basic real analysis - stony brook universityaknapp/download/b2-realanal...basic algebra advanced...

Basic Real Analysis

Digital Second EditionsBy Anthony W. Knapp

Basic Algebra

Advanced Algebra

Basic Real Analysis,with an appendix Elementary Complex Analysis

Advanced Real Analysis

Anthony W. Knapp

Basic Real AnalysisWith an Appendix Elementary Complex Analysis

Along with a Companion Volume Advanced Real Analysis

Digital Second Edition, 2016

Published by the AuthorEast Setauket, New York

Anthony W. Knapp81 Upper Sheep Pasture RoadEast Setauket, N.Y. 117331729, U.S.A.Email to: [email protected]: www.math.stonybrook.edu/aknapp

Title: Basic Real Analysis, with an appendix Elementary Complex AnalysisCover: An instance of the Rising Sun Lemma in Section VII.1.

Mathematics Subject Classification (2010): 2801, 2601, 4201, 5401, 3401, 3001, 3201.

First Edition, ISBN-13 978-0-8176-3250-2c2005 Anthony W. KnappPublished by Birkhauser Boston

Digital Second Edition, not to be sold, no ISBNc2016 Anthony W. KnappPublished by the author

All rights reserved. This file is a digital second edition of the above named book. The text, images,and other data contained in this file, which is in portable document format (PDF), are proprietary tothe author, and the author retains all rights, including copyright, in them. The use in this file of tradenames, trademarks, service marks, and similar items, even if they are not identified as such, is notto be taken as an expression of opinion as to whether or not they are subject to proprietary rights.All rights to print media for the first edition of this book have been licensed to Birkhuser Boston,c/o Springer Science+Business Media Inc., 233 Spring Street, New York, NY 10013, USA, andthis organization and its successor licensees may have certain rights concerning print media for thedigital second edition. The author has retained all rights worldwide concerning digital media forboth the first edition and the digital second edition.The file is made available for limited noncommercial use for purposes of education, scholarship, andresearch, and for these purposes only, or for fair use as understood in the United States copyright law.Users may freely download this file for their own use and may store it, post it online, and transmit itdigitally for purposes of education, scholarship, and research. They may not convert it from PDF toany other format (e.g., EPUB), they may not edit it, and they may not do reverse engineering with it.In transmitting the file to others or posting it online, users must charge no fee, nor may they includethe file in any collection of files for which a fee is charged. Any exception to these rules requireswritten permission from the author.Except as provided by fair use provisions of theUnited States copyright law, no extracts or quotationsfrom this file may be used that do not consist of whole pages unless permission has been granted bythe author (and by Birkhuser Boston if appropriate).The permission granted for use of the whole file and the prohibition against charging fees extend toany partial file that contains only whole pages from this file, except that the copyright notice on thispage must be included in any partial file that does not consist exclusively of the front cover page.Such a partial file shall not be included in any derivative work unless permission has been grantedby the author (and by Birkhuser Boston if appropriate).Inquiries concerning print copies of either edition should be directed to Springer Science+BusinessMedia Inc.

iv

To Susan

and

To My Children, Sarah and William,

and

To My Real-Analysis Teachers:

Salomon Bochner, William Feller, Hillel Furstenberg,

Harish-Chandra, Sigurdur Helgason, John Kemeny,

John Lamperti, Hazleton Mirkil, Edward Nelson,

Laurie Snell, Elias Stein, Richard Williamson

CONTENTS

Contents of Advanced Real Analysis xiDependence Among Chapters xiiPreface to the Second Edition xiiiPreface to the First Edition xvList of Figures xviiiAcknowledgments xixGuide for the Reader xxiStandard Notation xxv

I. THEORY OF CALCULUS IN ONE REAL VARIABLE 11. Review of Real Numbers, Sequences, Continuity 22. Interchange of Limits 133. Uniform Convergence 154. Riemann Integral 265. Complex-Valued Functions 416. Taylors Theorem with Integral Remainder 437. Power Series and Special Functions 458. Summability 549. Weierstrass Approximation Theorem 5910. Fourier Series 6211. Problems 78

II. METRIC SPACES 831. Definition and Examples 842. Open Sets and Closed Sets 923. Continuous Functions 964. Sequences and Convergence 985. Subspaces and Products 1036. Properties of Metric Spaces 1067. Compactness and Completeness 1098. Connectedness 1169. Baire Category Theorem 11810. Properties of C(S) for Compact Metric S 12211. Completion 12812. Problems 131

vii

viii Contents

III. THEORY OF CALCULUS IN SEVERAL REAL VARIABLES 1361. Operator Norm 1362. Nonlinear Functions and Differentiation 1403. Vector-Valued Partial Derivatives and Riemann Integrals 1474. Exponential of a Matrix 1495. Partitions of Unity 1526. Inverse and Implicit Function Theorems 1537. Definition and Properties of Riemann Integral 1628. Riemann Integrable Functions 1679. Fubinis Theorem for the Riemann Integral 17110. Change of Variables for the Riemann Integral 17311. Arc Length and Integrals with Respect to Arc Length 18112. Line Integrals and Conservative Vector Fields 19413. Greens Theorem in the Plane 20314. Problems 212

IV. THEORY OF ORDINARY DIFFERENTIAL EQUATIONSAND SYSTEMS 2181. Qualitative Features and Examples 2182. Existence and Uniqueness 2223. Dependence on Initial Conditions and Parameters 2294. Integral Curves 2345. Linear Equations and Systems, Wronskian 2366. Homogeneous Equations with Constant Coefficients 2437. Homogeneous Systems with Constant Coefficients 2468. Series Solutions in the Second-Order Linear Case 2539. Problems 261

V. LEBESGUEMEASURE AND ABSTRACTMEASURE THEORY 2671. Measures and Examples 2672. Measurable Functions 2743. Lebesgue Integral 2774. Properties of the Integral 2815. Proof of the Extension Theorem 2896. Completion of a Measure Space 2987. Fubinis Theorem for the Lebesgue Integral 3018. Integration of Complex-Valued and Vector-Valued Functions 3109. L1, L2, L, and Normed Linear Spaces 31510. Arc Length and Lebesgue Integration 32511. Problems 327

Contents ix

VI. MEASURE THEORY FOR EUCLIDEAN SPACE 3341. Lebesgue Measure and Other Borel Measures 3352. Convolution 3443. Borel Measures on Open Sets 3524. Comparison of Riemann and Lebesgue Integrals 3565. Change of Variables for the Lebesgue Integral 3586. HardyLittlewood Maximal Theorem 3657. Fourier Series and the RieszFischer Theorem 3728. Stieltjes Measures on the Line 3779. Fourier Series and the DirichletJordan Theorem 38410. Distribution Functions 38811. Problems 390

VII. DIFFERENTIATION OF LEBESGUE INTEGRALSON THE LINE 3951. Differentiation of Monotone Functions 3952. Absolute Continuity, Singular Measures, and

Lebesgue Decomposition 4023. Problems 408

VIII. FOURIER TRANSFORM IN EUCLIDEAN SPACE 4111. Elementary Properties 4112. Fourier Transform on L1, Inversion Formula 4153. Fourier Transform on L2, Plancherel Formula 4194. Schwartz Space 4225. Poisson Summation Formula 4276. Poisson Integral Formula 4307. Hilbert Transform 4358. Problems 442

IX. L p SPACES 4481. Inequalities and Completeness 4482. Convolution Involving L p 4563. Jordan and Hahn Decompositions 4584. RadonNikodym Theorem 4595. Continuous Linear Functionals on L p 4636. RieszThorin Convexity Theorem 4667. Marcinkiewicz Interpolation Theorem 4768. Problems 484

x Contents

X. TOPOLOGICAL SPACES 4901. Open Sets and Constructions of Topologies 4902. Properties of Topological Spaces 4963. Compactness and Local Compactness 5004. Product Spaces and the Tychonoff Product Theorem 5075. Sequences and Nets 5126. Quotient Spaces 5207. Urysohns Lemma 5238. Metrization in the Separable Case 5259. AscoliArzela and StoneWeierstrass Theorems 52610. Problems 529

XI. INTEGRATION ON LOCALLY COMPACT SPACES 5341. Setting 5342. Riesz Representation Theorem 5393. Regular Borel Measures 5534. Dual to Space of Finite Signed Measures 5585. Problems 566

XII. HILBERT AND BANACH SPACES 5701. Definitions and Examples 5702. Geometry of Hilbert Space 5763. Bounded Linear Operators on Hilbert Spaces 5854. HahnBanach Theorem 5875. Uniform Boundedness Theorem 5936. Interior Mapping Principle 5957. Problems 599

APPENDIX A. BACKGROUND TOPICS 603A1. Sets and Functions 603A2. Mean Value Theorem and Some Consequences 610A3. Inverse Function Theorem in One Variable 611A4. Complex Numbers 613A5. Classical Schwarz Inequality 614A6. Equivalence Relations 614A7. Linear Transformations, Matrices, and Determinants 615A8. Factorization and Roots of Polynomials 618A9. Partial Orderings and Zorns Lemma 623A10. Cardinality 627

Contents xi

APPENDIX B. ELEMENTARY COMPLEX ANALYSIS 631B1. Complex Derivative and Analytic Functions 632B2. Complex Line Integrals 636B3. Goursats Lemma and the Cauchy Integral Theorem 641B4. Cauchy Integral Formula 648B5. Taylors Theorem 654B6. Local Properties of Analytic Functions 656B7. Logarithms and Winding Numbers 660B8. Operations on Taylor Series 665B9. Argument Principle 669B10. Residue Theorem 673B11. Evaluation of Definite Integrals 675B12. Global Theorems in Simply Connected Regions 681B13. Global Theorems in General Regions 694B14. Laurent Series 696B15. Holomorphic Functions of Several Variables 701B16. Problems 704

Hints for Solutions of Problems 715Selected References 793Index of Notation 795Index 799

CONTENTS OFADVANCED REAL ANALYSIS

I. Introduction to Boundary-Value ProblemsII. Compact Self-Adjoint OperatorsIII. Topics in Euclidean Fourier AnalysisIV. Topics in Functional AnalysisV. DistributionsVI. Compact and Locally Compact GroupsVII. Aspects of Partial Differential EquationsVIII. Analysis on ManifoldsIX. Foundations of ProbabilityX. Introduction to Wavelets

DEPENDENCE AMONG CHAPTERS

Below is a chart of the main lines of dependence of chapters on prior chapters.The dashed lines indicate helpful motivation but no logical dependence. Apartfrom that, particular examplesmaymake use of information from earlier chaptersthat is not indicated by the chart. Appendix B is used in problems at the ends ofChapters IV, VI, and VIII, and it is used also in Section IX.6.

I, II, III in order

V IV

VI

VIII VII X

IX

IX.6 XI

XII

xii

PREFACE TO THE SECOND EDITION

In the years since publicationof thefirst editionofBasicRealAnalysis, many read-ers have reacted to the book by sending comments, suggestions, and corrections.People appreciated the overall comprehensive nature of the book, associating thisfeature in part with the large number of problems that develop so many sidelightsand applications of the theory. Some people wondered whether a way mightbe found for a revision to include some minimal treatment of Stokess Theoremand complex analysis, despite the reservations I expressed in the original prefaceabout including these topics.Along with the general comments and specific suggestions were corrections,

well over a hundred in all, that needed to be addressed in any revision. Many ofthe corrections were of minor matters, yet readers should not have to cope witherrors along with new material. Fortunately no results in the first edition neededto be deleted or seriously modified, and additional results and problems could beincluded without renumbering.For the first edition, the author granted a publishing license to Birkhauser

Boston that was limited to print media, leaving the question of electronic publi-cation unresolved. A major change with the second edition is that the question ofelectronic publication has now been resolved, and a PDF file, called the digitalsecondedition, is beingmade freely available to everyoneworldwide for personaluse. This file may be downloaded from the authors own Web page and fromelsewhere.The main changes to the first edition of Basic Real Analysis are as follows:

A careful treatment of arc length, line integrals, and Greens Theorem forthe plane has been added at the end of Chapter III. These aspects of StokessTheorem can be handled by the same kinds of techniques of real analysisas in the first edition. Treatment of aspects of Stokess Theorem in higherdimensions would require a great deal more geometry, for reasons given inSection III.13, and that more general treatment has not been included.

The core of a first course in complex analysis has been included asAppendixB.Emphasis is on those aspects of elementary complex analysis that are usefulas tools in real analysis. The appendix includes more than 80 problems, andsomestandard topics in complex analysis are developed in theseproblems. Thetreatment assumes parts of Chapters IIII as a prerequisite. How the appendixfits into the plan of the book is explained in the Guide for the Reader.

xiii

xiv Preface to the Second Edition

A new section in Chapter IX proves and applies the RieszThorin ConvexityTheorem, a fundamental result about L p spaces that takes advantage of ele-mentary complex analysis.

About20problemshavebeenaddedat the endsofChapters IXII.Chiefly theseare of three kinds: some illustrate the new topics of arc length, line integrals,and Greens Theorem; some make use of elementary complex analysis asin Appendix B to shed further light on results and problems in the variouschapters; and some relate to the topic of Banach spaces in Chapter XII .

The corrections sent by readers and by reviewers have been made. The mostsignificant such correction was a revision to the proof of Zorns Lemma, theearlier proof having had a gap.The material in Appendix B is designed as the text of part of a first course in

complex analysis. I taught such a course myself on one occasion. A course incomplex analysis invariably begins with some preliminary material, and that canbe taken fromChapters I to III; details appear in theGuide to theReader. AppendixB forms the core of the course, dealing with results having an analytic flavor,including the part of the theory due to Cauchy. The topic of conformal mapping,which has a more geometric flavor, has been omitted, and some instructors mightfeel obliged to include something on this topic in the course. Appendix B statesthe Riemann Mapping Theorem at one point but does not prove it; all the toolsneeded for its proof, however, are present in the appendix and its problems. Oftenan instructor will end a first course in complex analysis with material on infiniteseries and products of functions, or of aspects of the theory of special functions,or on analytic continuation. Supplementary notes on any such topics would benecessary.It was Benjamin Levitt, Birkhauser mathematics editor in New York, who

encouraged the writing of this second edition, whomade a number of suggestionsabout pursuing it, and who passed along comments from several anonymousreferees about the strengths and weaknesses of the book. I am especially gratefulto those readers who have sent me comments over the years. Many of thecorrections that were made were kindly sent to me either by S. H. Kim of SouthKoreaor by JacquesLarochelleofCanada. The correction to theproof ofTheorem1.35 was kindly sent by Glenn Jia of China. The long correction to the proof ofZorns Lemma resulted from a discussion with Qiu Ruyue. The typesetting wasdone by the program Textures using AMS-TEX, and the figures were drawn withMathematica.Just as with the first edition, I invite corrections and other comments from

readers. For as long as I am able, I plan to point to a list of known correctionsfrom my own homepage, www.math.stonybrook.edu/aknapp.

A. W. KNAPPFebruary 2016

PREFACE TO THE FIRST EDITION

This book and its companion volume, Advanced Real Analysis, systematicallydevelop concepts and tools in real analysis that are vital to every mathematician,whether pure or applied, aspiring or established. The two books together containwhat the young mathematician needs to know about real analysis in order tocommunicate well with colleagues in all branches of mathematics.The books are written as textbooks, and their primary audience is students who

are learning the material for the first time and who are planning a career in whichthey will use advanced mathematics professionally. Much of the material in thebooks corresponds to normal course work. Nevertheless, it is often the case thatcore mathematics curricula, time-limited as they are, do not include all the topicsthat one might like. Thus the book includes important topics that may be skippedin required courses but that the professional mathematician will ultimately wantto learn by self-study.The content of the required courses at each university reflects expectations of

what studentsneedbeforebeginning specializedstudyandworkona thesis. Theseexpectations vary from country to country and from university to university. Evenso, there seems to be a rough consensus aboutwhatmathematics a plenary lecturerat a broad international or national meeting may take as known by the audience.The tables of contents of the two books represent my own understanding of whatthat degree of knowledge is for real analysis today.

Key topics and features of Basic Real Analysis are as follows: Early chapters treat the fundamentals of real variables, sequences and seriesof functions, the theory of Fourier series for the Riemann integral, metricspaces, and the theoretical underpinnings of multivariable calculus and ordi-nary differential equations.

Subsequent chapters develop the Lebesgue theory in Euclidean and abstractspaces, Fourier series and the Fourier transform for the Lebesgue integral,point-set topology, measure theory in locally compact Hausdorff spaces, andthe basics of Hilbert and Banach spaces.

The subjects of Fourier series and harmonic functions are used as recurringmotivation for a number of theoretical developments.

The development proceeds from the particular to the general, often introducingexamples well before a theory that incorporates them.

xv

xvi Preface to the First Edition

More than 300 problems at the ends of chapters illuminate aspects of thetext, develop related topics, and point to additional applications. A separate55-page section Hints for Solutions of Problems at the end of the book givesdetailed hints for most of the problems, together with complete solutions formany.Beyond a standard calculus sequence in one and several variables, the most

important prerequisite for using Basic Real Analysis is that the reader alreadyknow what a proof is, how to read a proof, and how to write a proof. Thisknowledge typically is obtained from honors calculus courses, or from a coursein linear algebra, or from a first junior-senior course in real variables. In addition,it is assumed that the reader is comfortablewith amodest amount of linear algebra,including row reduction of matrices, vector spaces and bases, and the associatedgeometry. A passing acquaintance with the notions of group, subgroup, andquotient is helpful as well.Chapters IIV are appropriate for a single rigorous real-variables course and

may be used in either of two ways. For students who have learned about proofsfrom honors calculus or linear algebra, these chapters offer a full treatment of realvariables, leaving out only the more familiar parts near the beginningsuch aselementary manipulations with limits, convergence tests for infinite series withpositive scalar terms, and routine facts about continuity and differentiability. Forstudents who have learned about proofs from a first junior-senior course in realvariables, these chapters are appropriate for a second such course that begins withRiemann integration and sequences and series of functions; in this case the firstsection of Chapter I will be a review of some of the more difficult foundationaltheorems, and the course can conclude with an introduction to the Lebesgueintegral from Chapter V if time permits.Chapters V through XII treat Lebesgue integration in various settings, as well

as introductions to the Euclidean Fourier transform and to functional analysis.Typically this material is taught at the graduate level in the United States, fre-quently in oneof threeways: ThefirstwaydoesLebesgue integration inEuclideanand abstract settings and goes on to consider the Euclidean Fourier transform insome detail; this corresponds to Chapters VVIII. A second way does Lebesgueintegration in Euclidean and abstract settings, treats L p spaces and integration onlocally compact Hausdorff spaces, and concludes with an introduction to Hilbertand Banach spaces; this corresponds to Chapters VVII, part of IX, and XIXII.A third way combines an introduction to the Lebesgue integral and the EuclideanFourier transform with some of the subject of partial differential equations; thiscorresponds to some portion of Chapters VVI and VIII, followed by chaptersfrom the companion volume, Advanced Real Analysis.In my own teaching, I have most often built one course around Chapters IIV

and another around Chapters VVII, part of IX, and XIXII. I have normally

Preface to the First Edition xvii

assigned the easier sections of Chapters II and X as outside reading, indicatingthe date when the lectures would begin to use that material.More detailed information about how the book may be used with courses may

be deduced from the chart Dependence among Chapters on page xiv and thesection Guide to the Reader on pages xvxvii.The problems at the ends of chapters are an important part of the book. Some

of them are really theorems, some are examples showing the degree to whichhypotheses can be stretched, and a few are just exercises. The reader gets noindication which problems are of which type, nor of which ones are relativelyeasy. Each problem can be solved with tools developed up to that point in thebook, plus any additional prerequisites that are noted.

Two omissions from the pair of books are of note. One is any treatment ofStokess Theorem and differential forms. Although there is some advantage,when studying these topics, in having the Lebesgue integral available and inhaving developed an attitude that integration can be defined by means of suitablelinear functionals, the topic of Stokess Theorem seems to fit better in a bookabout geometry and topology, rather than in a book about real analysis.The other omission concerns the use of complex analysis. It is tempting to try

to combine real analysis and complex analysis into a single subject, but my ownexperience is that this combination does not work well at the level of Basic RealAnalysis, only at the level of Advanced Real Analysis.Almost all of the mathematics in the two books is at least forty years old, and I

make no claim that any result is new. The books are a distillation of lecture notesfrom a 35-year period of my own learning and teaching. Sometimes a problem atthe end of a chapter or an approach to the exposition may not be a standard one,but no attempt has been made to identify such problems and approaches. In thereverse direction it is possible that my early lecture notes have directly quotedsome source without proper attribution. As an attempt to rectify any difficultiesof this kind, I have included a section of Acknowledgments on pages xixxxof this volume to identify the main sources, as far as I can reconstruct them, forthose original lecture notes.I amgrateful toAnnKostant andStevenKrantz for encouraging this project and

for making many suggestions about pursuing it, and to Susan Knapp and DavidKramer for helping with the readability. The typesetting was by AMS-TEX, andthe figures were drawn with Mathematica.I invite corrections and other comments from readers. I plan to maintain a list

of known corrections on my own Web page.A. W. KNAPP

May 2005

LIST OF FIGURES

1.1. Approximate identity 601.2. Fourier series of sawtooth function 661.3. Dirichlet kernel 702.1. An open set centered at the origin in the hedgehog space 892.2. Open ball contained in an intersection of two open balls 933.1. Polygonal approximation for estimating arc length 1823.2. Cycloid 1933.3. A piecewise C1 curve that retraces part of itself 2003.4. Greens Theorem for an annulus 2063.5. Failure of inscribed triangles to give a useful notion of

surface area 2104.1. Graphs of solutions of some first-order ordinary differential

equations 2204.2. Integral curve of a vector field 2344.3. Graph of Bessel function J0(t) 2606.1. Construction of a Cantor function F 3817.1. Rising Sun Lemma 3969.1. Geometric description of points in the RieszThorin

Convexity Theorem 468B.1. First bisection of the rectangle R in Goursats Lemma 642B.2. Handling an exceptional point in Goursats Lemma 644B.3. Reduction steps in the proof of Theorem B.9 645B.4. Computation of

R ( z)

1 d over a standard circlewhen z is not the center 647

B.5. Construction of a cycle and a point a with n( , a) = 1 684B.6. Two polygonal paths 1 and 2 from z0 to z in the region U 686B.7. Boundary cycle: (a) as the difference of two cycles,

(b) as the sum of three cycles 696

xviii

ACKNOWLEDGMENTS

The author acknowledges the sources below as themain ones he used in preparingthe lectures from which this book evolved. Any residual unattributed directquotations in the book are likely to be from these.The descriptions below have been abbreviated. Full descriptions of the books

and Stone article may be found in the section Selected References at theend of the book. The item Fellers Functional Analysis refers to lectures byWilliam Feller at Princeton University for Fall 1962 and Spring 1963, and theitem Nelsons Probability refers to lectures by Edward Nelson at PrincetonUniversity for Spring 1963.This list is not to be confused with a list of recommended present-day reading

for these topics; newer books deserve attention.

CHAPTER I. Rudins Principles of Mathematical Analysis, ZygmundsTrigonometric Series.CHAPTER II. Fellers Functional Analysis, Kelleys General Topology,

Stones A generalized Weierstrass approximation theorem.CHAPTER III. Rudins Principles of Mathematical Analysis, Spivaks

Calculus on Manifolds.CHAPTER IV. CoddingtonLevinsons Theory of Ordinary Differential

Equations, Kaplans Ordinary Differential Equations.CHAPTER V. Halmoss Measure Theory, Rudins Principles of Mathematical

Analysis.CHAPTER VI. Rudins Principles of Mathematical Analysis, Rudins Real and

Complex Analysis, Sakss Theory of the Integral, SpivaksCalculus onManifolds,SteinWeisss Introduction to Fourier Analysis on Euclidean Spaces.CHAPTER VII. RieszNagys Functional Analysis, Zygmunds Trigonometric

Series.CHAPTER VIII. Steins Singular Integrals and Differentiability Properties of

Functions, SteinWeisss Introduction to Fourier Analysis on Euclidean Spaces.CHAPTER IX. DunfordSchwartzs Linear Operators, Fellers Functional

Analysis, Halmoss Measure Theory, Sakss Theory of the Integral, SteinsSingular Integrals and Differentiability Properties of Functions.

xix

xx Acknowledgments

CHAPTER X. Kelleys General Topology, Nelsons Probability.CHAPTER XI. Fellers Functional Analysis, Halmoss Measure Theory,

Nelsons Probability.CHAPTER XII. DunfordSchwartzs Linear Operators, Fellers Functional

Analysis, RieszNagys Functional Analysis.APPENDIX A. For Sections 1, 9, 10: DunfordSchwartzs Linear Operators,

HaydenKennisons ZermeloFraenkel Set Theory, Kelleys General Topology.APPENDIX B. Ahlfors, Complex Analysis, Gunnings Introduction to Holo-

morphic Functions of Several Variables.

GUIDE FOR THE READER

This section is intended to help the reader find out what parts of each chapter aremost important and how the chapters are interrelated. Further information of thiskind is contained in the abstracts that begin each of the chapters.The book pays attention to certain recurring themes in real analysis, allowing

a person to see how these themes arise in increasingly sophisticated ways. Ex-amples are the role of interchanges of limits in theorems, the need for certainexplicit formulas in the foundations of subject areas, the role of compactness andcompleteness in existence theorems, and the approach of handling nice functionsfirst and then passing to general functions.All of these themes are introduced in Chapter I, and already at that stage they

interact in subtle ways. For example, a natural investigation of interchanges oflimits in Sections 23 leads to the discovery of Ascolis Theorem, which is afundamental compactness tool for proving existence results. Ascolis Theoremis proved by the Cantor diagonal process, which has other applications tocompactness questions and does not get fully explained until Chapter X. Theconsequence is that, no matter where in the book a reader plans to start, everyonewill be helped by at least leafing through Chapter I.The remainder of this section is an overview of individual chapters and groups

of chapters.Chapter I. Every section of this chapter plays a role in setting up matters

for later chapters. No knowledge of metric spaces is assumed anywhere in thechapter. Section1will be a review for anyonewhohas alreadyhad a course in real-variable theory; the section shows how compactness and completeness addressall the difficult theorems whose proofs are often skipped in calculus. Section 2begins the development of real-variable theory at the point of sequences and seriesof functions. It contains interchange results that turn out to be special cases ofthe main theorems of Chapter V. Sections 89 introduce the approach of handlingnice functions before general functions, and Section 10 introduces Fourier series,which provided a great deal of motivation historically for the development of realanalysis and are used in this book in that same way. Fourier series are somewhatlimited in the setting of Chapter I because one encounters no class of functions,other than infinitely differentiable ones, that corresponds exactly to some class ofFourier coefficients; as a result Fourier series, with Riemann integration in use,

xxi

xxii Guide for the Reader

are not particularly useful for constructing new functions from old ones. Thisdefect will be fixed with the aid of the Lebesgue integral in Chapter VI.Chapter II. Now that continuity and convergence have been addressed on

the line, this chapter establishes a framework for these questions in higher-dimensional Euclidean space and other settings. There is no point in ad hocdefinitions for each setting, and metric spaces handle many such settings at once.Chapter X later will enlarge the framework from metric spaces to topologicalspaces. Sections 16 of Chapter II are routine. Section 7, on compactnessand completeness, is the core. The Baire Category Theorem in Section 9 is notused outside of Chapter II until Chapter XII, and it may therefore be skippedtemporarily. Section 10 contains the StoneWeierstrass Theorem, which is afundamental approximation tool. Section 11 is used in some of the problems butis not otherwise used in the book.Chapter III. This chapter does for the several-variable theory what Chapter I

has done for the one-variable theory. Themain results are the Inverse and ImplicitFunction Theorems in Section 6 and the change-of-variables formula for multipleintegrals in Section 10. The change-of-variables formula has to be regarded asonly a preliminary version, since what it directly accomplishes for the changeto polar coordinates still needs supplementing; this difficulty will be repaired inChapter VI with the aid of the Lebesgue integral. Section 4, on exponentials ofmatrices, may be skipped if linear systems of ordinary differential equations aregoing to be skipped in Chapter IV. Sections 1113 contain a careful treatmentof arc length, line integrals, and Greens Theorem for the plane. These sectionsemphasize properties of parametrized curves that are unchanged when the curveis reparametrized; length is an example. An important point to bear in mind isthat two curves are always reparametrizations of each other if they have the sameimage in the plane and they are both traced out in one-one fashion. This theoryis tidier if carried out in the context of Lebesgue integration, but its placement inthe text soon after Riemann integration is traditional. The difficulty with usingRiemann integrals arises already in the standard proof of Greens Theorem fora circle, which parametrizes each quarter of the circle twice, once with y interms of x and once with x in terms of y. The problem is that in each of theseparametrizations, the derivative of the one variable with respect to the other isunbounded, and thus arc length is not given by a Riemann integral. Some ofthe problems at the end of the chapter introduce harmonic functions; harmonicfunctions will be combined with Fourier series in problems in later chapters tomotivate and illustrate some of the development.Chapter IV provides theoretical underpinnings for the material in a traditional

undergraduate course in ordinary differential equations. Nothing later in the bookis logically dependent on Chapter IV; however, Chapter XII includes a discussionof orthogonal systems of functions, and the examples of these that arise in Chapter

Guide for the Reader xxiii

IV are helpful as motivation. Some people shy away from differential equationsand might wish to treat Chapter IV only lightly, or perhaps not at all. The subjectis nevertheless of great importance, and Chapter IV is the beginning of it. Aminimal treatment of Chapter IV might involve Sections 12 and Section 8, allof which visibly continue the themes begun in Chapter I.Chapters VVI treat the core of measure theoryincluding the basic conver-

gence theorems for integrals, the development of Lebesgue measure in one andseveral variables, Fubinis Theorem, the metric spaces L1 and L2 and L, andthe use of maximal theorems for getting at differentiation of integrals and othertheorems concerning almost-everywhere convergence. In Chapter V Lebesguemeasure in one dimension is introduced right away, so that one immediately hasthe most important example at hand. The fundamental Extension Theorem forgettingmeasures tobedefinedon -rings and -algebras is statedwhenneededbutis provedonly after thebasic convergence theorems for integrals havebeenproved;the proof in Sections 56 may be skipped on first reading. Section 7, on FubinisTheorem, is a powerful result about interchange of integrals. At the same timethat it justifies interchange, it also constructs a double integral; consequentlythe section prepares the way for the construction in Chapter VI of n-dimensionalLebesguemeasure from 1-dimensional Lebesguemeasure. Section 10 introducesnormed linear spaces along with the examples of L1 and L2 and L, and it goeson to establish some properties of all normed linear spaces. Chapter VI fleshesout measure theory as it applies to Euclidean space in more than one dimension.Of special note is the Lebesgue-integration version in Section 5 of the change-of-variables formula for multiple integrals and the RieszFischer Theorem inSection 7. The latter characterizes square-integrable periodic functions by theirFourier coefficients and makes the subject of Fourier series useful in constructingfunctions. Differentiation of integrals in approached in Section 6 of Chapter VIas a problem of estimating finiteness of a quantity, rather than its smallness; thedevice is the HardyLittlewood Maximal Theorem, and the approach becomes aroutine way of approaching almost-everywhere convergence theorems. Sections810 are of somewhat less importance and may be omitted if time is short;Section 10 is applied only in Section IX.6.Chapters VIIIX are continuations of measure theory that are largely indepen-

dent of each other. ChapterVII contains the traditional proof of the differentiationof integrals on the line via differentiation of monotone functions. No later chapteris logically dependent on Chapter VII; the material is included only because of itshistorical importance and its usefulness as motivation for the RadonNikodymTheorem in Chapter IX. Chapter VIII is an introduction to the Fourier transformin Euclidean space. Its core consists of the first four sections, and the rest may beconsidered as optional if Section IX.6 is to be omitted. Chapter IX concerns L pspaces for 1 p ; only Section 6 makes use of material from Chapter VIII.

xxiv Guide for the Reader

Chapter X develops, at the latest possible time in the book, the necessary partof point-set topology that goes beyond metric spaces. Emphasis is on productand quotient spaces, and on Urysohns Lemma concerning the construction ofreal-valued functions on normal spaces.Chapter XI contains one more continuation of measure theory, namely special

features ofmeasures on locally compactHausdorff spaces. It provides an examplebeyond L p spaces in which one can usefully identify the dual of a particularnormed linear space. These chapters depend on Chapter X and on the first fivesections of Chapter IX but do not depend on Chapters VIIVIII.ChapterXII is a brief introduction to functional analysis, particularly toHilbert

spaces, Banach spaces, and linear operators on them. The main topics are thegeometry of Hilbert space and the three main theorems about Banach spaces.Appendix B is the core of a first course in complex analysis. The prerequisites

from real analysis for reading this appendix consist of Sections 17 of Chapter I,Section 18 of Chapter II, and Sections 13, 56, and 1112 of Chapter III;Section 6 of Chapter III is used only lightly. According to the plan of the book,it is possible to read the text of Chapters IXII without using any of Appendix B,but results of Appendix B are applied in problems at the end of Chapters IV,VI, and VIII, as well as in one spot in Section IX.6, in order to illustrate theinterplay between real analysis and complex analysis. The problems at the endof Appendix B are extensive and are of particular importance, since the topics oflinear fractional transformations, normal families, and the relationship betweenharmonic functions and analytic functions are developed there and not otherwisein the book.

STANDARD NOTATION

Item Meaning

#S or |S| number of elements in S empty set{x E | P} the set of x in E such that P holdsEc complement of the set EE F, E F, E F union, intersection, difference of setsS

E,T

E union, intersection of the sets EE F, E F E is contained in F , E contains FE F, sS Xs products of sets(a1, . . . , an), {a1, . . . , an} ordered n-tuple, unordered n-tuplef : E F, x 7 f (x) function, effect of functionf g, f

E composition of f following g, restriction to E

f ( , y) the function x 7 f (x, y)f (E), f 1(E) direct and inverse image of a seti j Kronecker delta: 1 if i = j , 0 if i 6= jnk

binomial coefficientn positive, n negative n > 0, n < 0Z, Q, R, C integers, rationals, reals, complex numbersmax (and similarly min) maximum of finite subset of a totally ordered setPor

Qsum or product, possibly with a limit operation

countable finite or in one-one correspondence with Z[x] greatest integer x if x is realRe z, Im z real and imaginary parts of complex zz complex conjugate of z|z| absolute value of z1 multiplicative identity1 or I identity matrix or operatordim V dimension of vector spaceRn , Cn spaces of column vectorsdet A determinant of AAtr transpose of Adiag(a1, . . . , an) diagonal matrix= is isomorphic to, is equivalent to

xxv

Basic Real Analysis

CHAPTER I

Theory of Calculus in One Real Variable

Abstract. This chapter, beginning with Section 2, develops the topic of sequences and seriesof functions, especially of functions of one variable. An important part of the treatment is anintroduction to the problem of interchange of limits, both theoretically and practically. This problemplays a role repeatedly in real analysis, but its visibility decreases as more and more results aredeveloped for handling it in various situations. Fourier series are introduced in this chapter and arecarried along throughout the book as a motivating example for a number of problems in real analysis.Section 1 makes contact with the core of a first undergraduate course in real-variable theory.

Some material from such a course is repeated here in order to establish notation and a point of view.Omitted material is summarized at the end of the section, and some of it is discussed in a little moredetail in AppendixA at the end of the book. The point of view being established is the use of definingproperties of the real number system to prove the BolzanoWeierstrass Theorem, followed by theuse of that theorem to prove some of the difficult theorems that are usually assumed in a one-variablecalculus course. The treatment makes use of the extended real-number system, in order to allow supand inf to be defined for any nonempty set of reals and to allow lim sup and lim inf to be meaningfulfor any sequence.Sections 23 introduce the problem of interchange of limits. They show how certain concrete

problems can be viewed in this way, and they give a way of thinking about all such interchangesin a common framework. A positive result affirms such an interchange under suitable hypothesesof monotonicity. This positive result is by way of introduction to the topic in Section 3 of uniformconvergence and the role of uniform convergence in continuity and differentiation.Section 4 gives a careful development of the Riemann integral for real-valued functions of one

variable, establishing existence of Riemann integrals for bounded functions that are discontinuousat only finitely many points, basic properties of the integral, the Fundamental Theorem of Calculusfor continuous integrands, the change-of-variables formula, and other results. Section 5 examinescomplex-valued functions, pointing out the extent to which the results for real-valued functions inthe first four sections extend to complex-valued functions.Section 6 is a short treatment of the version of Taylors Theorem in which the remainder is given

by an integral. Section 7 takes up power series and uses them to define the elementary transcendentalfunctions and establish their properties. The power series expansion of (1+x)p for arbitrary complexp is studied carefully. Section 8 introduces Cesaro and Abel summability, which play a role in thesubject of Fourier series. A converse theorem to Abels theorem is used to exhibit the function |x | asthe uniform limit of polynomials on [1, 1]. The Weierstrass Approximation Theorem of Section 9generalizes this example and establishes that every continuous complex-valued function on a closedbounded interval is the uniform limit of polynomials.Section 10 introduces Fourier series in one variable in the context of the Riemann integral. The

main theorems of the section are a convergence result for continuously differentiable functions,Bessels inequality, the RiemannLebesgue Lemma, Fejers Theorem, and Parsevals Theorem.

1

2 I. Theory of Calculus in One Real Variable

1. Review of Real Numbers, Sequences, Continuity

This section reviews some material that is normally in an undergraduate coursein real analysis. The emphasis will be on a rigorous proof of the BolzanoWeierstrass Theorem and its use to prove some of the difficult theorems that areusually assumed in a one-variable calculus course. We shall skip over some easieraspects of an undergraduate course in real analysis that fit logically at the end ofthis section. A list of such topics appears at the end of the section.The system of real numbersRmay be constructed out of the system of rational

numbersQ, and we take this construction as known. The formal definition is thata real number is a cut of rational numbers, i.e., a subset of rational numbers thatis neither Q nor the empty set, has no largest element, and contains all rationalnumbers less than any rational that it contains. The idea of the construction isas follows: Each rational number q determines a cut q, namely the set of allrationals less than q. Under the identification of Q with a subset of R, the cutdefining a real number consists of all rational numbers less than the given realnumber.The set of cuts gets a natural ordering, given by inclusion. In place of , we

write . For any two cuts r and s, we have r s or s r , and if both occur,then r = s. We can then define in the expected way. The positivecuts r are those with 0 < r , and the negative cuts are those with r < 0.Once cuts and their ordering are in place, one can go about defining the usual

operations of arithmetic and proving that R with these operations satisfies thefamiliar associative, commutative, and distributive laws, and that these interactwith inequalities in the usual ways. The definitions of addition and subtractionare easy: the sum or difference of two cuts is simply the set of sums or differencesof the rationals from the respective cuts. For multiplication and reciprocals onehas to take signs into account. For example, the product of two positive cutsconsists of all products of positive rationals from the two cuts, as well as 0 and allnegative rationals. After these definitions and the proofs of the usual arithmeticoperations are complete, it is customary to write 0 and 1 in place of 0 and 1.An upper bound for a nonempty subset E of R is a real number M such that

x M for all x in E . If the nonempty set E has an upper bound, we can take thecuts that E consists of and form their union. This turns out to be a cut, it is anupper bound for E , and it is all upper bounds for E . We can summarize thisresult as a theorem.

Theorem 1.1. Any nonempty subset E of R with an upper bound has a leastupper bound.

The least upper bound is necessarily unique, and the notation for it is supxE xor sup {x | x E}, sup being an abbreviation for the Latin word supremum,

1. Review of Real Numbers, Sequences, Continuity 3

the largest. Of course, the least upper bound for a set E with an upper boundneed not be in E ; for example, the supremum of the negative rationals is 0, whichis not negative.A lower bound for a nonempty set E ofR is a real numberm such that x m

for all x E . If m is a lower bound for E , thenm is an upper bound for the setE of negatives of members of E . Thus E has an upper bound, and Theorem1.1 shows that it has a least upper bound supxE x . Thenx is a greatest lowerbound for E . This greatest lower bound is denoted by infyE y or inf {y | y E},inf being an abbreviation for infimum. We can summarize as follows.

Corollary 1.2. Any nonempty subset E ofRwith a lower bound has a greatestlower bound.

A subset ofR is said to be bounded if it has an upper bound and a lower bound.Let us introduce notation and terminology for intervals of R, first treating thebounded ones.1 Let a and b be real numbers with a b. The open intervalfrom a to b is the set (a, b) = {x R | a < x < b}, the closed interval isthe set [a, b] = {x R | a x b}, and the half-open intervals are the sets[a, b) = {x R | a x < b} and (a, b] = {x R | a < x b}. Each of theabove intervals is indeed bounded, having a as a lower bound and b as an upperbound. These intervals are nonempty when a < b or when the interval is [a, b]with a = b, and in these cases the least upper bound is b and the greatest lowerbound is a.Open sets in R are defined to be arbitrary unions of open bounded intervals,

and a closed set is any set whose complement inR is open. A set E is open if andonly if for each x E , there is an open interval (a, b) such that x (a, b) E .In this case we of course have a < x < b. If we put = min{x a, b x},then we see that x lies in the subset (x , x + ) of (a, b). The open interval(x , x + ) equals

y R

|y x | <

. Thus an open set in R is any set E

such that for each x E , there is a number > 0 such thaty R

|y x | <

lies in E . A limit point x of a subset F of R is a point of R such that anyopen interval containing x meets F in a point other than x . For example, the set[a, b) {b+ 1} has [a, b] as its set of limit points. A subset of R is closed if andonly if it contains all its limit points.Now let us turn to unbounded intervals. To provide notation for these, we shall

make use of two symbols+ and thatwill shortly be defined to be extendedreal numbers. If a is in R, then the subsets (a,+) = {x R | a < x},(, a) = {x R | x < a}, (,+) = R, [a,+) = {x R | a x},and (, a] = {x R | x a} are defined to be intervals, and they are allunbounded. The first three are open sets of R and are considered to be open

1Bounded intervals are called finite intervals by some authors.


intervals, while the last three are closed sets and are considered to be closedintervals. Specifically the middle set R is both open and closed.One important consequence of Theorem 1.1 is the archimedean property of

R, as follows.

Corollary 1.3. If a and b are real numbers with a > 0, then there exists aninteger n with na > b.

PROOF. If, on the contrary, na b for all integers n, then b is an upper boundfor the set of allna. LetM be the least upperboundof the set {na | n is an integer}.Using that a is positive, we find that a1M is a least upper bound for the integers.Thus n a1M for all integers n, and there is no smaller upper bound. However,the smaller number a1M 1 must be an upper bound, since saying n a1Mfor all integers is the same as saying n1 a1M1 for all integers. We arriveat a contradiction, and we conclude that there is some integer n with na > b.

The archimedean property enables one to see, for example, that any twodistinct real numbers have a rational number lying between them. We provethis consequence as Corollary 1.5 after isolating one step as Corollary 1.4.

Corollary 1.4. If c is a real number, then there exists an integer n such thatn c < n + 1.

PROOF. Corollary 1.3 with a = 1 and b = c shows that there is an integer Mwith M > c, and Corollary 1.3 with a = 1 and b = c shows that there is aninteger m with m > c. Then m < c < M , and it follows that there exists agreatest integer n with n c. This n must have the property that c < n + 1, andthe corollary follows.

Corollary 1.5. If x and y are real numbers with x < y, then there exists arational number r with x < r < y.

PROOF. By Corollary 1.3 with a = y x and b = 1, there is an integer Nsuch that N (y x) > 1. This integer N has to be positive. Then 1N < y x .By Corollary 1.4 with c = Nx , there exists an integer n with n Nx < n + 1,hence with nN x 0 be given. If there were no integer N with aN > a , then a would bea smaller upper bound, contradiction. Thus such an N exists. For that N , n Nimplies a < aN an a < a + . Thus n N implies |an a| < .Since is arbitrary, limn an = a. If the given sequence {an} is monotonedecreasing, we argue similarly with a = infn an .

In working with sup and inf, it will be quite convenient to use the notationsupxE x evenwhen E is nonempty but not bounded above, and to use the notation


infxE x evenwhen E is nonempty but not bounded below. We introduce symbols+ and, plus andminus infinity, for this purpose and extend the definitionsof supxE x and infxE x to all nonempty subsets E of R by taking

supxE

x = + if E has no upper bound,

infxE

x = if E has no lower bound.

To work effectively with these new pieces of notation, we shall enlarge R to aset R called the extended real numbers by defining

R = R {+} {}.

An ordering onR is defined by taking < r < + for every member r ofRand by retaining the usual ordering withinR. It is immediate from this definitionthat

infxE

x supxE

x

if E is any nonempty subset ofR. In fact, we can enlarge the definitionsof infxE xand supxE x in obvious fashion to include the case that E is any nonemptysubset of R, and we still have inf sup. With the ordering in place, we canunambiguously speak of open intervals (a, b), closed intervals [a, b], and half-open intervals [a, b) and (a, b] in R even if a or b is infinite. Under ourdefinitions the intervals of R are the intervals of R that are subsets of R, even ifa or b is infinite. If no special mention is made whether an interval lies in R orR, it is usually assumed to lie in R.The next step is to extend the operations of arithmetic to R. It is important

not to try to make such operations be everywhere defined, lest the distributivelaws fail. Letting r denote any member of R and a and b be any members of R,we make the following new definitions:

Multiplication: r(+) = (+)r =

+ if r > 0,0 if r = 0, if r < 0,

r() = ()r =

if r > 0,0 if r = 0,+ if r < 0,

(+)(+) = ()() = +,

(+)() = ()(+) = .


Addition: r + (+) = (+) + r = +,r + () = () + r = ,

(+) + (+) = +,

() + () = .

Subtraction: a b = a + (b) whenever the right side is defined.

Division: a/b = 0 if a R and b is ,a/b = b1a if b R with b 6= 0 and a is .

The only surprise in the list is that 0 times anything is 0. This definition will beimportant to us when we get to measure theory, starting in Chapter V.It is now a simple matter to define convergence of a sequence inR. The cases

that need addressing are that the sequence is inR and that the limit is+ or.We say that a sequence {an} inR tends to+ if for any positive numberM , thereexists an integer N such that an M for all n N . The sequence tends to if for any negative number M , there exists an integer N such that an Mfor all n N . It is important to indicate whether convergence/divergence of asequence is being discussed inR or inR. The default setting isR, in keepingwithstandard terminology in calculus. Thus, for example, we say that the sequence{n}n1 diverges, but it converges in R (to +).With our new definitions every monotone sequence converges in R.For a sequence {an} inR or even inR, we now introducemembers lim supn an

and lim infn an of R. These will always be defined, and thus we can apply theoperations lim sup and lim inf to any sequence in R. For the case of lim supwe define bn = supkn ak as a sequence in R. The sequence {bn} is monotonedecreasing. Thus it converges to infn bn in R. We define2

lim supn

an = infnsupkn

ak

as a member of R, and we define

lim infn

an = supninfkn

ak

as a member of R. Let us underscore that lim sup an and lim inf an always exist.However, one or both may be even if an is in R for every n.

Proposition 1.7. The operations lim sup and lim inf on sequences {an} and{bn} in R have the following properties:

(a) if an bn for all n, then lim sup an lim sup bn and lim inf an lim inf bn ,

2The notation lim was at one time used for lim sup, and lim was used for lim inf.


(b) lim inf an lim sup an ,(c) {an} has a subsequence converging in R to lim sup an and another sub-

sequence converging in R to lim inf an ,(d) lim sup an is the supremum of all subsequential limits of {an} in R, and

lim infn is the infimum of all subsequential limits of {an} in R,(e) if lim sup an < +, then lim sup an is the infimum of all extended real

numbers a such that an a for only finitely many n, and if lim inf an >, then lim inf an is the supremum of all extended real numbers a suchthat an a for only finitely many n,

(f) the sequence {an} in R converges in R if and only if lim inf an =lim sup an , and in this case the limit is the common value of lim inf an andlim sup an .

REMARK. It is enough to prove the results about lim sup, since lim inf an = lim sup(an).

PROOFS FOR lim sup.(a) From al bl for all l, we have al supkn bk if l n. Hence supln al

supkn bk . Then (a) follows by taking the limit on n.(b) This follows by taking the limit onn of the inequality infkn ak supkn ak .(c) We divide matters into cases. The main case is that a = lim sup an is in R.

Inductively, for each l 1, choose N nl1 such that | supkN ak a| < l1.Then choose nl > nl1 such that |anl supkN ak | < l1. Together theseinequalities imply |anl a| < 2l1 for all l, and thus liml anl = a. Thesecond case is that a = lim sup an equals +. Since supkn ak is monotonedecreasing in n, we must have supkn ak = + for all n. Inductively for l 1,we can choose nl > nl1 such that anl l. Then liml anl = +. Thethird case is that a = lim sup an equals . The sequence bn = supkn ak ismonotone decreasing to . Inductively for l 1, choose nl > nl1 such thatbnl l. Then anl bnl l, and liml anl = .(d) By (c), lim sup an is one subsequential limit. Let a = limk ank be an-

other subsequential limit. Put bn = supln al . Then {bn} converges to lim sup anin R, and the same thing is true of every subsequence. Since ank suplnk al =bnk for all k, we can let k tend to infinity and obtain a = limk ank limk bnk = lim sup an .(e) Since lim sup an < +, we have supkn ak < + for n greater than or

equal to some N . For this N and any a > supkN ak , we then have an a onlyfinitely often. Thus there exists a R such that an a for only finitely many n.On the other hand, if a0 is a real number< lim sup an , then (c) shows that an a0for infinitely many n. Hence

lim sup an inf {a | an a for only finitely many a}.


Arguing by contradiction, suppose that< holds in this inequality, and let a00 be areal number strictly in between the two sides of the inequality. Then supkn ak 0 be given. By (e), an a+only finitely often, and an a only finitely often. Thus |an a| < forall n sufficiently large. In other words, lim an = a as asserted. The other casesare that a = + or a = , and they are completely analogous to each other.Suppose for definiteness that a = +. Since lim inf an = +, the monotoneincreasing sequence bn = infkn ak converges in R to +. Given M , chooseN such that bn M for n N . Then also an M for n N , and an convergesin R to +. This completes the proof.

With Proposition1.7 as a tool, we can nowprove theBolzanoWeierstrassThe-orem. The remainder of the section will consist of applications of this theorem,showing that Cauchy sequences in R converge in R, that continuous functionson closed bounded intervals of R are uniformly continuous, that continuousfunctions on closed bounded intervals are bounded and assume their maximumand minimum values, and that continuous functions on closed intervals take onall intermediate values.

Theorem 1.8 (BolzanoWeierstrass). Every bounded sequence in R has aconvergent subsequence with limit in R.PROOF. If the given bounded sequence is {an}, form the subsequence noted

in Proposition 1.7c that converges in R to lim sup an . All quantities arising inthe formation of lim sup an are in R, since {an} is bounded, and thus the limit isin R.

A sequence {an} in R is called a Cauchy sequence if for any > 0, thereexists an N such that |an am | < for all n and m that are N .

EXAMPLE. Every convergent sequence in R with limit in R is Cauchy. In fact,let a = lim an , and let > 0 be given. Choose N such that n N implies|an a| < . Then n,m N implies

|an am | |an a| + |a am | < + = 2.

Hence the sequence is Cauchy.


In the above example and elsewhere in this book, we allow ourselves the luxuryof having our final bound come out as a fixed multiple M of , rather than itself. Strictly speaking, we should have introduced 0 = /M and aimed for0 rather than . Then our final bound would have been M0 = . Since thetechnique for adjusting a proof in this way is always the same, we shall not addthese extra steps in the future unless there would otherwise be a possibility ofconfusion.This convention suggests a handy piece of terminologythat a proof as in the

above example, in which M = 2, is a 2 proof. That name conveys a great dealof information about the proof, saying that one should expect two contributionsto the final estimate and that the final bound will be 2.

Theorem 1.9 (Cauchy criterion). Every Cauchy sequence in R converges to alimit in R.PROOF. Let {an} be Cauchy in R. First let us see that {an} is bounded. In

fact, for = 1, choose N such that n,m N implies |an am | < 1. Then|am | |aN | + 1 for m N , and M = max{|a1|, . . . , |aN1|, |aN | + 1} is acommon bound for all |an|.Since {an} is bounded, it has a convergent subsequence {ank }, say with limit

a, by the BolzanoWeierstrass Theorem. The subsequential limit has to satisfy|a| M within R, and thus a is in R.Finally let us see that lim an = a. In fact, if > 0 is given, choose N such

that nk N implies |ank a| < . Also, choose N 0 N such that n,m N 0implies |an am | < . If n N 0, then any nk N 0 has |an ank | < , andhence

|an a| |an ank | + |ank a| < + = 2.

This completes the proof.

Let f be a function with domain an interval and with range in R. The intervalis allowed to be unbounded, but it is required to be a subset of R. We saythat f is continuous at a point x0 of the domain of f within R if for each > 0, there is some > 0 such that all x in the domain of f that satisfy|x x0| < have | f (x) f (x0)| < . This notion is sometimes abbreviated aslimxx0 f (x) = f (x0). Alternatively, one may say that f (x) tends to f (x0) asx tends to x0, and one may write f (x) f (x0) as x x0.Amathematically equivalent definition is that f is continuous at x0 if whenever

a sequence has xn x0 within the domain interval, then f (xn) f (x0). Thislatter version of continuity will be shown in Section II.4 to be equivalent to theformer version, given in terms of continuous limits, in greater generality than justfor R, and thus we shall not stop to prove the equivalence now. We say that f iscontinuous if it is continuous at all points of its domain.


We say that the a function f as above is uniformly continuous on its domainif for any > 0, there is some > 0 such that | f (x) f (x0)| < whenever xand x0 are in the domain interval and |x x0| < . (In other words, the conditionfor the continuity to be uniform is that can always be chosen independently ofx0.)

EXAMPLE. The function f (x) = x2 is continuous on (,+), but it isnot uniformly continuous. In fact, it is not uniformly continuous on [1,+).Assuming the contrary, choose for = 1. Thenwemust have

(x+ 2 )

2x2 < 1

for all x 1. But(x + 2 )

2 x2 = x +

2

4 x , and this is 1 for x 1.

Theorem 1.10. A continuous function f from a closed bounded interval [a, b]into R is uniformly continuous.PROOF. Fix > 0. For x0 in the domain of f , the continuity of f at x0 means

that it makes sense to define

x0() = min1, sup

0 > 0

|x x0| < 0 and x in the domainof f imply | f (x) f (x0)| <

.

If |x x0| < x0(), then | f (x) f (x0)| < . Put () = infx0[a,b] x0().Let us see that it is enough to prove that () > 0. If x and y are in [a, b] with|x y| < (), then |x y| < () y(). Hence | f (x) f (y)| < asrequired.Thus we are to prove that () > 0. If () = 0, then, for each integer

n > 0, we can choose xn such that xn () < 1n . By the BolzanoWeierstrassTheorem, there is a convergent subsequence, say with xnk x 0. Along thissubsequence, xnk () 0. Fix k large enough so that |xnk x

0| < 12x 0(2 ). Then

| f (xnk ) f (x 0)| < 2 . Also, |x xnk | 0

such that |x1 x2| < implies | f (x1) f (x2)| < whenever x1 and x2 bothlie in [a, b]. Then choose an integer n such that 2n(b a) < , and considerthe value of f at the points pk = a + k2n(b a) for 0 k 2n . Sincepk+1 pk = 2n(b a) < , we have | f (pk+1) f (pk)| < = 2 1.Consequently if f (pk) 1, then

f (pk+1) f (pk) + | f (pk+1) f (pk)| < 1 + (2 1) = 2,and hence f (pk+1) 1. Now f (p0) = f (a) = 1. Thus induction showsthat f (pk) 1 for all k 2n . However, for k = 2n , we have p2n = b, andf (b) = > 1, and we have arrived at a contradiction.

Further topics. Here a number of other topics of an undergraduate course in real-variabletheory fit well logically. Among these are countable vs. uncountable sets, infinite series and testsfor their convergence, the fact that every rearrangement of an infinite series of positive terms has thesame sum, special sequences, derivatives, the Mean Value Theorem as in Section A2 of AppendixA, and continuity and differentiability of inverse functions as in Section A3 of Appendix A.We shallnot stop here to review these topics, which are treated in many books. One such book is RudinsPrinciples of Mathematical Analysis, the relevant chapters being 1 to 5. In Chapter 2 of that book,only the first few pages are needed; they are the ones where countable and uncountable sets arediscussed.

2. Interchange of Limits 13

2. Interchange of Limits

Let {bi j } be a doubly indexed sequence of real numbers. It is natural to ask forthe extent to which

limilimjbi j = lim

jlimibi j ,

more specifically to ask how to tell, in an expression involving iterated limits,whether we can interchange the order of the two limit operations. We can viewmatters conveniently in terms of an infinite matrix

b11 b12 b21 b22...

. . .

.

The left-hand iterated limit, namely limi limj bi j , is obtained by forming the limitof each row, assembling the results, and then taking the limit of the row limitsdown through the rows. The right-hand iterated limit, namely limj limi bi j , isobtained by forming the limit of each column, assembling the results, and thentaking the limit of the column limits through the columns. If we use the particularinfinite matrix

1 1 1 1 0 1 1 1 0 0 1 1 0 0 0 1 ...

. . .

,

thenwe see that the first iterated limit depends only on the part of thematrix abovethe main diagonal, while the second iterated limit depends only on the part of thematrix below the main diagonal. Thus the two iterated limits in general have noreason at all to be related. In the specific matrix that we have just considered,they are 1 and 0, respectively. Let us consider some examples along the samelines but with an analytic flavor.

EXAMPLES.

(1) Let bi j =j

i + j. Then limi limj bi j = 1, while limj limi bi j = 0.

(2) Let Fn be a continuous real-valued function onR, and suppose that F(x) =lim Fn(x) exists for every x . Is F continuous? This is the same kind of question.It asks whether limtx F(t)

?= F(x), hence whether

limtx

limn

Fn(t)?= lim

nlimtx

Fn(t).


If we take fk(x) =x2

(1+ x2)kfor k 0 and define Fn(x) =

Pnk=0 fk(x), then

each Fn is continuous. The sequence of functions {Fn} has a pointwise limit

F(x) =X

k=0x2

(1+ x2)k. The series is a geometric series, and we can easily

calculate explicitly the partial sums and the limit function. The latter is

F(x) = 0 if x = 01+ x2 if x 6= 0.

It is apparent that the limit function is discontinuous.(3) Let { fn} be a sequence of differentiable functions, and suppose that f (x) =

lim fn(x) exists for every x and is differentiable. Is lim f 0n(x) = f 0(x)? Thisquestion comes down to whether

limn

limtx

fn(t) fn(x)t x

?= lim

txlimn

fn(t) fn(x)t x

.

An example where the answer is negative uses the sine and cosine functions,which are undefined in the rigorous development until Section 7 on power series.

The example has fn(x) =sin nxpn

for n 1. Then limn fn(x) = 0, so that

f (x) = 0 and f 0(x) = 0. Also, f 0n(x) =pn cos nx , so that f 0n(0) =

pn does

not tend to 0 = f 0(0).

Yet we know many examples from calculus where an interchange of limits isvalid. For example, in calculus of two variables, the first partial derivatives ofnice functionspolynomials, for examplecan be computed in either order withthe same result, and double integrals of continuous functions over a rectangle canbe calculated as iterated integrals in either order with the same result. Positivetheorems about interchanging limits are usually based on some kind of uniformbehavior, in a sense that we take up in the next section. A number of positiveresults of this kind ultimately come down to the following general theorem aboutdoubly indexed sequences that are monotone increasing in each variable. InSection 3 we shall examine the mechanism of this theorem closely: the proofshows that the equality in question is supi supj bi j = supj supi bi j and that itholds because both sides equal supi, j bi j .

Theorem 1.13. Let bi j be members ofR that are 0 for all i and j . Supposethat bi j is monotone increasing in i , for each j , and is monotone increasing in j ,for each i . Then

limilimjbi j = lim

jlimibi j ,

with all the indicated limits existing in R.

3. Uniform Convergence 15

PROOF. Put Li = limj bi j and L 0j = limi bi j . These limits exist in R, sincethe sequences in question are monotone. Then Li Li+1 and L 0j L 0j+1, andthus

L = limiLi and L 0 = lim

jL 0j

both exist in R. Arguing by contradiction, suppose that L < L 0. Then we canchoose j0 such that L 0j0 > L . Since L

0j0 = limi bi j0 , we can choose i0 such that

bi0 j0 > L . Then we have L < bi0 j0 Li0 L , contradiction. Similarly theassumption L 0 < L leads to a contradiction. We conclude that L = L 0.

Corollary 1.14. If al j are members of R that are 0 and are monotoneincreasing in j for each l, then

limj

X

lal j =

X

llimjal j

in R, the limits existing.

REMARK. This result will be generalized by the Monotone ConvergenceTheorem when we study abstract measure theory in Chapter V.

PROOF. Put bi j =Pi

l=1 al j in Theorem 1.13.

Corollary 1.15. If ci j are members of R that are 0 for all i and j , thenX

i

X

jci j =

X

j

X

ici j

in R, the limits existing.

REMARK. This result will be generalized by Fubinis Theorem when we studyabstract measure theory in Chapter V.

PROOF. This follows from Corollary 1.14.

3. Uniform Convergence

Let us examine more closely what is happening in the proof of Theorem 1.13, inwhich it is proved that iterated limits can be interchangedunder certain hypothesesof monotonicity. One of the iterated limits is L = limi limj bi j , and the claim isthat L is approached as i and j tend to infinity jointly. In terms of a matrix whose


entries are the various bi j s, the pictorial assertion is that all the terms far downand to the right are close to L:

All terms hereare close to L

.

To see this claim, let us choose a row limit Li0 that is close to L and then take anentry bi0 j0 that is close to Li0 . Then bi0 j0 is close to L , and all terms down and tothe right from there are even closer because of the hypothesis of monotonicity.To relate this behavior to something uniform, suppose that L < +, and let

some > 0 be given. We have just seen that we can arrange to have |Lbi j | < whenever i i0 and j j0. Then |Li bi j | < whenever i i0, providedj j0. Also, we have limj bi j = Li for i = 1, 2, . . . , i01. Thus |Li bi j | < for all i , provided j j 00, where j 00 is some larger index than j0. This is thenotion of uniform convergence that we shall define precisely in a moment: anexpression with a parameter (i in our case) has a limit (on the variable j in ourcase) with an estimate independent of the parameter. We can visualize matters asin the following matrix:

j j 00

i

All terms hereare close to Lion all rows.

!

.

The vertical dividing line occurs when the column index j is equal to j 00, and allterms to the right of this line are close to their respective row limits Li .Let us see the effect of this situation on the problem of interchange of limits.

The above diagram forces all the terms in the shaded part of

//////

!

to

be close to one number if lim Li exists, i.e., if the row limits are tending to alimit. If the other iterated limit exists, then it must be this same number. Thusthe interchange of limits is valid under these circumstances.Actually, we can get by with less. If, in the displayed diagram above, we

assume that all the column limits L 0j exist, then it appears that all the columnlimits with j j 00 have to be close to the Li s. From this we can deduce that thecolumn limits have a limit L 0 and that the row limits Li must tend to the limitof the column limits. In other words, the convergence of the rows in a suitableuniform fashion and the convergence of the columns together imply that both


iterated limits exist and they are equal. We shall state this result rigorously asProposition 1.16, which will become a prototype for applications later in thissection.Let S be a nonempty set, and let f and fn , for integers n 1, be functions

from S to R. We say that fn(x) converges to f (x) uniformly for x in S if forany > 0, there is an integer N such that n N implies | fn(x) f (x)| < forall x in S. It is equivalent to say that supxS | fn(x) f (x)| tends to 0 as n tendsto infinity.

Proposition 1.16. Let bi j be real numbers for i 1 and j 1. Suppose that(i) Li = limj bi j exists in R uniformly in i , and(ii) L 0j = limi bi j exists in R for each j .

Then(a) L = limi Li exists in R,(b) L 0 = limj L 0j exists in R,(c) L = L 0,(d) the double limit on i and j of bi j exists and equals the common value of

the iterated limits L and L 0, i.e., for each > 0, there exist i0 and j0 suchthat |bi j L| < whenever i i0 and j j0,

(e) L 0j = limi bi j exists in R uniformly in j .

REMARK. In applicationswe shall sometimes have extra information, typicallythe validity of (a) or (b). According to the statement of the proposition, however,the conclusions are valid without using this extra information as an additionalhypothesis.PROOF. Let > 0 be given. By (i), choose j0 such that |bi j Li | < for all

i whenever j j0. With j j0 fixed, (ii) says that |bi j L 0j | < whenever i is some i0 = i0( j). For j j0 and i i0( j), we then have

|Li L 0j | |Li bi j | + |bi j L0j | < + = 2.

If j 0 j0 and i i0( j 0), we similarly have |Li L 0j 0 | < 2. Hence if j j0,j 0 j0, and i max{i0( j), i0( j 0)}, then

|L 0j L0j 0 | |L

0j Li | + |Li L

0j 0 | < 2 + 2 = 4.

In other words, {L 0j } is a Cauchy sequence. By Theorem 1.9, L 0 = limj L 0j existsin R. This proves (b).Passing to the limit in our inequality, we have |L 0j L 0| 4 when j j0

and in particular when j = j0. If i i0( j0), then we saw that |bi j0 Li | < and |bi j0 L 0j0 | < . Hence i i0( j0) implies

|Li L 0| |Li bi j0 | + |bi j0 L0j0 | + |L

0j0 L

0| < + + 4 = 6.


Since is arbitrary, L = limi Li exists and equals L 0. This proves (a) and (c).Since limi Li = L , choose i1 such that |Li L| < whenever i i1. If i i1

and j j0, we then have

|bi j L| |bi j Li | + |Li L| < + = 2.

This proves (d).Let i1 and j0 be as in the previous paragraph. We have seen that |L 0jL 0j 0 | < 4

for j j0. By (b), |L 0j L 0| 4 whenever j j0. Hence (c) and the inequalityof the previous paragraph give

|bi j L 0j | |bi j L| + |L L0| + |L 0 L 0j | < 2 + 0+ 4 = 6

whenever i i1 and j j0. By (ii), choose i2 i1 such that |bi j L 0j | < 6whenever j {1, . . . , j01} and i i2. Then i i2 implies |bi j L 0j | < 6 forall j whenever i i2.

In checking for uniform convergence, we often do not have access to explicitexpressions for limiting values. One device for dealing with the problem is auniform version of the Cauchy criterion. Let S be a nonempty set, and let { fn}n1be a sequence of functions from S toR. We say that { fn(x)} is uniformlyCauchyfor x S if for any > 0, there is an integer N such that n N and m Ntogether imply | fn(x) fm(x)| < for all x in S.

Proposition 1.17 (uniform Cauchy criterion). A sequence { fn} of functionsfrom a nonempty set S to R is uniformly Cauchy if and only if it is uniformlyconvergent.

PROOF. If { fn} is uniformly convergent to f , we use a 2 argument, just asin the example before Theorem 1.9: Given > 0, choose N such that n Nimplies | fn(x) f (x)| < . Then n N and m N together imply

| fn(x) fm(x)| | fn(x) f (x)| + | f (x) fm(x)| < + = 2.

Thus { fn} is uniformly Cauchy.Conversely suppose that { fn} is uniformlyCauchy. Then { fn(x)} is Cauchy for

each x . Theorem 1.9 therefore shows that there exists a function f : S R suchthat limn fn(x) = f (x) for each x . We prove that the convergence is uniform.Given > 0, choose N , as is possible since { fn} is uniformly Cauchy, such thatn N and m N together imply | fn(x) fm(x)| < . Letting m tend to shows that | fn(x) f (x)| for n N . Hence limn fn(x) = f (x) uniformlyfor x in S.


In practice, uniform convergence often arises with infinite series of functions,and then the definition and results about uniform convergence are to be applied tothe sequenceof partial sums. If the series is

Pk=1 ak(x), onewants

Pn

k=m ak(x)

to be small for all m and n sufficiently large. Some of the standard tests forconvergence of series of numbers yield tests for uniform convergence of series offunctions just by introducing a parameter and ensuring that the estimates do notdepend on the parameter. We give two clear-cut examples. One is the uniformalternating series test or Leibniz test, given in Corollary 1.18. A generalizationis the handy test given in Corollary 1.19.

Corollary 1.18. If for each x in a nonempty set S, {an(x)}n1 is a mono-tone decreasing sequence of nonnegative real numbers such that limn an(x) = 0uniformly in x , then

Pn=1 (1)nan(x) converges uniformly.

PROOF. The hypotheses are such thatPn

k=m (1)kak(x) supx |am(x)|

whenever n m, and the uniform convergence is immediate from the uniformCauchy criterion.

Corollary 1.19. If for each x in a nonempty set S, {an(x)}n1 is a monotonedecreasing sequence of nonnegative real numbers such that limn an(x) = 0uniformly in x and if {bn(x)}n1 is a sequence of real-valued functions on Swhose partial sums Bn(x) =

Pnk=1 bk(x) have |Bn(x)| M for some M and all

n and x , thenP

n=1 an(x)bn(x) converges uniformly.

PROOF. If n m, summation by parts gives

nX

k=mak(x)bk(x) =

n1X

k=mBk(x)(ak(x) ak+1(x)) + Bn(x)an(x) Bm1(x)am(x),

as one can check by expanding out the right side. Let > 0 be given, and chooseN such that ak(x) for all x whenever k N . If n m N , then

nX

k=mak(x)bk(x)

n1X

k=m|Bk(x)|(ak(x) ak+1(x)) + M + M

Mn1X

k=m(ak(x) ak+1(x)) + 2M

Mam(x) + 2M

3M,

and the uniform convergence is immediate from the uniform Cauchy criterion.


A third consequence can be considered as a uniform version of the result thatabsolute convergence implies convergence. In practice it tends to be fairly easyto apply, but it applies only in the simplest situations.

Proposition 1.20 (Weierstrass M test). Let S be a nonempty set, and let { fn}be a sequence of real-valued functions on S such that | fn(x)| Mn for all x inS. Suppose that

Pn Mn < +. Then

Pn=1 fn(x) converges uniformly for x in

S.

PROOF. If n m N , thenPn

k=m fk(x)

Pnk=m | fk(x)|

Pnk=m Mk ,

and the right side tends to 0 uniformly in x as N tends to infinity. Therefore theresult follows from the uniform Cauchy criterion.

EXAMPLES.(1) The series

X

n=1

1n2

xn

converges uniformly for1 x 1 by theWeierstrass M test with Mn = 1/n2.(2) The series

X

n=1(1)n

x2 + nn2

converges uniformly for 1 x 1, but the M test does not apply. To seethat the M test does not apply, we use the smallest possible Mn , which is Mn =supx

(1)n x2+nn2 | =

n+1n2 . The series

P n+1n2 diverges, and hence the M test

cannot apply for any choice of the numbers Mn . To see the uniform convergenceof the given series, we observe that the terms strictly alternate in sign. Also,

x2 + nn2

x2 + (n + 1)

(n + 1)2because

x2

n2

x2

(n + 1)2and

1n

1

n + 1.

Finallyx2 + nn2

n + 1n2

0

uniformly for1 x 1. Hence the series converges uniformly by the uniformLeibniz test (Corollary 1.18).

Having developed some tools for proving uniform convergence, let us applythe notion of uniform convergence to interchanges of limits involving functionsof a real variable. For a point of reference, recall the diagrams of interchanges oflimits at the beginning of the section. We take the column index to be n and think


of the row index as a variable t , which is tending to x . We make assumptionsthat correspond to (i) and (ii) in Proposition 1.16, namely that { fn(t)} convergesuniformly in t as n tends to infinity, say to f (t), and that fn(t) converges to somelimit fn(x) as t tends to x . With fn(x) defined as this limit, fn is continuousat x . In other words, the assumptions are that the sequence { fn} is uniformlyconvergent to f and each fn is continuous.

Theorem 1.21. If { fn} is a sequence of real-valued functions on [a, b] that arecontinuous at x and if { fn} converges to f uniformly, then f is continuous at x .REMARKS. This is really a consequence of Proposition 1.16 except that one of

the indices, namely t , is regarded as continuous and not discrete. Actually, there isa subtle simplification here, by comparison with Proposition 1.16, in that { fn(x)}at the limiting parameter x is being assumed to tend to f (x). This correspondsto assuming (b) in the proposition, as well as (i) and (ii). Consequently the proofof the theorem will be considerably simpler than the proof of Proposition 1.16.In fact, the proof will be our first example of a 3 proof. In many applicationsof Theorem 1.21, the given sequence { fn} is continuous at every x , and then theconclusion is that f is continuous at every x .PROOF. We write

| f (t) f (x)| | f (t) fn(t)| + | fn(t) fn(x)| + | fn(x) f (x)|.Given > 0, choose N large enough so that | fn(t) f (t)| < for all t whenevern N . With such an n fixed, choose some of continuity for the functionfn , the point x , and the number . Each term above is then < , and hence| f (t) f (x)| < 3. Since is arbitrary, f is continuous at x .

Theorem 1.21 in effect uses only conclusion (c) of Proposition 1.16, whichconcerns the equality of the two iterated limits. Conclusion (d) gives a strongerresult, namely that the double limit exists and equals each iterated limit. Thestrengthened version of Theorem 1.21 is as follows.

Theorem 1.210. If { fn} is a sequence of real-valued functions on [a, b] thatare continuous at x and if { fn} converges to f uniformly, then for each > 0,there exist an integer N and a number > 0 such that

| fn(t) f (x)| < whenever n N and |t x | < .PROOF. If > 0 is given, choose N such that | fn(t) f (t)| < /2 for all

t whenever n N , and choose in the conclusion of Theorem 1.21 such that|t x | < implies | f (t) f (x)| < /2. Then

| fn(t) f (x)| | fn(t) f (t)| + | f (t) f (x)| < 2 +2 =

whenever n N and |t x | < . Theorem 1.210 follows.


In interpreting our diagrams of interchanges of limits to get at the statement ofTheorem 1.21, we took the column index to be n and thought of the row index asa variable t , which was tending to x . It is instructive to see what happens whenthe roles of n and t are reversed, i.e., when the row index is n and the columnindex is the variable t , which is tending to x . Again we have fn(t) convergingto f (t) and limtx fn(t) = fn(x), but the uniformity is different. This timewe want the uniformity to be in n as t tends to x . This means that the ofcontinuity that corresponds to can be taken independent of n. This is the notionof equicontinuity, and there is a classical theorem about it. The theorem isactually stronger than Proposition 1.16 suggests, since the theorem assumes lessthan that fn(t) converges to f (t) for all t .Let F = { f | A} be a set of real-valued functions on a bounded interval

[a, b]. We say that F is equicontinuous at x [a, b] if for each > 0, there issome > 0 such that |tx | < implies | f (t) f (x)| < for all f F. The setF of functions is pointwise bounded if for each t [a, b], there exists a numberMt such that | f (t)| Mt for all f F. The set is uniformly equicontinuous on[a, b] if it is equicontinuous at each point x and if the can be taken independentof x . The set is uniformly bounded on [a, b] if it is pointwise bounded at eacht [a, b] and the bound Mt can be taken independent of t .

Theorem 1.22 (Ascolis Theorem). If { fn} is a sequence of real-valued func-tions on a closed bounded interval [a, b] that is equicontinuous at each point of[a, b] and pointwise bounded on [a, b], then

(a) { fn} is uniformly equicontinuous and uniformly bounded on [a, b],(b) { fn} has a uniformly convergent subsequence.

PROOF. Since each fn is continuous at each point, we know from Theorems1.10 and 1.11 that each fn is uniformly continuous and bounded. The proof of(a) amounts to an argument that the estimates in those theorems can be arrangedto apply simultaneously for all n.First consider the questionof uniformboundedness. Choose, byTheorem1.11,

some xn in [a, b] with | fn(xn)| equal to Kn = supx[a,b] | fn(x)|. Then choose asubsequence on which the numbers Kn tend to supn Kn in R. There will be noloss of generality in assuming that this subsequence is ourwhole sequence. Applythe BolzanoWeierstrass Theorem to find a convergent subsequence {xnk } of {xn},say with limit x0. By pointwise boundedness, find Mx0 with | fn(x0)| Mx0 forall n. Then choose some of equicontinuity at x0 for = 1. As soon as k is largeenough so that |xnk x0| < , we have

Knk = | fnk (xnk )| | fnk (xnk ) fnk (x0)| + | fnk (x0)| < 1+ Mx0 .

Thus 1+ Mx0 is a uniform bound for the functions fn .


The proof of uniform equicontinuity proceeds in the same spirit but takeslonger to write out. Fix > 0. The uniform continuity of fn for each n meansthat it makes sense to define

n() = min1, sup

0 > 0

| fn(x) fn(y)| < whenever |xy| < 0and x and y are in the domain of fn

.

If |x y| < n(), then | fn(x) fn(y)| < . Put () = infn n(). Let us seethat it is enough to prove that () > 0: If x and y are in [a, b] with |xy| < (),then |x y| < () n(). Hence | fn(x) fn(y)| < as required.Thus we are to prove that () > 0. If () = 0, then we first choose an

increasing sequence {nk} of positive integers such that nk () < 1k , and we nextchoose xk and yk in [a, b] with |xk yk | < 1/k and | fnk (xk) fnk (yk)| .Applying the BolzanoWeierstrass Theorem, we obtain a subsequence {xkl } of{xk} such that {xkl } converges, say to x0. Then

lim supl

|ykl x0| lim supl

|ykl xkl | + lim supl

|xkl x0| = 0+ 0 = 0,

so that {ykl } converges to x0. Now choose, by equicon

basic real analysis - stony brook universityaknapp/download/b2-realanal...basic algebra advanced...

Documents