Table Of ContentA COURSE OF
HIGHER
MATHEMATICS
V. I. Smirnov
Volume V
INTEGRATION
AND
FUNCTIONAL
ANALYSIS
A C O U R SE OF
H i g h er M a t h e m a t i cs
VOLUME V
V. I. SMIRNOV
Translated by
D. E. B R O WN
Translation edited by
I. N. SNEDDON
Simson Professor in Mathematics
University of Glasgow
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Library of Congress Catalog Card Nimiber 63-10134
This translation has been made from the Russian Edition of V. I. Smirnov's
book Kypc eucme٧ MameMammu (Kurs vysshei matematiki), published in 1960
by Fizmatgiz, Moscow
MADE IN QREAT BRITAIN
INTRODUCTION
THIS is the final volume of Prof. Smirnov's five-volume course of
higher mathematics, about whose history some remarks were made
in the Introduction to Vol. I of the present English edition.
The first Russian edition of this volume, published in 1947, enjoyed
the distinction of being the first book in any language on the theory
of integration and the elements of functional analysis to be written
specifically with the needs of theoretical physicists in mind. Indeed
nearly twenty years after its pubhcation its only rivals would appear
to be works by other Russian authors.
Functional analysis arose as the result of generalizing various con
cepts and methods of classical branches of mathematics. Although it
has become (in the manner characteristic of contemporary mathe
matics) a very abstract discipHne, its general results can be used to
derive the solution of particular problems in classical analysis and in
apphed mathematics. Its successes have been such that it is difficult
to imagine that a strong hght cannot be cast on the solution of almost
any problem in mathematical analysis by the use of the concepts and
techniques of functional analysis. Large areas of the modern theories
of approximation, differential equations and mathematical physics
are dominated by these methods and so research workers in physics
and engineering need to become famihar with the ideas of functional
analysis. They will find a clear and authoritative introduction to
these topics in this volume, but it should not be regarded as of use
to them only; students of pure mathematics will find here an account
not only of the essentials of a flourishing branch of modern pure
mathematics but also of its Unks with the past and of the motivation
of much of the recent abstract work in the subject.
I. N. SNEDDON
ix
PREFACE
IN MODERN theoretical treatments of mathematical physics great
importance attaches to the theory of functions of a real variable,
the various functional spaces and the general theory of operators.
These subjects provide the essential material for the present book,
which is based on the fifth volume of my Course of Higher Mathe
matics, pubUshed in 1947.
The branches of the theory of functions of a real variable in the
present book include the theory of the classical Stieltjes integral, the
Lebesgue-Stieltjes integral and the theory of completely additive
set functions.
The first chapter discusses the theory of the classical Stieltjes
integral, and also considers the more general definition of the Stieltjes
integral over an interval of any type, based on the equality of the
upper and lower Darboux integrals with a subdivision of the basic
interval into intervals of any type. The Fourier-Stieltjes and Cauchy-
Stieltjes integrals are taken as examples of the classical Stieltjes
integral, and inversion formulae are established for these. The Stieltjes
integral is also defined for the plane case.
The space G of continuous functions is also discussed in Chapter I,
and the general form of hnear functionals in this space is established.
The second chapter deals with the foundations of the metric theory
of functions of a real variable and the Lebesgue-Stieltjes integral.
The whole of the theory is expounded for the case of a plane and the
possibihty of its obvious generalization to the case of 7i-dimensional
Euclidean space is indicated. The theory of measure is built up on
the basis of any non-negative, additive, normal function, defined on
semi-open two-dimensional intervals. The Lebesgue-Stieltjes integral
of a bounded function is defined on the basis of the coincidence of
the upper and lower Darboux integrals when the basic measurable
set is subdivided into measurable sets. Chapter II ends with a detailed
discussion of an averaging process for functions and the properties
of the mean functions, when the averaging kernel is subject to certain
conditions. Wide use is subsequently made of the averaging process.
XI
Xii PBEFACE
The third chapter deals with the theory of completely additive
set functions. After proving the initial theorems, the theorem on
the decomposition of a completely additive set function into a singular
and an absolutely continuous part is stated without proof, and the
fundamental facts relating to this decomposition are discussed. The
case of a single independent variable is treated in detail. Also, an
absolutely continuous set function is studied in the general case,
and the formula established for changing the variables in a multi
dimensional Lebesgue-Stieltjes integral.
The third chapter ends with a proof of the above-mentioned theorem
on decomposing a completely additive set function into two terms.
Furthermore, the concept of Hellinger integral is introduced in the
multi-dimensional case, and its properties are investigated. In particu
lar, the connection is established between the Hellinger integral and
the Lebesgue-Stieltjes integral. The case of the one-dimensional
Hellinger integral is analyzed in detail. All the proofs at the end of
Chapter III are based on a preliminary detailed treatment of the
properties of completely additive set functions [78, 79].
The fourth chapter contains an exposition of the foundations of
the general theory of metric and normed spaces. It ends with a
detailed discussion of generalized derivatives, embedding theorems
for the various function spaces, and the theory of functionals in
the space of continuously differentiable functions. All these questions
are related to S. L. Sobolev's well-known investigations. They are
dealt with in his monograph Some Applications Of Functional Analysis
To Mathematical Physics (Nekotorye primeneniya funktsional'nogo
analiza í matematicheskoi fizike) (1950).
Generalized derivatives are defined in two ways - with the aid of
the formula for integration by parts and by means of the closure
of functions with continuous derivatives; the equivalence of these
definitions is proved. Special attention is paid to the case of a star-
shaped domain. Furthermore, the complete normed functional spaces
WfD) and W^¿\D) are introduced; the first of these consists of the
functions φ(χ) that are defined in the domain D and have all generalized
derivatives of order Z, where φ(χ) and the derivatives in question
belong to Lp(D)y whilst the second space consists of the functions
φ(χ) that have all generalized derivatives up to and including order /.
It is subsequently proved that, for a wide class of domains Z), Wp\D)
and W^\D) consist of the same set of functions, and that the norms
introduced into them are equivalent. Moreover, fairly simple proofs
PREFACE XUl
are given for space W^p{D) of theorems that are particular cases of
the embedding theorems for W^p(D).
These theorems are first formulated, then a complete proof
of them is given in fine print, on the basis of Sobolev's integral
form. All this material is closely related to the above-mentioned
monograph.
The final fifth chapter deals with the general theory of Hilbert
space, the whole of the treatment being first given for the case of
bounded operators. Fredholm's theorems are proved for linear
equations with completely continuous operators. They have been
stated without proof for normed spaces.
The relevant integral forms in terms of the differential solutions
are given with the aid of HelHnger integrals for self-conjugate operators
on a continuous spectrum. Examples are given of the application of
the general theory of bounded operators in and 2^2·
The final section of the fifth chapter is devoted to the theory of
unbounded operators in Hilbert space. After proving the general
theorems, numerous examples are given of differential operators with
one and several independent variables. The general theory of extension
of closed symmetric operators is followed by a discussion of the
special case of semi-bounded operators, and in particular, of their
Friedrichs extensions.
The publication of a sixth volume is envisaged, dealing with certain
problems of the modem theory of differential operators with one and
several independent variables.
In addition to specialized articles, I have made use of numerous
books in preparing the present volume. The chief titles are as follows:
V. I. Glivenko, The Stieltjes Integral (Integral Stilt*esa); I. P. Natanson,
Theorie der Funktionen einer reellen Verδnderlichen] Saks, Theory Of The
Integral (Teoriya intégrala); de la Vallée-Poussin, Integrales de Lebesgue.
Fonctions d'ensembles. Classes de Baire; Stone, Linear Transformations
in Hilbert Space and their Applications to Analysis; N. I. Akhiezer
and I. M. Glazman, Theory of Linear Operators (Teoriya lineinykh
operatorov); A. 1. Plesner, Spectral Theory of Linear Operators, I
(SpektraFnaya teoriya lineinykh operatorov, I) (Uspekhi matema-
ticheskhikh nauk, t. IX, 1941); N. I. Akhiezer, Infinite Jacobian
Matrices and the Problem of Moments (Beskonechneye matritsy
Jakobi i problema momentov) (loe. cit.); S. L. Sobolev, Some Applicat
ions of Functional Analysis to Mathematical Physics (Nekotorye
primeneniya funktsionaFnogo anahza í matematicheskoi fizike).
XIV PREFACE
I want to thank S. M. Lozinskii for reading the original manuscript
and making a number of valuable suggestions.
The treatment of numerous problems in the second part of this
book is due to Prof. O. A. Ladyzhenskaya, who is the associate
author of the second part. I discussed in detail with her the plan of
this book.
M. S. Birman gave great assistance in preparing the second part
of the book. He is responsible for the exposition of the sections
dealing with embedding theorems [114-118] and with the theory of
small perturbations of the spectrum [198]. He gave valuable advice
on the spectra of symmetric operators and their extensions, as also
on the treatment in Chapter IV.
Let me express my indebtedness to O. A. Ladyzhenskaya and
M. S. Birman. Without their help I should not have been able to
carry the work through to the end.
The first three chapters were read by G. P. Akilov, from whom
I obtained a number of valuable suggestions regarding the treatment
of certain problems. I tender him my sincere thanks.
V. SMIRNOV
CHAPTER I
THE STIELTJES INTEGRAL
1. Sets and their powers. The various concepts of integral play a
large part in the application of mathematical analysis to present-day
science, and we shall discuss in our first two chapters the theory of
integration in a more general form than previously. As a preliminary,
the present section contains a certain amount of elementary set
theory, which is supplementary to that given in [IV; 16].
Suppose we have two sets and A2, consisting of objects of any
type (elements). The sets are said to have the same power if a one-to-
one correspondence can be established between the elements of
and the elements of -^2, i.e. a correspondence in which a definite
element of A2 is associated with each element of A^, and conversely,
each element of A2 is associated with one and only one element of A^,
An infinite set (i.e. a set containing an infinite number of elements)
is described as denumerable if it has the same power as the set of
all positive integers, i.e. if its elements can be enumerated by means
of positive integers: a^, ... Two denumerable sets have the
same power. Let us examine some properties of denumerable sets.
We consider the part of a denumerable set containing an infinite set
of elements Up^, ap^, ..., where Pi, P2> · · · is an increasing sequence
of positive integers. The elements of this new set are also numbered.
The number of each element is the subscript of p. In other words, they
are numbered in order of increasing subscripts p^, p^^ An infinite
part of a denumerable set is therefore a denumerable set. We now
take two denumerable sets: A(a-^, a^y ag, ...), consisting of elements
(^v ^> ^3» · · · ^3» · · ·)> consisting of elements 6^, h^j 63, ...;
we form their sum, i.e. we combine the elements of both sets into a
single set C. The new set G thus obtained is generally called the sum
of sets A and B, This new set is also denumerable. For we only need
to arrange the elements of set G say in the following order: a^, δ^,
«2, b^, ..., in order to see that G is denumerable. If there are identical
elements a/^, 6/, we have to take one of them and strike out the re
mainder. A similar argument applies for the sum of a finite number of
2 THE STIBLTJES INTEGRAL [1
denumerable sets, i.e. the sum of a finite number of denumerable sets
is a denumerable set.
Suppose we have a denumerable set of denumerable sets. The
elements of all these sets can be denoted by a letter with two integral
indices a!f^. The upper index indicates the number of the set to which
the element belongs, and the lower the number which the element
has in the denumerable set to which it belongs. There is no difficulty
in enumerating all the elements a^^\ We take as the first element
the one in which both indices are unity: a^^\ We then take the elements
in which the sum of the indices is 3, and arrange them in order of
increasing upper index. We thus obtain a^^\ αψ^ as the second and
third elements of the sum of sets. We now take the elements in which
the sum of the indices is 4, and arrange them in order of increasing
upper index: a^^\ a^^\ \ This gives the fourth, fifth and sixth
elements of the sum of sets. It may be seen on continuing this con
struction that the sum of a denumerable number of denumerable
sets is a denumerable set. This assertion would obviously still hold
if certain of the component sets were finite instead of denume
rable.
Let A be an infinite set. We choose any element of it and assign
it the number one. The remainder of the set will be infinite, as before.
We choose any element from it and assign it the number 2. On proceed
ing in this way, it will be seen that a denumerable set can be ex
tracted from any infinite set. The set remaining after such extraction
may be either empty, i.e. contain no element at all, or may be finite,
or infinite. Let us show that, if this remaining set is infinite, it has
the same power as the original set, i.e. the following assertion holds:
if, after extracting a denumerable set Ρ from an infinite set Ay an
infinite set Β remains, sets A and Β have the same power. We extract
from the infinite set -B a further denumerable set and let C be
the remaining set. The original set A is now split into three sets: A=
= Ρ + Q + C, of which the set C may be empty or may be infinite,
whilst sets Ρ and Q are denumerable sets. We had A = Ρ + Β
prior to the second extraction. A one-to-one correspondence is readily
established between the elements of A and B; for we have A =
= Ρ + Q + C Sind Β = Q + C, The sum Ρ + Q of denumerable sets
is a denumerable set, so that a one-to-one correspondence can be
established between the elements of Ρ + Q and Q. We put every
element of the set O in correspondence with itself. A one-to-one
correspondence will thus be established between the elements of A
