Elementary Real and Complex Analysis (Dover Books on Mathematics)

526 Pages·1996·16.13 MB·English

by  Georgi E. Shilov

Elementary Real and
Complex Analysis
Georgi E. Shilov
TRANSLATED AND EDITED BY
Richard A. Silverman
REVISED ENGLISH EDITION
DOVER PUBLICATIONS, INC.
New York
Copy rzgh t
Copyright O 1973 by the Massachusetts Institute of Technology.
All rights reserved under Pan American and International Copyright
Conventions.
Bibliographical Note
This Dover edition, first published in 1996, is an unabridged, corrected
republication of the work first published in English by The MIT Press,
Cambridge, Massachusetts, 1973, as Volume 1 of the two-volume course
"Mathematical Analysis."
Library of Congress Catalopng-in-PublicationD ata
Shilov, G. E. (GeorgiI Evgen'evich)
[Matematicheskii analiz. Chasti 1-2. English]
Elementary real and complex analysis / Gcorgi E. Shilov ; revised
English edition translated and edited by Richard A. Silverman.
p. cm.
Originally published in English: Cambridge, Mass. : MIT Press, 1973.
Includes index.
ISBN 0-486-68922-0( pbk.)
1. Mathematical analysis. I. Silverman, Richard A. 11. Title,
QA3OO.S4552 1996
5 15-dc20 95-37030
CIP
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
Contents
Preface
1 Real Numbers
1.1. Set-Theoretic Preliminaries
1.2. Axioms for the Real Number System
1.3. Consequences or the Addition Axioms
1.4. Consequences of the Multiplication Axioms
1.5. Consequences of the Order Axioms
1.6. Consequences of the Least Upper Bound Axiom
1.7. The Principle of Archimedes and Its Consequences
1.8. The Principle of Nested Intervals
1.9. The Extended Real Number System
Problenls
2 Sets
2.1. Operations on Sets
2 2. Equivalence of Sets
2.3. Countable Sets
2.4. Uncountable Sets
2.5. Mathematical Structures
2.6. n-Dimensional Space
2.7. Complex Numbers
2.8. Functions and Graphs
ProbIerns
3 Metric Spaces
.
3.1 Definitions and Examples
3.2. Open Sets
3.3. Convergent Sequences and Homeomarphisms
3.4. Limit Points
3.5. Closed Sets
3.6. Dense Sets and Closures
3.7. C:omplete Metric Spaces
3.8. Completion of a Metric Space
3.9. Compactness
Problems
Contents
4 Limits
4.1. Basic Concepts
4.2. Some General Theorems
4.3. Limits of Numerical Functions
4.4. Upper and Lower Limits
4.5. Nondecreasing and Nonincreasing Functions
4.6. Limits of Numerical Sequences
4.7. Limits of Vector Functions
Problems
5 Continuous Functions
5.1. Continuous Functions on a Metric Space
5.2. Continuous Numerical Functions on the Real Line
5.3. Monotonic Functions
5.4. The Logarithm
5.5. The Exponential
5.6. Trigonometric Functions
5.7. Applications of Trigonometric Functions
5.8. Continuous Vector Functions of a Vector Variable
5.9. Sequences of Functions
Problems
6 Series
6.1. Numerical Series
6.2. Absolute and Conditional Convergence
6.3. Operations on Series
6.4. Series of Vectors
6.5. Series of Functions
6.6. Power Series
Problems
7 The Derivative
7.1. Definitions and Examples
7.2. Properties of Differentiable Functions
Con tents
7.3. The Differential
7.4. Mean Value Theorems
7.5. Concavity and Inflection Points
7.6. L'Hospital's Rules
Problems
8 Higher Derivatives
8.1. Definitions and Examples
8.2. Taylor's Formula
8.3. More on Concavity and Inflection Points
8.4. Another Version of Taylor's Formula
8.5. Taylor Series
8.6. Complex Exponentials and Trigonometric Functions
8.7. Hyperbolic Functions
Problems
9 The Integral
9.1. Definitions and Basic Properties
9.2. Area and Arc Length
9.3. Antiderivatives and Indefinite Integrals
9.4. Technique of Indefinite Integration
9.5. Evaluation of Definite Integrals
9.6. More on Area
9.7. More on Arc Length
9.8. Area of a Surface of Revolution
9.9. Further Applications of Integration
9.10. Integration of Sequences of Functions
9.1 1. Parameter-Dependent Integrals
9.12. Line Integrals
Problems
10 Analytic Functions
10.1. Basic Concepts
10.2. Line Integrals of Complex Functions
10.3. Cauchy's Theorem and Its Consequences
Contents
10.4. Residues and Isolated Singular Points
10.5. Mappings and Elementary Functions
Problems
11 Improper Integrals
1 1.1. Improper Integrals of the First Kind
1 1.2. Convergence of Improper Integrals
11.3. Improper Integrals of the Second and Third Kinds
11.4. Evaluation of Improper Integrals by Residues
1 1.5. Parameter-Dependent Improper Integrals
11.6. The Gamma and Beta Functions
Problems
Appendix A
Elementary Symbolic Logic
Appendix B
Measure and Integration on a Compact Metric Space 484
Selected Hints and Answers 489
Preface
It was with great delight that I learned of the imminent publication of an
English-language edition of my introductory course on mathematical
analysis under the editorship of Dr. R. A. Silverman. Since the literature
already includes many fine books devoted to the same general subject
matter, I would like to take this opportunity to point out the special features
of my approach.
Mathematical analysis is a large "continent" concerned with the con-
cepts of function: derivative, and integral. At present this continent consists
of many "countries" such as differential equations (ordinary and partial),
integral. equations, functions of a complex variable, differential geometry,
calculus of variations, etc. But even though the subject matter of mathe-
matical analysis can be regarded as well-established, notable changes in its
structure are still under way, In Goursat's classical "cours d'analyse" of
the twenties all of analysis is portrayed on a kind of "great plain," on a single
level of abstraction. In the books of our day, however, much attention is
paid to the appearance in analysis of various "stages" of abstraction, i.e.,
to various "structures" (Bourbaki's term) characterizing the mathematico-
logical foundations of the original constructions. This emphasis on founda-
tions clarifies the gist of the ideas involved, thereby freeing mathematics
from concern with the idiosyncracies of each object under consideration.
At the same time, an understanding of the nub of the matter allows one to
take account immediately of new objects of a different individual nature
hut of exactly the same "structural depth."
Consider, for example, Picard's proof of the existence and uniqueness of
the solution of a differential equation in which the desired function is suc-
cessively approximated on a given interval by other functions in accordance
with certain rules. This proof had been known for some time when Banach
and others formulated the "fixed point method." The Iatter plainly reveals
the nub of Picard's proof, namely the presence of a contraction opera tor in
a certain metric space. In this regard, the specific context of Picard's prob-
lem, i.e., numerical functions on an interval, a differential equation, etc.,
turns out to be quite irrelevant. As a result, the fixed point method not
only makes the "geometrical" proof of Picard's theorem more transparent,
but, by further developing the key idea of Picard's proof, even leads to the
proof of existence theorems involving neither functions on an interval nor
a differential equation. Considerations of the same kind apply equally well
to the geometry of Hilbert space, the study of differentiable functionals,
and many other topics.
Analysis presented from this point of view can be found, for example, in
the superb books by J. DieudonnC. However it seems to me that Dieudonnk's
books, for all their formal perfection, require that the reader's "mathemati-
Preface
cal I. Q." be too high. Thus, for my part, I have tried to accomodate the
interests of a larger population oft hose concerned with mathematics. There-
fore in many cases where DieudonnC instantly and almost miraculously
produces deep classical results from general considerations, so that the
reader can only take off his hat in silent admiration, the reader of my course
is invited to climb with me from the foothills of elementary topics to succes-
sive levels of abstraction and then look down from above on the various
valleys which now come into his field of view. Perhaps this approach is
thornier, hut in any event the mathematical traveler will therebv acquire
the training needed for further exploration on his own.
The present course begins with a systematic study of the real numbers,
understood to be a set of objects satisfying certain definite axioms. There
are other approaches to the theory of real numbers where things I take as
axioms are proved, starting from set theorv and the axioms for the natural
numbers (for example, a rigorous treatment in this vein can be found in
Landau's famous course). Both treatments have a key deficiency, namely
the absence of a proof of the compatibility of the axioms. Evidently modern
mathematics lacks a construction of the real numbers which is free of this
shortcoming. The whole question, far from being a mere technicality, in-
volves the very foundations of mathematical thought. In any event, this
being the case, it is really not very important where one starts a general
treatment of analysis, and my choice is governed by the consideration that
the starting point bear as close a resemblance as possible to analytic con-
structions proper.
The concepts of a mathematical structure and an isomorphism are in-
troduced in Chapter 2, after a brief digression on set theory, and a proof of
the uniqueness of the structure of real numbers (to within an isomor-
phism) is given as an illustration. Two other structures are then introduced,
namely n-dimensional space and the field of complex numbers. After a
detailed treatment of metric spaces in Chapter 3, a general theory of limits
is developed in Chapter 4. The starting point of this theory is taken to be
on the one hand a set Eequipped with a "direction," i.e., a system ofsubsets
of E with an empty intersection (this notion, closely related to the "fiiters"
of H. Cartan, is more restricted than that of a filter but is entirely adequate
for the purposes of analysis), and on the other hand a function defined on
E taking values in a metric space. All the limits considered in analysis, from
limits of a numerical sequence to the notions of the derivative and integral,
are comprised in this scheme. Chapter 5 is concerned first with some theo-
rems on continuous numerical functions on the real line and then with the
of functional equations to introduce the logarithm (from which the
exponential is obtained by inversion) and the trigonometric functions. The
Preface
algebra and topology of complex numbers and the f'rndamental theorem
of algebra are presented as applications. C:hapter 6 is on infinite series,
deaIing not only with numerical series but also with series whose terms are
vectors and functions (including power series). Chapters 7 and 8 treat dif-
ferential calculus proper, with Taylor's series leading to a natural extension
of real analysis into the complcx domain. Chapter 9 presents the general
theory of Riemann integration, together with a number of its applications.
The further development of analysis requires the technique of analytic
functions, which is considered in detail in Chapter 10. Finally Chapter 11
is devoted to improper integrals, and makes full usc of the technique of
analytic functions at our disposal.
110~'
Each chapter. is equipped with a set of problems; hints and answers to
most of these problems appear at the end of the book. To a certain extent,
the problems help to develop necesyary technical skill, but they are prima-
rily intended to iIlustrate and amplify the material in the text.