An Introduction to Analysis

306 Pages·2012·28.569 MB·English

by  Arlen Brown, Carl Pearcy

154
Graduate Texts in Mathematics
Editorial Board
1.H. Ewing F.W. Gehring P.R. Halmos
Graduate Texts in Mathematics
T AKEUnlZARING. Introduction to Axiomatic 33 HIRSCH. Differential Topology.
Set Theory. 2nd ed. 34 SPITZER. Principles of Random Walk. 2nd ed.
2 OXTOBY. Measure and Category. 2nd ed. 35 WERMER. Banach Algebras and Several
3 SCHAEFFER. Topological Vector Spaces. Complex Variables. 2nd ed.
4 HILTON/STAMMBACfI. A Course in 36 KELLEy/NAMIOKA et al. Linear Topological
Homological Algebra. Spaces.
5 MAc LANE. Categories for the Working 37 MONK. Mathematical Logic.
Mathematician. 38 GRAUBRT/FRrrzscHE. Several Complex
6 HUGHES!PIPER. Projective Planes. Variables.
7 SERRE. A Course in Aritlunl:tic. 39 ARVESON. An Invitation to C*-Algebras.
8 TAKEUnlZARING. Axiomatic Set Theory. 40 KEMENy/SNELL/KNAPP. Denumerable Markov
9 HUMPHREYS. Introduction to Lie Algebras Chains. 2nd ed.
and Representation Theory. 41 APOSTOL. Modular Functions and Dirichlet
10 COHEN. A Course in Simple Homotopy Series in Number Theory. 2nd ed.
Theory. 42 SERRE. Linear Representations of Finite
11 CONWAY. Functions of One Complex Groups.
Variable. 2nd ed. 43 GILLMAN/JERISON. Rings of Continuous
12 BEALS. Advanced Mathematical Analysis. Functions.
13 ANDERSON/FuLLER. Rings and Categories of 44 KENDIG. Elementary Algebraic Geometry.
Modules. 2nd ed. 45 LoEVE. Probability Theory I. 4th ed.
14 GOLUBITSKy/GUILEMIN. Stable Mappings and 46 LOEVE. Probability Theory II. 4th ed.
Their Singularities. 47 MOISE. Geometric Topology in Dimensions 2
15 BERBERIAN. Lectures in Functional Analysis and 3.
and Operator Theory. 48 SACHslWu. General Relativity for
16 WINTER. The Structure of Fields. Mathematicians.
17 ROSENBLATT. Random Processes. 2nd ed. 49 GRUENBERGIWEIR. Linear Geometry. 2nd ed.
18 HALMos. Measure Theory. 50 EDWARDS. Fermat's Last Theorem.
19 HALMOS. A Hilbert Space Problem Book. 51 KLINGENBERG. A Course in Differential
2nd ed. Geometry.
20 HUSEMOLLBR. Fibre Bundles. 3rd ed. 52 HARTSHORNE. Algebraic Geometry.
21 HUMPHREYS. Linear Algebraic Groups. 53 MANIN. A Course in Mathematical Logic.
22 BARNEs/MACK. An Algebraic Introduction to 54 GRAVERIWATKINS. Combinatorics with
Mathematical Logic. Emphasis on the Theory of Graphs.
23 GREUB. Linear Algebra. 4th ed. 55 BROWN/PEARCY. Introduction to Operator
24 HOLMES. Geometric Functional Analysis and Theory I: Elements of Functional Analysis.
Its Applications. 56 MASSEY. Algebraic Topology: An
25 HEWITT/STROMBERG. Real and Abstract Introduction.
Analysis. 57 CROWELL/Fox. Introduction to Knot Theory.
26 MANES. Algebraic Theories. 58 KoBLITZ. p-adic Numbers, p-adic Analysis,
27 KELLEY. General Topology. and Zeta-Functions. 2nd ed.
28 ZARISKIISAMUEL. Commutative Algebra. 59 LANG. Cyclotomic Fields.
Vol.I. 60 ARNOLD. Mathematical Methods in Classical
29 ZARISKIISAMUEL. Commutative Algebra. Mechanics. 2nd ed.
Vol.lI. 61 WHITEHEAD. Elements of Homotopy Theory.
30 JACOBSON. Lectures in Abstract Algebra I. 62 KARGAPOwvIMBRLzJAKOV. Fundamentals of
Basic Concepts. the Theory of Groups.
31 JACOBSON. Lectures in Abstract Algebra II. 63 BOLWBAS. Graph Theory.
Linear Algebra. 64 EDWARDS. Fourier Series. Vol. I. 2nd ed.
32 JACOBSON. Lectures in Abstract Algebra III.
Theory of Fields and Galois Theory. continued qfter Index
Arlen Brown Carl Pearcy
An Introduction
to Analysis
Springer-Science+Business Media, LLC
Arlen Brown Carl Pearcy
460 Kenwood Place Department of Mathematics
Bloomington, IN 47401 Texas A&M University
USA College Station, TX 77843-3368
USA
Editorial Board
J.H. Ewing F. W. Gehring P.R. Halmos
Department of Department of Department of
Mathematics Mathematics Mathematics
Indiana University University of Michigan Santa Clara University
Bloomington, IN 47405 Ann Arbor, MI 48109 Santa Clara, CA 95053
USA USA USA
With 7 Illustrations
Mathematics Subject Classifications (1991): 46-01, llAxx
Library of Congress Cataloging-in-Publication Data
Brown, Arlen, 1926-
An introduction to analysis / Arlen Brown, Carl Pearcy.
p. cm. - (Graduate texts in mathematics; 154)
Includes bibliographical references and index.
ISBN 978-1-4612-6901-4 ISBN 978-1-4612-0787-0 (eBook)
DOI 10.1007/978-1-4612-0787-0
I. Mathematical analysis. I. Pearcy, Carl M., 1935-
II. Title. III. Series.
QA300.B73 1994
515 - dc20 94-22509
Printed on acid-free paper.
© 1995 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc in 1995
Softcover reprint of the hardcover 1s t edition 1995
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer-Science+Business Media, LLC), except for brief excerpts in co
nnection with reviews or scholarly analysis. Use in connection with any form of information storage and
retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known
or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even
if the former are not especially identified, is not to be taken as a sign that such names, as
understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely
by anyone.
Production managed by Bill Imbornoni; manufacturing supervised by Genieve Shaw.
Photocomposed pages prepared from the authors' TeX file.
987654321
ISBN 978-1-4612-6901-4
Preface
As its title indicates, this book is intended to serve as a textbook for an
introductory course in mathematical analysis. In preliminary form the
book has been used in this way at the University of Michigan, Indiana
University, and Texas A&M University, and has proved serviceable. In
addition to its primary purpose as a textbook for a formal course, however,
it is the authors' hope that this book will also prove of value to readers
interested in studying mathematical analysis on their own. Indeed, we
believe the wealth and variety of examples and exercises will be especially
conducive to this end.
A word on prerequisites. With what mathematical background might a
prospective reader hope to profit from the study of this book? Our con
scious intent in writing it was to address the needs of a beginning graduate
student in mathematics, or, to put matters slightly differently, a student
who has completed an undergraduate program with a mathematics ma
jor. On the other hand, the book is very largely self-contained and should
therefore be accessible to a lower classman whose interest in mathematical
analysis has already been awakened.
The contents of the book may be briefly summarized. Chapters 1
through 3 constitute an overview of the preliminary material on which
the rest of the book is built, viz., set theory, the number systems, and lin
ear algebra. In no case do we imagine that this brief summary of material
can serve as the reader's initial encounter with these ideas. Rather we have
gathered together here the basic terminology and facts to be employed in
all that follows. In particular, in Chapters 2 and 3 we introduce only mate
rial that is assumed to be already familiar to the reader, though perhaps in
different form, and these two chapters may in most cases be treated quite
lightly. Chapter 1, on the other hand, dealing with the rudiments of set
theory, acquaints the reader with inductive proofs based on the maximum
principle in its various forms, and is deserving of more careful attention.
In Chapters 4 and 5 we present the essentials from the theory of trans
finite numbers. This treatment, while concise, presents all of the ideas and
results that will actually be employed in the sequel, and is, in any case,
fuller than is to be found in most other texts. In this connection we note
that the various number systems, formally introduced in Chapter 2, actu-
v
Preface
ally make a few brief cameo appearances in Chapter 1 as well. This minor
logical embarrassment could easily be averted, of course, but only at the
cost of unwelcome circumlocutions.
Chapters 6 through 8 constitute the heart of the book. In them we
explore in thoroughgoing fashion the structure of various metric spaces
and the mappings defined on or taking values in such spaces. The topics
and facts adduced are largely standard, though our choice of examples,
problems, and manner of presentation may make some modest claim to
freshness if not to novelty, but many of these lines of inquiry are pursued
in greater detail than will be found in most other recent texts.
The final chapter (Chapter 9) consists of a treatment of general topology.
In this chapter we equip the reader with the full panoply of topological
equipment needed for the transition from the world of classical analysis,
set in metric spaces, to "modem" or "abstract" analysis, the realm of
maximal ideal spaces, kernel-hull topologies, etc.
In formulating the sets of problems that follow each chapter we have
followed current practice. Each problem, or part of a problem, is, in effect,
a theorem to be proved, and it is our intention that the solutions should be
written out with that in mind. Thus a problem posed as a simple yes-or
no question has for its proper solution not a simple yes-or-no answer, but
rather an argument showing which is, in fact, correct. Similarly, a problem
posed as a statement of fact is really a disguised invitation to the reader
to establish the validity of that fact. No conscious attempt was made to
grade the problems according to difficulty, but they are arranged in loosely
chronological order, so that the first problems in each chapter relate to
the earlier parts of that chapter and subsequent problems to later parts.
Thus the earlier problems in anyone chapter do tum out, in general, to
be somewhat easier than the later ones. (The problem sets are an integral
part of the text; an independent reader is advised to begin to look into the
problem set at the end of a chapter as soon as he begins the perusal of
the chapter itself, just as he would do if assigned homework problems in a
formal classroom setting.)
Finally, the authors take this opportunity to express their appreciation
to the Mathematics Department of Texas A&M University for its support
during the preparation of the manuscript. In particular, the existence of
the associated 'lEX file is due almost entirely to the efforts of Professor
N. W. Naugle, a leading expert in this area, and Ms. Jan Want, who
cheerfully and conscientiously produced the entire file.
ARLEN BROWN
CARL PEARCY
June 1994
vi
Contents
Preface v
1 The rudiments of set theory 3
2 Number systems 25
3 Linear analysis 46
4 Cardinal numbers 65
5 Ordinal numbers 80
6 Metric spaces 96
7 Continuity and limits 135
8 Completeness and compactness 174
9 General topology 224
Bibliography 277
Index 279
vii
An Introduction
to Analysis
1
The rudiments of set theory
Sets and relations
We assume the reader to be familiar with the basic concepts of set and
element (or member or point) of a set, as well as with the idea of a subset
of a set, and the notions of union and intersection of a collection of sets.
We write x E A to mean that x is an element of a set A, x fj. A to mean
that x is not an element of A, and B c A (or A ::J B) to mean that B is
a subset of A. We also use the standard notation U and n for unions and
intersections, respectively.
If p( ) is some predicate that is either true or false for every element of
some set X, then the notation {x EX: p( x)} will be used to denote the
subset of X consisting of all those elements of X for which p(x) is true. If
A and B are sets, we write A \B for the difference
A\B={XEA:xfj.B},
AVB for the symmetric difference AVB = (A\B) U (B\A), and A x B
for the (Cartesian) product consisting of the set of all ordered pairs (a, b)
where a E A, bE B. It will also be convenient to reserve certain symbols
throughout the book for certain sets. Thus the empty set will consistently
be denoted by 0, the singleton on an element x, i.e., the set whose sole
element is x, by {x}, the doubleton having x and y as its only elements
by {x, y}, etc. The set of all positive integers will be denoted by N, the
set of all nonnegative integers by No, and the set of all integers by Z.
Similarly, we consistently use the symbols Q, JR, and C to denote the systems
3