin xjx exceeds 1 in absolute value, so the equation sin xjx = b
has no roots at all when \b\ > 1).
1
2 MATHEMATICAL ANALYSIS
As a rule we shall denote sets by capital letters A, B, C, ...,
and their elements by small ones. The notation a £ A (or A 9 a)
indicates that a is an element of the set A; a (J A (or A J) a) means
that a is not an element of the set A. The notation A a B (or
BID A) indicates that each element of the set A is an element of
the set B; in this case the set A is called a subset of the set B. The
largest subset of the set B is evidently the set B itself; the smallest
is the empty set. Any other subset of B is known to contain some,
but not all, of the elements of B. Each such subset is called a
proper subset. The symbols £, 3, a, ID are called inclusion signs.
If the inclusions A a B, B a A both hold, then each element of
the set A is an element of the set B, while conversely each element
of B is an element of A; thus in this case A and B consist of the
same elements, that is, they coincide with one another. This is
denoted by the equation
A = B.
Sets can be formally denoted in the following distinct ways.
The simplest consists in enumerating all the elements of a
set, e.g. A = {1, 2, ..., n, ...}. An alternative form frequently
used appeals to the properties of the elements of a set; thus,
A = {x: x2 — 1 < 0} is the set of all x for which the inequality
following the colon is satisfied.
2. OPERATIONS ON SETS
We consider here three simple operations which can be carried
out on sets: union, intersection, and forming the complement.
We describe first the operation of forming the union of sets.
Let the sets A, B, C, ... be given. We consider the totality of all
elements, each of which belongs to at least one of A, B, C, ... This
aggregate is a new set, which is termed the union of the sets
A,B,C ...
Thus the union of the set
A = {6, 7, 8, ...}
(all the natural numbers greater than 5) and the set
B = {3, 6, 9, ...}
SETS 3
(all the natural numbers that are multiples of 3) is the set
S = {3, 6, 7, 8, 9, 10, ...}
(all the natural numbers with the exception of 1, 2, 4, and 5).
We introduce now the operation of forming the intersection of
sets. The intersection of the sets A, B, C, ... is defined as the
totality of elements belonging to each of the specified sets.
Thus, in the preceding example, the intersection of the sets
A = {6,7,8,9, 10, ...}
B = {3,6,9, 12, ...}
is the set
D = {6,9, 12, ...}.
It may happen that the sets A, B, C, ... do not have a single
element in common. In that case their intersection is the empty
set, and the sets A, B, C, ... are said to be non-intersecting or
disjoint. For example, the sets of integers
A = {1,2}, B = {2,3}, C= {1,3}
are disjoint (although when taken in pairs they have common
elements).
Unions and intersections can be formed not only for finite
collections of sets but also for infinite ones. For example, it is
possible to form the union of the sets of points of all the straight
lines in the plane that pass through a given point 0. This union
will obviously be the set of all points in the plane. The inter-
section of the specified sets will consist in the single point 0.
The union S of sets A, B, C, ... is sometimes called their sum
and is written in the form S = A + B + C + •••; the intersection
D is also termed the product and is denoted by D = ABC ...
There is some motivation for such "arithmetical'' terminology.
For instance, for any three sets A, B, C the equation
(A + B)C = AC + BC
holds. We shall give a proof of this equality as a simple but typical
example of set-theoretic reasoning.
As we remarked, two sets are considered equal if each element
of one of them is at the same time an element of the other. Thus
4 MATHEMATICAL ANALYSIS
we have to show that each element x of (A + B) C (the left-hand
side) is an element of AC -f BG (the right-hand side) and con-
versely that each element y of A C + BCis contained in (A -f B) C.
First let x belong to (A + B) C. Being an element of the inter-
section of the sets A + B and C, it must belong to each of them;
thus we have
x £A + B and x £ C.
Since x is contained in the union of A and B, it is certainly con-
tained in one of them, say in A. But the inclusions x £ A, x £ C,
imply x £AC, whence x £AC + BC. And if # is contained, not
in A, but in B, then in the same way x £ BC, x £ A C + BC, as
required. Conversely if y belongs to the sum AC + BC, then it
belongs either to AC or to BC, y £BC say. But then y £B
and y £ C; further it follows from y £ B that y £ A -\- B and finally
that y £ (A + B) C. The case y £ A G is treated analogously and
the proof is complete.
It is to be observed, however, that by no means all arithmetical
rules carry over to operations with sets. For example, we have
for sets A, B, C, the formulae
A + A = A,
AA=A,
A + BC = (A + B)(A + C)
which are quite unlike the usual arithmetical equations. We suggest
that the reader satisfy himself as to the accuracy of these for-
mulae.
We shall indicate a few further symbols for sums and inter-
sections of sets. For the union of sets we employ the symbols 2J
and U, so that, for example, the notation
CO CO
S = £A or S = UA,
V
v=l v—1
denotes the union of the sets A A , ..., A , ...
l9 2 v
For the intersection of sets we employ the symbols 77 and f], so
that, for example
CO CO
I> = 11A or D = fl A
v
denotes the intersection of the sets A A , ..., A , ...
1} 2 v
SETS 5
We turn now to the operation of forming the complement.
If the set B is a subset of the set A, then the totality of those
elements of A that do not belong to B is termed the complement
of B with respect to A and is denoted by CB or A — B.
We note the obvious formula
(A - B) + B = A.
Note that for two arbitrary sets A and B, the result
(A + B) - B = A
is generally false; it is true only when A and B have no common
elements.
Of more complicated results we remark the following, which
will be of frequent occurrence:
GZB=nCB; (1)
r v
V V
it can be read as follows: the complement of a union of sets is the
intersection of their complements.
We shall give a proof of this result. Let x£C 2 B ; then
v
V
x (£ 2J B\ this implies that, for any v, x (£ B , i.e. x £ CB \ but
V v V
V
then x £ JJ OB . Conversely, if x £ JJ CB , then, for any v,
v V
V
x£CB, i.e. x$B for any v; but then X$2JB> i-e- %€.C2JB >
v v V V
as required. v v
Operating again with C on both sides of equation (1) and letting
A = CB , we obtain the result
v V
ZCA = CTIA, (2)
V
V V
i.e. the complement of an intersection of sets is the union of their
complements.
The results cited can be combined in the form of a general rule:
the complement symbol C can be interchanged with either of the
symbols ^ and 77 provided that each of the latter is substituted for
the other.
3. EQUIVALENCE OF SETS
We wish now to establish a rule by which it would be possible
to compare different sets with respect to the number of elements
in them.