Table Of ContentHA.ASER
SULLIVAN
real analysis
The University
Series
in Undergraduate
Mathematics
NUNC COGNOSCO EX PARTE
TRENT UNIVERSITY
LIBRARY
PRESENTED BY
BRUCE IVES
SUZANt.'F. BARRETT
Real Analysis
THE UNIVERSITY SERIES IN MATHEMATICS
PATRICK SUPPES lnlroduclion lo Logic
PAUL R. HALMOS Fini1e-Dimensional Veclor Spaces, 2nd Ed.
JOHN G. KEMENY AND J. LAURIE SNELL
Finile Markov Chains
PATRICK SUPPES Axiomalic Set Theory
PAUL R. HALMOS Naive Set Theory
JOHN L. KELLEY lntroduclion to Modern Algebra
IVAN NIVEN Calculus: 1\ n lnlrocluc1ory Approach
A. SEIDENBERG lcclurcs in Projeclive Geometry
MAYNARD J. MA>ISFIELD lnlroduction 10 Topclog}
FRANK M. STEWART lntroduclion 10 linear Algebra
LEON W. COHEN AND GERTRUDE EliRLICII
The Structure or the Real Number System
ELLIOTT MENDELSON lniroduction 10 ~talhcmatical Logic
HERMAN MEYER Prccalculus Ma1hcn,a1ics
ALBER I G. FADELL Calculus with Analylic Geometry
JOHN L. KELLEY Algebra: A .\1odcrn ln1roduc1ion
ANNITA TULLER A Modern Introduction 10 Geometrics
K. W. GRUENBERG ANO A. J. WEIR Linear Gcomelry
HOWARD LEVI Polynomials, Power Series, and C'alcullt\
ALBERT G. FADELL
Vector Calculu< and Differcnlial Equa1ion,
EDWARD R. FADELL AND ALBERT G. FADELL Calculus
NORMAN 8. HAASER AND JOSEPH A. SIJLLIVAN
Real Analy.is
Real Analysis
NORMAN B. HAASER
University of Notre Dame
JOSEPH A. SULLIVAN
Boston College
Van Nostrand Reinhold Company
New York/ Cincinnati I Toronto I London I 1\1.elbourm;
To Beatrice and Mary
Van Nostrand Reinhold Company Regional 001ces:
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Copyright © 1971 by Litton Educational Publishing, Inc.
Library of Congrc~~ Catalog Card Number: 77-14666S
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Preface
This book is a text for a first course in abstract analy,is. Although the
topics treated have been traditionally found in first year graduate courses,
the presentation is such as to make them :1ccc,sibtc to undergraduates with
a good background in the calculus or functions or one m,d s~vcral vari
ables. Preliminary versions of th<.' text have been used hoth in courses for
undergraduates and in courses for graduate students. 1n designing the
course, due consideration has been given to tile rccommendaiions of the
Committee on the Undergraduate Program in Mathematics of the Mathe
matical Association of America for a course in Real Analysis as de
scribed in their reports: Preg,.aduate Prl'JJaration of Res!!arc/1 Matliema
ticicms (M;iy !'963) and Pn·1iaratir111 for Graduate Study in Mathematics
(November 1965).
As a text for a first course in abstract analysis, it is intenJed to provide
a smooth transition from calculus to more advanced work in analysis. In
outline the course consists of a study or the familiar concepts or calculus
such as convergence. continuity, differentiation, and integration in a more
general and abstract setting. This serves to reinrorce and deepen the
reader's understanding of the basic concrpts of a11alysis and. at the same
time, to provide a familiarity with the abstract approach to analysis which
is valuable in many areas of applied mathematics and essential to the study
of advanced analysis.
The first three chapters- Sets and Relations, The Real Number System,
and Linear Spaces-are preliminary in nature and provide a foundation
for what follows. Although they consist to a considerable extent in the
introduction of notation and review of definitions and elementary results,
there are a number of topics that are likely to be new to the reader. The
introduction of the real numbers via Cauchy ~equencc~ of rationals not
only establishes the basic properties of this number system, but also pro
vides a model for the completion of an arbitrary metric space. The treat
ment of linear spaces emphasizes those aspects pertinent to analysis which
might not be treated adequately in a Linear Algebra course.
The main body of the text starts with Chapter 4, Metric Spaces. ln the
context of metric ~paces, we study the properties of completeness. com
pactness, and connectedness, as well as the continuity of functions. In
the next chapter we illustrate the utility of metric spaces by proving a
fixed-point theorem in this setting and tht11 using it to obtain results on
the existence o[ solutions of differential equations, integral equations, and
systems of linear algebraic equations.
V
Prdare
The study of integration begins in Chapter 6 with a review of the Rie
mann integral. Then the Lcbesque integral for [unctions from R• to R is
developed via the Daniell approach. This method of development is used
because it makes good contact "'ith the Riemann integral and, also,
because it gets us to the central ideas more quickly than docs the develOJ>
ment via measure theory.
Returning to a study or abstract spaces we consider normed linear
spaces wluch combine the properties of linear spaces and metric spaces.
The primary model for these spaces is the Euclidean space R• and im
portant examples considered are the sequence spaces I, nnd the function
spaces .!t',. In the setting of normed linear spaces, we consider differentia
tion and some results on approximation including the Stone-Weierstrass
Theorem.
In Chapter 9 the Fundamental Theorems of Calculus arc obtained for
the Lebesgue integral or functions from R to R. Here is number of con
cepts arc introduced which lead in a natural way to a discu,sion of the
Stieltjcs integral which is considered in Chapter 10.
In the final chapter we treat inner product spaces and orthogonal bases
for such spaces. As examples or orthogonal bases for certmn inner product
spaces, we consider the comple, exponcn1ial and trigonometric sequence~.
the Legendre polynomials, and the Hennite polynomials Other examples
occur in problems. Finally, poinl\1 be covergcncc of Fourier series and
Fourier integrals is discussed.
We wish lo thank several anonymous reviewers who read an early set
or class notes upon which part or this book was based and who made
many helpful suggestions. We also express our gratitude to our students
who aided us as we sought better methods of presenting the material.
Finally, we wish to thank Mrs. Verna Osborne for her care and patience
in producing an excellent typescript from the manuscript.
Norman B. Haascr
Joseph A. Sullivan
Contents
Preface / v
1 Sets and Relations
1 fntroduction /
2 Sets / l
3 Relations, Functions / 4
4 Partial Orders and Equivalence Relations ; JI
5 Countable Sets / 14
6. Uncountable Sets / 19
2 The Real Number System
1 Introduction / 22
2 Ordered Rings and Fields / 23
3 Cauchy Sequences / 26
4 The Real Numbers / 29
5 Completeness of the Real Number System I 35
6 The Complex Numbers / 38
3 Linear Spaces
1 Introduction / 41
2 Linear Spaces / 41
3 Hamel Dases / 45
4 Linear Transformations I 50
5 Algebras / 53
4 Metric Spaces
1 Introduction / 56
2 Metric Spaces / 56
3 Open and Closed Sets / 61
4 Continuity / 64
5 Topological Spaces / 68
6 Convergence and Completeness / 72
7 Completion of a Metric Space / 77
8 Compactness / 79
9 Sequential Compactness / 82
TU
Content•
10 Heine-Borel and Arzclit-Ascoli Theorems / 85
11 Connectedness / 88
5 A 1'ixcd-Point Theorem; Applications
I Introduction / 93
2 A Fixed-Point Theorem / 93
3 Linear Algebraic Equations / 96
4 Integral Equations / 100
5 DifTerential Equations I J0 4
6 The Lcl,c\guc lnlegral
1 Introduction / 107
2 The Riemann Integral / 108
3 Step Functions / 11]
4 Sets of Measure Zero / 114
5 The Riemann Integral Continued / I 17
6 Extension of the fntegral of Step Functions / I 20
7 1 he Lebesgue Integral / 124
8 Some Convergence Theorems 128
9 Fubini's Theorem / 132
10 Measurable Functions / 135
1 I Complex-Valued Functions / 140
12 Measurable Sets / 142
13 Measure / 144
1-t The Lebesgue lntcgr.:il over a Measurable Set I 149
15 The Daniell Integral / 152
7 Normed Linear Sp:tC(!S
1 Introduction / I 57
2 1 he Holder and Minh.owski Inequalities ; 157
3 Normed Linea,· Spaces / 160
4 Examples of l'\ormcd Linear Spaces / 163
5 Linear Tramfonnations . 166
6 Isomorphisms / 171
7 rinite-dimensional Spaces ' 174
8 Banach Spaces / 177
9 Series / 181
10 The Space of Bounded Functions 183
8 Approximation
1 Introduction / 187
2 The Stone-Weierstrass 'I hcorem / 187
