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HA.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: New Yol'k Cincinnati Chicago Millbrae Dallas Van Nowand Reinhold Company International Office, London Toronto ~lelbourne Copyright © 1971 by Litton Educational Publishing, Inc. Library of Congrc~~ Catalog Card Number: 77-14666S All righL< reserved. No part of this work covered by the copyrights hereon may be reproduced or used in any form or by any mcan~- graphic, cle<:lronic, or mechanical. including photocopying, recordint, taping, or information storage and retrieval ~ystcm!.-without written permission of lbe publisher. Manufactured in the United States of America. Publi,he<l by Van No\lrand Reinhold Company 450 West 33rd Street, New York, N.Y. IOOOI 10 9 8 7 6 S 4 3 2 I 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