A Course of Mathematical Analysis Volume I

466 Pages·9.21 MB·English

by  S. M. Nikolsky

S. M. NIKOLSKY
A Course
of Mathematical
Analysis
Volume
1
MIR PUBLISHERS MOSCOW
Academician S.M NIKOLSKY
a State Prize winner, the
author of more than 130
scienti�c papers and several
monographs including
Quadrature Formulas,
Approximation of Functions of
Several Variables and
Embedding Theorems and
Integral Representation of
Functions and Embedding
Theorems (co-author).
He contributed many
fundamental results to the
theory of approximation of
functions, variational methods
for solving boundary-value
problems and functional
analysis. For his monograph
Approximation of Functions of
Several Variables and
Embedding Theorems he was
awarded the Chebyshev Prize
of the USSR Academy of
Sciences.
C. M. HHKOJIfeCKHM
KYPC
MATEMATHMECKOrO
AHAJIH3A
TOM 1
H3AATEnbCTBO «HAyKA»
MOCKBA
S. M . N IK O L SK Y
Member, USSR Academy of Sciences
A Course
o f Mathematical
Analysis
Volume 1
Translated from the Russian
by
V. M. VOLOSOV, D. Sc.
MIR PUBLISHERS
MOSCOW

First published 1977
Revised from the 1975 Russian edition
Ha anz.iuucKOM mbtKe
© M3.naTejn.CTB0 «Hayica», 1975
© English translation, Mir Publishers, 1977
Preface to the English Edition
The major part of this two-volume textbook, with the exception of some
supplementary material, stems from the course in mathematical analysis
given by the author for many years at the Moscow Physico-technical
Institute.
The first chapter is an introduction. It treats of fundamental notions of
mathematical analysis using an intuitive concept of a limit. Ultimately, it
even establishes, with the aid of visual interpretation and some considera­
tions of a physical character, the relationship between the derivative and the
integral and gives some elements of differentiation and integration tech­
niques necessary to those readers who are simultaneously studying physics.
The second chapter is devoted to the notion of a real number which is
interpreted on the basis of its representation as an infinite decimal. The part
of this chapter given in small print may be omitted by the reader in his first
acquaintance with the subject-matter.
It is my belief, which incidentally coincides with the traditional point of
view, that the fundamentals of mathematical analysis should be first pre­
sented for functions of one independent variable and only after that extended
to functions of several variables although this leads to some unavoidable
recapitulations. On the other hand, in a comprehensive course intended for
students in the fields of mathematics and physics it is quite possible to pass
from functions of one variable not to the cases of two and three variables
but directly to n variables. Here the whole question rests merely on a suitable
choice of notation. However, such notation has already been elaborated in
scientific journals and monographs and has proved expedient, and now it
only remains for the authors of textbooks to accept it. Such an approach
provides the necessary prerequisites for the second half of the course where
in the study of such topics as the Fourier series and the Fourier integral the
6 PREFACE TO THE ENGLISH EDITION
reader should acquire a good understanding of the concept of an infinite-
dimensional function space.
The notions of an /i-dimensional Euclidean space, of an arbitrary space
with scalar product and of a Banach space are introduced in the course
at a rather early stage. They are then widely used but only to an extent
required for the realization of the plan of the book.
The presentation of the subject-matter of the book is based mainly on the
notion of the Riemann integral. To save the reader time and effort I have
always tried, when possible, to give similar proofs of the analogous theorems
in the one-dimensional and many-dimensional cases.
There arises the subtle question of the completeness of the spaces L and
Ln. In this connection I do not introduce abstract elements which substitute
for the Lebesgue integrable functions simply confining myself in the general
discussion of the question to an explanation of the corresponding fact in
terms of the Lebesgue integral.
Incidentally, the textbook includes a supplementary section (Chapter 19,
Vol. 2) devoted to the Lebesgue integral. I hope that many of the readers
will be interested enough to look through it. The concept of the Lebesgue
integral is essential for modem mathematical physics. Without using the
Lebesgue integral it is quite impossible to study the direct variational
methods of mathematical physics.
Chapters 17 and 18 (Vol. 2) also contain supplementary material. Chapter
18 deals with such important notions of modern calculus as the Sobolev
regularization of functions and the partition of unity.
Chapter 17 is devoted to differentiable manifolds and differential forms.
It culminates in the proof of the Stokes theorem for the //-dimensional
space. This chapter may serve as a test of the reader’s grasp of the material
of the book.
My intention was to help the reader to proceed more easily to the study
of mathematical physics. A number of supplementary topics included in the
course have been chosen particularly in consideration of their use in mathe­
matical physics. There still remain a number of pedagogical problems to be
solved in teaching the theory of functions of several variables. It is hoped
that this book will contribute to the purpose.
I owe a great deal to two books that have helped me greatly.
One is Course d’analyse infinitesimal by Ch. J. de la Vailed Poissin, a
book which I studied assiduously in my student days.
The other is An Introduction to the Theory of Functions of a Real Variable
7
preface to THE HNGUSH WirnnN
by P. S. Alexandrov and Ar * K ^ ^ ^ h o s e many lectures 1 have
also attended. I wish to express my deepestgratitude to these distinguished
scientists and especially to A.N. Kolmogorov to whom I am greatly in­
debted for scientific inspiration.
In the prefaces to the first and second Russian editions of the book i
mentioned a number of persons and organizations from whom I received
valuable advice in the preparation of the present course. These were the
Chairs of Mathematics of the Moscow Physico-technical Institute and of
the Moscow Institute of Electronic Engineering and also O. V. Besov,
A. A. Vasharin, I. N. Vekua, E. A. Volkov, R. V. Gamkrclidze, A. A.
Dezin, L. D. Kudryavtsev, P. I. Lizorkin and Yu. S. Nikolsky.
To all of them I once again express my warmest gratitude.
Finally, I wish to express my appreciation to the Mir Publishers English
Department and to the translator of the book Professor V. M. Volosov for
their most efficient handling of the publication of this English translation.*
S. M. Nikolsky
The present translation incorporates suggestions made by the author. — Tr.