Undergraduate Texts in

Editors S. Axler F. W. Gehring K.A. Ribet

Springer-Science+Business Media, LLC Undergraduate Texts in Mathematics

Anglin: Mathematics: A Concise History Chung: Elementary Probability Theory and Philosophy. with Stochastic Processes. Third Readings in Mathematics. edition. AnglinlLambek: The Heritage of CoxlLittle/O'Shea: Ideals, Varieties, Thales. and Algorithms. Second edition. Readings in Mathematics. Croom: Basic Concepts of Algebraic Apostol: Introduction to Analytic Topology. Number Theory. Second edition. Curtis: : An Introductory Armstrong: Basic Topology. Approach. Fourth edition. Armstrong: Groups and Symmetry. Devlin: The Joy of Sets: Fundamentals Axler: Linear Algebra Done Right. of Contemporary Set Theory. Second edition. Second edition. Beardon: Limits: A New Approach to Dixmier: General Topology. Real Analysis. Driver: Why Math? BaklNewman: Complex Analysis. EbbinghauslFlumffhomas: Second edition. Mathematical Logic. Second edition. BanchofflWermer: Linear Algebra Edgar: , Topology, and Fractal Through Geometry. Second edition. Geometry. Berberian: A First Course in Real Elaydi: An Introduction to Difference Analysis. Equations. Second edition. Bix: Conics and Cubics: A Exner: An Accompaniment to Higher Concrete Introduction to Algebraic Mathematics. Curves. Exner: Inside . Bremaud: An Introduction to FinelRosenberger: The Fundamental Probabilistic Modeling. Theory of Algebra. Bressoud: Factorization and Primality Fischer: Intermediate Real Analysis. Testing. FlaniganlKazdan: Calculus Two: Linear Bressoud: Second Year Calculus. and Nonlinear Functions. Second Readings in Mathematics. edition. Brickman: Mathematical Introduction Fleming: Functions of Several Variables. to Linear Programming and Game Second edition. Theory. Foulds: Combinatorial Optimization for Browder: Mathematical Analysis: Undergraduates. An Introduction. Foulds: Optimization Techniques: An Buchmann: Introduction to Introduction. Cryptography. Franklin: Methods of Mathematical Buskes/van Rooij: Topological Spaces: Economics. From Distance to Neighborhood. Frazier: An Introduction to Wavelets Callahan: The Geometry of Spacetime: Through Linear Algebra. An Introduction to Special and General Gamelin: Complex Analysis. Relavitity. Gordon: Discrete Probability. Carter/van Brunt: The Lebesgue• HairerlWanner: Analysis by Its History. Stieitjes Integral: A Practical Readings in Mathematics. Introduction. Halmos: Finite-Dimensional Vector Cederberg: A Course in Modem Spaces. Second edition. Geometries. Second edition. Halmos: Naive Set Theory. Childs: A Concrete Introduction to Higher Algebra. Second edition. (continued after index) Andrew Browder

Mathematical Analysis An Introduction

, Springer Andrew Browder Mathematics Department Providence, RI 02912 USA

Editorial Board

S. Axler F.W. Gehring K.A. Ribet Mathematics Department Mathematics Department Mathematics Department San Francisco State Bast Hali University of California University University of Michigan at Berkeley San Francisco, CA 94132 Ann Arbor, MI 48109 Berkeley, CA 94720-3840 USA USA USA

With five iIlustrations.

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 1st 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 ad• vanced 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 sub• specialties, but most of it can be placed roughly into three categories: al• gebra, geometry, and analysis. In fact, almost all mathematics done today is a mixture of algebra, geometry and analysis, and some of the most in• teresting 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 measur• ing, 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 differ• ential 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 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 ap• plications. ) 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 defini• tions, 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 log• ical 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 cryp• tic; 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 hand• drawn, 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 55 3.1 Limits and Continuity ... 55 3.2 Two Fundamental Theorems 59 3.3 Uniform Continuity .... 61 3.4 Sequences of Functions . . 62 3.5 The Exponential Function 65 3.6 Trigonometric Functions 67 3.7 Exercises 69 3.8 Notes ... 72

4 Differentiation 74 4.1 Derivatives...... 74 4.2 Derivatives of Some Elementary Functions 76 4.3 Convex Functions . . . . 78 4.4 The Differential Calculus 81 4.5 L'Hospital's Rule .... 86 4.6 Higher Order Derivatives 88 4.7 Analytic Functions 90 4.8 Exercises 93 4.9 Notes ...... 95

5 The Riemann Integral 98 5.1 Riemann Sums ... 98 5.2 Existence Results 102 5.3 Properties of the Integral. 107 5.4 Fundamental Theorems of Calculus 110 5.5 Integrating Sequences and Series . 113 5.6 Improper Integrals 114 5.7 Exercises 118 5.8 Notes 121

6 Topology 123 6.1 Topological Spaces 123 6.2 Continuous Mappings. 126 6.3 Metric Spaces . . . . . 127 6.4 Constructing Topological Spaces . 131 6.5 Sequences... 135 6.6 Compactness.. 140 6.7 Connectedness. 147 6.8 Exercises 150 6.9 Notes .... 153

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

7.3 Fourier Series . 161 7.4 Weyl's Theorem 170 7.5 Exercises 171 7.6 Notes ...... 173

8 Differentiable Maps 175 8.1 Linear Algebra...... 176 8.2 Differentials...... 182 8.3 The Mean Value Theorem 185 8.4 Partial Derivatives .... 186 8.5 Inverse and Implicit Functions 191 8.6 Exercises 196 8.7 Notes ...... 199

9 Measures 201 9.1 Additive Set Functions 202 9.2 Countable Additivity . 204 9.3 Outer Measures . . . . 208 9.4 Constructing Measures 211 9.5 Metric Outer Measures 213 9.6 Measurable Sets 215 9.7 Exercises 219 9.8 Notes ...... 221

10 Integration 223 10.1 Measurable Functions ...... 223 10.2 Integration ...... 226 10.3 Lebesgue and Riemann Integrals. 231 10.4 Inequalities for Integrals 233 10.5 Uniqueness Theorems .. 237 10.6 Linear Transformations . 240 10.7 Smooth Transformations 241 10.8 Multiple and Repeated Integrals . 244 10.9 Exercises 247 10.10 Notes ...... 251

11 Manifolds 253 11.1 Definitions...... 253 11.2 Constructing Manifolds. 258 11.3 Tangent Spaces 260 11.4 Orientation. 262 11.5 Exercises 265 11.6 Notes .... 267 xiv Contents

12 Multilinear Algebra 269 12.1 Vectors and Tensors. 269 12.2 Alternating Tensors . 272 12.3 The Exterior Product. 277 12.4 Change of Coordinates 280 12.5 Exercises 282 12.6 Notes ..... 283

13 Differential Forms 285 13.1 Tensor Fields ...... 285 13.2 The Calculus of Forms . 286 13.3 Forms and Vector Fields 288 13.4 Induced Mappings. . . . 290 13.5 Closed and Exact Forms 291 13.6 Tensor Fields on Manifolds. 293 13.7 Integration of Forms in R n . 294 13.8 Exercises 295 13.9 Notes ...... 296

14 Integration on Manifolds 297 14.1 Partitions of Unity ...... 297 14.2 Integrating k-Forms ...... 300 14.3 The Brouwer Fixed Point Theorem 305 14.4 Integrating Functions on a Manifold. 307 14.5 Vector Analysis . . . 312 14.6 Harmonic Functions. 314 14.7 Exercises 318 14.8 Notes. 321

References 323

Index 325