Undergraduate Texts in Mathematics

Springer-Science+Business Media, LLC

Editors

S. Axler F. W. Gehring

K.A. Ribet

Undergraduate Texts in Mathematics

Anglin: Mathematics: A Concise History and Philosophy. Readings in Mathematics.

AnglinlLambek: The Heritage of Thales. Readings in Mathematics.

Apostol: Introduction to Analytic Number Theory. Second edition.

Armstrong: Basic Topology. Armstrong: Groups and Symmetry. Axler: Linear Algebra Done Right.

Second edition. Beardon: Limits: A New Approach to

Real Analysis. BaklNewman: Complex Analysis.

Second edition. BanchofflWermer: Linear Algebra

Through Geometry. Second edition. Berberian: A First Course in Real

Analysis. Bix: Conics and Cubics: A Concrete Introduction to Algebraic Curves. Bremaud: An Introduction to

Probabilistic Modeling. Bressoud: Factorization and Primality

Testing. Bressoud: Second Year Calculus.

Readings in Mathematics. Brickman: Mathematical Introduction

to Linear Programming and Game Theory.

Browder: Mathematical Analysis: An Introduction.

Buchmann: Introduction to Cryptography. Buskes/van Rooij: Topological Spaces:

From Distance to Neighborhood. Callahan: The Geometry of Spacetime:

An Introduction to Special and General Relavitity.

Carter/van Brunt: The Lebesgue­Stieitjes Integral: A Practical Introduction.

Cederberg: A Course in Modem Geometries. Second edition.

Childs: A Concrete Introduction to Higher Algebra. Second edition.

Chung: Elementary Probability Theory with Stochastic Processes. Third edition.

CoxlLittle/O'Shea: Ideals, Varieties, and Algorithms. Second edition.

Croom: Basic Concepts of Algebraic Topology.

Curtis: Linear Algebra: An Introductory Approach. Fourth edition.

Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory. Second edition.

Dixmier: General Topology. Driver: Why Math? EbbinghauslFlumffhomas:

Mathematical Logic. Second edition. Edgar: Measure, Topology, and Fractal

Geometry. Elaydi: An Introduction to Difference

Equations. Second edition. Exner: An Accompaniment to Higher

Mathematics. Exner: Inside Calculus. FinelRosenberger: The Fundamental

Theory of Algebra. Fischer: Intermediate Real Analysis. FlaniganlKazdan: Calculus Two: Linear

and Nonlinear Functions. Second edition.

Fleming: Functions of Several Variables. Second edition.

Foulds: Combinatorial Optimization for Undergraduates.

Foulds: Optimization Techniques: An Introduction.

Franklin: Methods of Mathematical Economics.

Frazier: An Introduction to Wavelets Through Linear Algebra.

Gamelin: Complex Analysis. Gordon: Discrete Probability. HairerlWanner: Analysis by Its History.

Readings in Mathematics. Halmos: Finite-Dimensional Vector

Spaces. Second edition. Halmos: Naive Set Theory.

(continued after index)

Andrew Browder

Mathematical Analysis An Introduction

, Springer

Andrew Browder Mathematics Department Brown University Providence, RI 02912 USA

Editorial Board

S. Axler Mathematics Department San Francisco State

University San Francisco, CA 94132 USA

With five iIlustrations.

F.W. Gehring Mathematics Department Bast Hali University of Michigan Ann Arbor, MI 48109 USA

K.A. Ribet Mathematics Department University of California

at Berkeley Berkeley, CA 94720-3840 USA

Mathematics Subject Classifications (2000): 26-01, 26Axx, 54Cxx, 28Axx

Library ofCongress Cataloging-in-Publication Data Browder, Andrew.

Mathematical analysis : an introduction / Andrew Browder. p. cm. - (Undergraduate texts in mathematics)

IncIudes bibliographical references and index. ISBN 978-1-4612-6879-6 ISBN 978-1-4612-0715-3 (eBook) DOI 10.1007/978-1-4612-0715-3 1. Mathematical analysis. 1. Title. II. Series.

QA300.B727 1996 515-dc20 95-44877

Printed on acid-free paper.

© 1996 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc in 1996 Softcover reprint of the hardcover 1 st edition 1996 AlI rights reserved. This work may not be translated or copied in whole or in part without the writlen permission of the publisher (Springer-Science+ Business Media, LLC,) except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production managed by Robert Wexler; manufacturing supervised by Jacqui Ashri. Photocomposed copy prepared from the author's LaTeX files.

9 8 7 6 5 4 3 (Corrected third printing, 2001)

ISBN 978-1-4612-6879-6 SPIN 10792722

For Anna

Preface

This is a textbook suitable for a year-long course in analysis at the ad­vanced undergraduate or possibly beginning-graduate level. It is intended for students with a strong background in calculus and linear algebra, and a strong motivation to learn mathematics for its own sake. At this stage of their education, such students are generally given a course in abstract algebra, and a course in analysis, which give the fundamentals of these two areas, as mathematicians today conceive them.

Mathematics is now a subject splintered into many specialties and sub­specialties, but most of it can be placed roughly into three categories: al­gebra, geometry, and analysis. In fact, almost all mathematics done today is a mixture of algebra, geometry and analysis, and some of the most in­teresting results are obtained by the application of analysis to algebra, say, or geometry to analysis, in a fresh and surprising way. What then do these categories signify? Algebra is the mathematics that arises from the ancient experiences of addition and multiplication of whole numbers; it deals with the finite and discrete. Geometry is the mathematics that grows out of spatial experience; it is concerned with shape and form, and with measur­ing, where algebra deals with counting. Analysis might be described as the mathematics that deals with the ideas of the infinite and the infinitesimal; more specifically, it is the word used to describe the great web of ideas that has grown in the last three centuries from the discovery of the differ­ential and integral calculus. Its basic arena is the system of real numbers, a mathematical construct which combines algebraic concepts of addition, multiplication, etc., with the geometric concept of a line, or continuum.

There is no general agreement on what an introductory analysis course

viii Preface

should include. I have chosen four major topics: the calculus of functions of one variable, treated with modern standards of rigor; an introduction to general topology, focusing on Euclidean space and spaces of functions; the general theory of integration, based on the concept of measure; and the calculus, differential and integral, for functions of several variables, with the inverse and implicit function theorems, and integration over manifolds. Inevitably, much time and effort go into giving definitions and proving technical propositions, building up the basic tools of analysis. I hope the reader will feel this machinery is justified by some of its products displayed here. The theorems of Dirichlet, Liouville, Weyl, Brouwer, and Riemann's Dirichlet principle for harmonic functions, for instance, should need no applications to be appreciated. (In fact, they have a great number of ap­plications. )

An ideal book of mathematics might uphold the standard of economy of expression, but this one does not. The reader will find many repetitions here; where a result might have been proved once and subsequently referred to, I have on occasion simply given the old argument again. My justification is found in communications theory, which has shown mathematically that redundancy is the key to successful communication in a noisy channel. I have also on occasion given more than one proof for a single theorem; this is done not because two proofs are more convincing than one, but because the second proof involves different ideas, which may be useful in some new context.

I have included some brief notes, usually historical, at the end of each chapter. The history is all from secondary sources, and is not to be relied on too much, but it appears that many students find these indications of how things developed to be interesting. A student who wants to learn the material in the early chapters of this book from a historical perspective will find Bressoud's recent book [1] quite interesting.

While this book is meant to be used for a year-long course, I myself have never managed to include everything here in such a course. A year and a half might be reasonable, for students with no previous experience with rigorous analysis. In different years, I have omitted different topics, always regretfully. Every topic treated here meets one of two tests: it is either something that everybody should know, or else it is just too beautiful to leave out. Nevertheless, life is short, and the academic year even shorter, and anyone who teaches with this book should plan on leaving something out. I expect that most teachers who use this book might also be tempted to include some topic that I have not treated, or develop further some theme that is touched on lightly here.

Those students whose previous mathematical experience is mostly with calculus are in for some culture shock. They will notice that this book is only about 30% as large as their calculus text, and contains about one-tenth as many exercises. (So far, so good.) But it will quickly become clear that some of the problems are quite demanding; I believe that (certainly at this

Preface ix

level) more is learned by spending hours, if necessary, on a few problems, sometimes a single problem, than in routinely dispatching a dozen exercises, all following the same pattern. I hope the reader is not discouraged by difficulty, but rewarded by difficulty overcome.

This book consists of theorems, propositions, and lemmas (these words all mean the same thing), along with definitions and examples. Most of these are set off formally as Theorem, Proposition, etc., but some defini­tions, examples, and theorems are in fact sneaked into the text between the formally announced items. I have been persuaded to number the Theorems, Examples, Definitions, etc. by one sequence. Thus Example 4.4 refers to the fourth item in Chapter 4, where an item could be either a Theorem, Proposition, Lemma, Definition, or Example. It would have been more log­ical to have it refer to the fourth example in this chapter, but it would have made navigation more difficult.

One of the important things that one learns in a course at this level is how to write a mathematical proof. It is quite difficult to prescribe what constitutes a proper proof. It should be a clear and compelling argument, that forces a reader (who has accepted previous theorems and understands the hypotheses) to accept its assertions. It should be concise, but not cryp­tic; it should be detailed, but not verbose. We learn to do it by imitating models. Here are two models from ancient Greece. Throughout this book, the symbol I will mark the conclusion of a proof.

Theorem. There are infinitely many primes.

Proof. If Pl, P2, ... ,Pn are primes, let N = P1P2 ... Pn + 1. Then N is not divisible by any Pj, j = 1,2, ... , n, so either N is a prime, or N is divisible by some prime other than Pl, P2, ... ,Pn. In either case, there are at least n + 1 primes. I

Note that this proof assumes a knowledge of what a prime number is, and a previously obtained result that every integer N > 1 is divisible by some prime. Note also that the last sentence of the proof has been omitted. It might be either to the effect that the hypothesis that the set of all primes can be listed in a finite sequence Pl, P2, ... ,Pn has led to a contradiction, or that we have given the recipe for finding a new prime for each natural number, so that the sequence 2, 3, 7, 43, ... , can be extended indefinitely.

Theorem. The square on the hypotenuse of a right triangle is the sum of the squares on the two shorter sides.

Proof. If the sides of the triangle are a, b, and c, with c the hypotenuse, then the square of side a + b can be dissected in two ways, as shown below. Removing the four copies of the triangle present in each dissection, the theorem follows. I

This is one of the rare occasions when I would accept a picture as a proof. (Having once been shown the proof, by a carefully drawn diagram,

x Preface

a

a

Figure 1. The Pythagorean Theorem.

that every triangle is isosceles, I could never again believe that pictures don't lie.) In this argument, the picture is clear and convincing. By the way, there are few pictures in what follows, and they are all simple and hand­drawn, meant to serve as a guide to simple ideas that are (unfortunately but necessarily) being expressed in awkward or complicated notation. I would urge the readers (of this or any other mathematics) to make their own sketches at all times, preferably crude and schematic.

Acknowledgments. It is impossible to list all the writers and individuals who have influenced me in the writing of this book, but for me the model of analysis textbooks at this level has always been Rudin's Principles of Mathematical Analysis [11]. The second half ofthis book (which was written first) was greatly influenced by Spivak's Calculus on Manifolds [13], the first clear and simple introduction to Stokes' theorem in its modern form. I was fortunate to read in manuscript Munkres' excellent book Analysis on Manifolds [10] while I was writing that earlier version, and profited from it.

Many students found typographical errors and other infelicities in the class notes which formed the first version of the first half of the book, and I want to thank especially Andrew Brecher, Greg Friedman, Ezra Miller, and Max Minzner for their detailed examination, and their suggestions for that part. Eva Kallin made many useful criticisms of an earlier version of the notes on which the second half of this book is based. I am grateful to Xiang-Qian Chang, who read the entire manuscript, and pointed out a great number of rough places.

The mistakes that remain are all my own. I would appreciate hearing about them.

22 August 2000 With this second printing, I have had the opportunity to correct a num­

ber of typographical errors, infelicities, rough spots, and mistakes. I thank the following for pointing them out to me: William Beckner, Mark Bruso, Xiang-Qian Chang, Eva Kallin, Tom Koornwinder, and Mark McKee.

A.B.

Contents

Preface vii

1 Real Numbers 1 1.1 Sets, Relations, Functions 1 1.2 Numbers 4 1.3 Infinite Sets 6 1.4 Incommensurability 8 1.5 Ordered Fields . 10 1.6 Functions on R 17 1.7 Intervals in R 19 1.8 Algebraic and Transcendental Numbers. 20 1.9 Existence of R . 21 1.10 Exercises 23 1.11 Notes. 26

2 Sequences and Series 28 2.1 Sequences 28 2.2 Continued Fractions. 36 2.3 Infinite Series 39 2.4 Rearrangements of Series 45 2.5 Unordered Series 47 2.6 Exercises 50 2.7 Notes. 53

xii Contents

3 Continuous Functions on Intervals 3.1 Limits and Continuity ... 3.2 Two Fundamental Theorems 3.3 Uniform Continuity .... 3.4 Sequences of Functions . . 3.5 The Exponential Function 3.6 Trigonometric Functions 3.7 Exercises 3.8 Notes ...

4 Differentiation 4.1 Derivatives.................. 4.2 Derivatives of Some Elementary Functions 4.3 Convex Functions . . . . 4.4 The Differential Calculus 4.5 L'Hospital's Rule .... 4.6 Higher Order Derivatives 4.7 Analytic Functions 4.8 Exercises 4.9 Notes ....... .

5 The Riemann Integral 5.1 Riemann Sums ... 5.2 Existence Results 5.3 Properties of the Integral. 5.4 Fundamental Theorems of Calculus 5.5 Integrating Sequences and Series . 5.6 Improper Integrals 5.7 Exercises 5.8 Notes

6 Topology 6.1 Topological Spaces 6.2 Continuous Mappings. 6.3 Metric Spaces . . . . . 6.4 Constructing Topological Spaces . 6.5 Sequences... 6.6 Compactness.. 6.7 Connectedness. 6.8 Exercises 6.9 Notes ....

55 55 59 61 62 65 67 69 72

74 74 76 78 81 86 88 90 93 95

98 98

102 107 110 113 114 118 121

123 123 126 127 131 135 140 147 150 153

7 Function Spaces 155 7.1 The Weierstrass Polynomial Approximation Theorem 155 7.2 Lengths of Paths .................... 159

7.3 Fourier Series . 7.4 Weyl's Theorem 7.5 Exercises 7.6 Notes ..... .

8 Differentiable Maps 8.1 Linear Algebra. . . . . . . 8.2 Differentials........ 8.3 The Mean Value Theorem 8.4 Partial Derivatives .... 8.5 Inverse and Implicit Functions 8.6 Exercises 8.7 Notes ............. .

9 Measures 9.1 Additive Set Functions 9.2 Countable Additivity . 9.3 Outer Measures . . . . 9.4 Constructing Measures 9.5 Metric Outer Measures 9.6 Measurable Sets 9.7 Exercises 9.8 Notes ..... .

10 Integration 10.1 Measurable Functions ...... . 10.2 Integration ............ . 10.3 Lebesgue and Riemann Integrals. 10.4 Inequalities for Integrals 10.5 Uniqueness Theorems .. 10.6 Linear Transformations . 10.7 Smooth Transformations 10.8 Multiple and Repeated Integrals . 10.9 Exercises 10.10 Notes ............... .

11 Manifolds 11.1 Definitions........ 11.2 Constructing Manifolds. 11.3 Tangent Spaces 11.4 Orientation. 11.5 Exercises 11.6 Notes ....

Contents xiii

161 170 171 173

175 176 182 185 186 191 196 199

201 202 204 208 211 213 215 219 221

223 223 226 231 233 237 240 241 244 247 251

253 253 258 260 262 265 267

xiv Contents

12 Multilinear Algebra 12.1 Vectors and Tensors. 12.2 Alternating Tensors . 12.3 The Exterior Product. 12.4 Change of Coordinates 12.5 Exercises 12.6 Notes .....

13 Differential Forms 13.1 Tensor Fields ..... . 13.2 The Calculus of Forms . 13.3 Forms and Vector Fields 13.4 Induced Mappings. . . . 13.5 Closed and Exact Forms 13.6 Tensor Fields on Manifolds. 13.7 Integration of Forms in R n .

13.8 Exercises 13.9 Notes ......... .

14 Integration on Manifolds 14.1 Partitions of Unity ........ . 14.2 Integrating k-Forms ........ . 14.3 The Brouwer Fixed Point Theorem 14.4 Integrating Functions on a Manifold. 14.5 Vector Analysis . . . 14.6 Harmonic Functions. 14.7 Exercises 14.8 Notes.

References

Index

269

269 272 277 280 282 283

285 285 286 288 290 291 293 294 295 296

297 297 300 305 307 312 314 318 321

323

325