Introduction to Mathematical Analysis

759 Pages·1996·19.174 MB·English

by  Steven A. Douglass

Introduction to
Mathematical Analysis
Steven A. Douglass
A
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ADDISON-WESLEY PUBLISHING COMPANY
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Original edition, entitled Introduction to Mathematical Analysis, 1st Edition,
0201508974 by Douglass, Steven A., published by Pearson Education, Inc.,
publishing as Addison-Wesley Higher Education, Copyright © 1996.
All rights reserved. No part of this book may be reproduced or transmitted in
any form or by any means, electronic or mechanical, including photocopying,
recording or by any information storage retrieval system, without permission
from Pearson Education, Inc.
KOREA edition published by PEARSON EDUCATION KOREA LTD,
Copyright © 2010
This edition is manufactured in KOREA, and is authorized for sale only in
KOREA.
ISBN 978-89-450-4022-0
ISBN 89-450-4022-6
To
Dr. Anna Penk
upon her retirement.
An inspiring teacher, steadfast
friend, and a woman of integrity.

Preface
Year after year, when teaching the sequence of courses in undergraduate analysis,
I have noticed an ever-widening gulf among the texts available on the market, the
expected content of these courses, and the observed capabilities of students in my
classes. The classic texts I most admire set a standard rather beyond the reach of
many of my students; other texts seem, for various reasons, not quite suitable for
my classes. My goal has been to write a book that is mathematically rigorous, that
treats those topics generally considered to comprise the central core of analysis at
the junior/senior level, and that is accessible to a broad spectrum of these beginning
students. I offer this text as my attempt to achieve these goals and to bridge this gap.
This text is suitable for use in a one-semester, two-quarter, or year-long course in
analysis at the undergraduate level. For students at this stage of development it is
rigorous throughout.
Mastery of the material presented here provides a solid initial training for
students intending to proceed to graduate study in pure or applied mathematics.
Students of engineering and the physical sciences, for whom these courses in anal­
ysis are terminal, will receive an essential and pragmatic introduction to classic,
nineteenth-century mathematics.
I have attempted to soften the hard edge of the mathematics by adopting a
somewhat gentle style of presentation, especially in the early chapters. As the student
progresses into the text, that style gradually hardens to more nearly approximate the
usual terse mode of modem mathematical discourse.
Having found that most of my students are intrigued by the historical devel­
opment of mathematics and by the characters in the vast, illustrious cast, I have
included brief historical discussions in the text body to serve as motivation. Most
students know little of the history of our discipline and often are delighted to realize
v
vi Preface
that mathematics is a purely human endeavor and that all of us have traversed the
path from the simple to the complex.
An ample number of examples serve to illustrate the standard techniques and
maneuvers used in analysis and to guide the beginner in the composition of careful
proofs. I have included a rich supply of exercises ranging in degree of difficulty
from the straightforward to the challenging.
Opinions differ, of course, as to what must be included in an introductory
analysis text and how that material ought to be presented. Since no one text can
adequately treat this vast subject (and still be portable), inevitable compromises are
required. Several features of this text deserve mention.
We begin, as we must, with the completeness of R and promptly obtain the
Bolzano-Weierstrass theorem. These essential topological concepts form an unfa­
miliar and uncharted region for the beginning student of analysis. Moreover, begin­
ners tend to think almost automatically in terms of sequences rather than in terms
of the more subtle concept of the continuum. Therefore we immediately discuss
sequences in R and prove that the real numbers are Cauchy complete.
In Chapter 2 we generalize these concepts to R" in order to emphasize that the
language being developed applies, with only the slightest of adjustments, as well to
n-dimensional space as to the real line. This approach prepares the student to learn
the basic facts of point set topology in Euclidean spaces and the various equivalent
formulations of completeness. Connectedness is treated briefly. Chapter 2 concludes
with the crucial concept of compactness.
The decision to treat the topology of Euclidean spaces in this early chapter is
the first significant compromise. The topology of R must be discussed early on; the
same theorems must eventually be obtained for R". Rather than repeat the identical
arguments in a later chapter and thereby increase the bulk of the final product, I have
chosen to state and prove these topological theorems for R” with the understanding
that, by letting n = 1, all the proofs remain valid as written.
Instructors who wish to postpone treatment of multivariate analysis can selec­
tively omit much of the material in Chapter 2. In this case, Section 2.1 would be
omitted and throughout Section 2.2 R" would be read as R; the notation has been
designed so the proofs are valid in either case. Section 2.3 would be omitted except
for Cantor’s nested interval theorem (Theorem 2.3.1 and Corollary 2.3.2), which is
proved independently for this purpose. Likewise Sections 2.4 and 2.5 can be used by
replacing R" by R. The Heine-Borel theorem is proved in R before its n-dimensional
version is discussed. Later, when undertaking analysis in R”, instructors can return
briefly to Chapter 2 to cover the necessary topological facts.
Once the topological preliminaries are established, the way is clear to treat
continuity in generality. Chapter 3 deals with continuity in topological terms so that
the groundwork is laid for an efficient treatment of functions of a vector variable in
Chapters 8 and 9. Again, instructors who wish to postpone the analysis of functions
of a vector variable can read R for R" and defer those portions of Chapter 3 applicable
only in R". The full import of continuity tends to be downplayed in the teaching
of introductory calculus courses in the interest of getting on with more “practical”
topics; Chapter 3 is intended to rectify this tendency.
Preface Vll
The remainder of the text is partitioned into three general areas. Chapters 4
through 7 treat functions of a single real variable. Chapters 8 through 10 discuss
functions of a vector variable. Chapters 11 through 14 present the representation of
functions by infinite series and integrals. An instructor who so wishes can proceed
directly from Chapter 6 to Chapter 11 and return to Chapter 8 at a later time.
The results of paramount importance in Chapter 4 are, of course, the Mean Value
Theorem and Taylor’s theorem. Chapter 5 treats functions of bounded variation.
This material is used in Chapters 6 and 7 and therefore is not optional. These
are functions well known to the masters of analysis in the nineteenth century that
deserve to be included in the education of every student of mathematics. Functions
of bounded variation provide an accessible model of functions more general than
the “nice” functions encountered in elementary calculus. As such, they provide a
tool to enlarge the student’s field of mathematical vision. Moreover, many of the
functions that arise in serious applications of mathematics in physics, chemistry,
and engineering are of bounded variation.
Many instructors, faced with the inevitable time constraints, will cover the
Riemann integral in Chapter 6 but will consider Chapter 7 to be optional. The text
has been structured so that this latter chapter can be omitted without disrupting
the study of subsequent chapters. It can be used as a resource for self-study by
those students who want a deeper understanding of integration. Anyone intending
to continue in graduate study in pure or applied mathematics will find Chapter 7
of value. The extension of integration theory to include Lebesgue’s integral seems
easy and natural to one already familiar with Stieltjes’s generalization of Riemann’s
work.
Chapters 8 and 9 present the differential calculus of real and vector-valued
functions of a vector variable. These chapters especially require familiarity with
linear algebra. Chapter 10 treats double and triple integrals. Since multiple integrals,
although important, are used only in a subordinate way in this text, just the basic
facts are proved. A classic and accessible proof of the role played by the Jacobian
in the transformation of multiple integrals is included.
Chapters 11 through 14 comprise the heart of the text; they treat infinite series,
improper integrals, and the representation of functions by power series, integrals,
and Fourier series. These topics introduce the student to one of the central goals
of analysis: the development of tools for the careful analysis of the behavior of
functions. Here too a compromise was required. This aspect of analysis is vast. To
remain within the usual constraints on time and space, I have chosen to present the
central core of basic facts and several classic examples but, reluctantly, to omit some
favorite topics. For example, rather than attempt to treat both the Laplace and Fourier
transforms (and therefore be forced to treat each somewhat superficially), I have
chosen to discuss the Laplace transform in detail, including the convolution product,
and have left treatment of the Fourier transform for in a course in complex analysis.
Someone who has worked through the details presented here will be able to apply
these methods to the study of other transforms. Chapter 14 includes Jordan’s elegant
conditions for the pointwise convergence of Fourier series and Fejer’s treatment of
the Cesaro summability of these series.
viii Preface
This book would not have seen the light of day without the continuous encour­
agement of my friends and colleagues to whom I owe a great debt. I especially want
to acknowledge the inspiring example of Dr. Anna Penk, who loves mathematics
and taught it with passion and who graciously suggested valuable improvements
to the various drafts of this text. I must also acknowledge my long-suffering stu­
dents who struggled, sometimes under duress, with the early versions, whose candid
responses helped me reshape my thinking and thereby improve the final product.
They reminded me, sometimes quite pointedly, that successful teaching requires
close attention to the student’s point of view. I appreciate the advice given by those
who carefully reviewed the manuscript: Ralph Grimaldi, Rose-Hulman Institute of
Technology; Ho Kuen Ng, San Jose State University; Harvey Greenwald, California
Polytechnic State University; Ken Johnson, University of Georgia; Jutta Hausen,
University of Houston; John D’Angelo, University of Illinois, Urbana-Champaign;
Howard Sherwood, University of Central Florida; Mark McConnell, Oklahoma
State University; Peter Colwell, Iowa State University; Eric Hayashi, San Francisco
State University; John Schiller, Temple University. Typographical errors, like spi­
der mites, seem to thrive and swarm in the process of writing a book; I’ve tried
to snag them all. Any that remain and any mathematical errors that persist are my
responsibility alone.
I owe special thanks to Laurie Rosatone, my editor at Addison-Wesley. She
nursed this project from its inception to the present, boosting my morale with her
enthusiasm, wit, and commitment and guiding me through the rough parts. Many
of the improvements from the original draft to the final product are the result of her
wise counsel. Her assistant, Ranjani Srinivasan, kept me smiling with her sprightly
good cheer and generous assistance. Kathy Manley, responsible for the production
phase, helped enormously with the complex task of transforming my work into
polished form. My sincere thanks to you all.
SAD