com ADVANCED CALCULUS An Introduction to Linear Analysis Leonard F. Richardson ~WILEY ~INTERSCIENCE A JOHN WILEY & SONS, INC.com ADVANCED CALCULUS www.com ADVANCED CALCULUS An Introduction to Linear Analysis Leonard F. Richardson ~WILEY ~INTERSCIENCE A JOHN WILEY & SONS, INC.com Copyright© 2008 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section I 07 or I 08 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fcc to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.
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Includes bibliographical references and index.R53 2008 515--dc22 2008007377 Printed in Mexico 10 9 8 7 6 5 4 3 2 www.com To Joan, Daniel, and Joseph www.com CONTENTS Preface Xlll Acknowledgments XIX Introduction xxi PART I ADVANCED CALCULUS IN ONE VARIABLE 1 Real Numbers and Limits of Sequences 3 1.1 The Real Number System 3 Exercises 7 1.2 Limits of Sequences & Cauchy Sequences 8 Exercises 12 1.3 The Completeness Axiom and Some Consequences 13 Exercises 18 1.4 Algebraic Combinations of Sequences 19 Exercises 21 1.5 The Bolzano-Weierstrass Theorem 22 Exercises 24 1.6 The Nested Intervals Theorem 24 vii www.com viii CONTENTS Exercises 26 1.7 The Heine-Borel Covering Theorem 27 Exercises 30 1.8 Countability of the Rational Numbers 31 Exercises 35 1.9 Test Yourself 37 Exercises 37 2 Continuous Functions 39 2.1 Limits of Functions 39 Exercises 43 2.2 Continuous Functions 46 Exercises 49 2.3 Some Properties of Continuous Functions 50 Exercises 53 2.4 Extreme Value Theorem and Its Consequences 55 Exercises 60 2.5 The Banach Space C[a, b] 61 Exercises 66 2.6 Test Yourself 67 Exercises 67 3 Riemann Integral 69 3.1 Definition and Basic Properties 69 Exercises 74 3.2 The Darboux Integrability Criterion 76 Exercises 81 3.3 Integrals of Uniform Limits 83 Exercises 87 3.4 The Cauchy-Schwarz Inequality 90 Exercises 93 3.5 Test Yourself 95 Exercises 95 4 The Derivative 99 4.1 Derivatives and Differentials 99 Exercises 103 4.2 The Mean Value Theorem 105 www.com CONTENTS ix Exercises 109 4.3 The Fundamental Theorem of Calculus 110 Exercises 112 4.4 Uniform Convergence and the Derivative 114 Exercises 116 4.5 Cauchy's Generalized Mean Value Theorem 117 Exercises 121 4.6 Taylor's Theorem 122 Exercises 125 4.7 Test Yourself 126 Exercises 126 5 Infinite Series 127 5.1 Series of Constants 127 Exercises 132 5.2 Convergence Tests for Positive Term Series 134 Exercises 137 5.3 Absolute Convergence and Products of Series 138 Exercises 146 5.4 The Banach Space l 1 and Its Dual Space 148 Exercises 153 5.5 Series of Functions: The Weierstrass M-Test 154 Exercises 157 5.6 Power Series 158 Exercises 161 5.7 Real Analytic Functions and c= Functions 162 Exercises 167 5.8 Weierstrass Approximation Theorem 169 Exercises 173 5.9 Test Yourself 174 Exercises 174 PART II ADVANCED TOPICS IN ONE VARIABLE 6 Fourier Series 179 6.1 The Vibrating String and Trigonometric Series 180 Exercises 183 6.2 Euler's Formula and the Fourier Transform 184 Exercises 190 www.3 Bessel's Inequality and lz 192 Exercises 196 6.4 Uniform Convergence & Riemann Localization 197 Exercises 204 6.5 L 2 -Convergence & the Dual of l 2 205 Exercises 208 6.6 Test Yourself 212 Exercises 212 7 The Rlemann-Stieltjes Integral 215 7.1 Functions of Bounded Variation 216 Exercises 220 7.2 Riemann-Stieltjes Sums and Integrals 223 Exercises 227 7.3 Riemann-Stieltjes Integrability Theorems 228 Exercises 230 7.4 The Riesz Representation Theorem 231 Exercises 239 7.5 Test Yourself 241 Exercises 241 PART Ill ADVANCED CALCULUS IN SEVERAL VARIABLES 8 Euclidean Space 245 8.1 Euclidean Space as a Complete Norrned Vector Space 245 Exercises 249 8.2 Open Sets and Closed Sets 252 Exercises 254 8.3 Compact Sets 256 Exercises 258 8.4 Connected Sets 259 Exercises 261 8.5 Test Yourself 263 Exercises 263 9 Continuous Functions on Euclidean Space 265 9.1 Limits of Functions 265 Exercises 268 www.com CONTENTS xi 9.2 Continuous Functions 270 Exercises 272 9.3 Continuous Image of a Compact Set 274 Exercises 276 9.4 Continuous Image of a Connected Set 278 Exercises 279 9.5 Test Yourself 280 Exercises 280 10 The Derivative in Euclidean Space 283 10.1 Linear Transformations and Norms 283 Exercises 286 10.2 Differentiable Functions 289 Exercises 295 10.3 The Chain Rule in Euclidean Space 298 10.1 The Mean Value Theorem 300 10.2 Taylor's Theorem 301 Exercises 303 10.4 Inverse Functions 305 Exercises 309 10.5 Implicit Functions 311 Exercises 317 10.6 Tangent Spaces and Lagrange Multipliers 322 Exercises 327 10.7 Test Yourself 328 Exercises 328 11 Riemann Integration in Euclidean Space 331 11.1 Definition of the Integral 331 Exercises 336 11.2 Lebesgue Null Sets and Jordan Null Sets 338 Exercises 341 11.3 Lebesgue's Criterion for Riemann Integrability 342 Exercises 344 11.4 Fubini's Theorem 346 Exercises 349 11.5 Jacobian Theorem for Change of Variables 351 Exercises 355 www.com Xii CONTENTS 11.6 Test Yourself 357 Exercises 357 Appendix A: Set Theory 359 A. I Terminology and Symbols 359 Exercises 363 A.2 Paradoxes 363 Problem Solutions 365 References 379 Index 381 www.com PREFACE Why this Book was Written The course known as Advanced Calculus (or Introductory Analysis) stands at the summit of the requirements for senior mathematics majors. An important objective of this course is to prepare the student for a critical challenge that he or she will face in the first year of graduate study: the course called Analysis I, Lebesgue Measure and Integration, or Introductory Functional Analysis. We live in an era of rapid change on a global scale.
And the author and his de- partment have been testing ways to improve the preparation of mathematics majors for the challenges they will face. During the past quarter century the United States has emerged as the destination of choice for graduate study in mathematics. The influx of well-prepared, talented students from around the world brings considerable benefit to American graduate programs. The international students usually arrive better prepared for graduate study in mathematics-in particular better prepared in analysis-than their typical U.
There are many reasons for this, in- cluding (a) school systems abroad that are oriented toward teaching only the brightest students, and (b) the self-selection that is part of a student taking the step of travel abroad to study in a foreign culture. The presence of strongly prepared international students in the classroom raises the level at which courses are taught. Thus it is appropriate at the present time, in the early years of the new millennium, for college and university mathematics departments to xiii www.com XiV PREFACE reconsider their advanced calculus courses with an eye toward preparing graduates for the international environment in American graduate schools. This is a challenge, but it is also an opportunity for American students and international students to learn side-by-side with, and also about, one another.
It is more important than ever to teach undergraduate advanced calculus or analysis in such a way as to prepare and reorient the student for graduate study as it is today in mathematics. Another recent change is that applied mathematics has emerged on a large scale as an important component of many mathematics departments. In applied and numerical mathematics, functional analysis at the graduate level plays a very important role. Yet another change that is emerging is that undergraduates planning careers in the secondary teaching of mathematics are being required to major in mathematics instead of education.
These students must be prepared to teach the next generation of young people for the world in which they will live. Whether or not the mathematics major is planning an academic career, he or she will benefit from better preparation in advanced calculus for careers in the emerging world. The author has taught mathematics majors and graduate students for thirty-seven years. He has served as director of his department's graduate program for nearly two decades.
All the changes described above are present today in the author's department. This book has been written in the hope of addressing the following needs. Students of mathematics should acquire a sense of the unity of mathematics. Hence a course designed for senior mathematics majors should have an in- tegrative effect.
Such a course should draw upon at least two branches of mathematics to show how they may be combined with illuminating effect. Students should learn the importance of rigorous proof and develop skill in coherent written exposition to counter the universal temptation to engage in wishful thinking. Students need practice composing and writing proofs of their own, and these must be checked and corrected. The fundamental theorems of the introductory calculus courses need to bees- tablished rigorously, along with the traditional theorems of advanced calculus, which are required for this purpose.
The task of establishing the rigorous foundations of calculus should be en- livened by taking this opportunity to introduce the student to modern mathe- matical structures that were not presented in introductory calculus courses. Students should learn the rigorous foundations of calculus in a manner that reorient<; thinking in the directions taken by modern analysis. The classic theorems should be couched in a manner that reflects the perspectives of modem analysis.com PREFACE XV Features of this Text The author has attempted to address these needs presented above in the following manner. The two parts of mathematics that have been studied by nearly every math- ematics major prior to the senior year are introductory calculus, including calculus of several variables, and linear algebra.
Thus the author has chosen to highlight the interplay between the calculus and linear algebra, emphasizing the role of the concepts of a vector space, a linear transformation (including a linear functional), a norm, and a scalar product. For example, the customary theorem concerning uniform limits of continuous functions is interpreted as a completeness theorem for C[a, b] as a vector space equipped with the sup-norm. The elementary properties of the Riemann integral gain coherence expressed as a theorem establishing the integral as a bounded linear functional on a con- venient function-space. Similarly, the family of absolutely convergent series is presented from the perspective that it is a complete normed vector space equipped with the h -norm.
Many exercises are offered for each section of the text. These are essential to the course. An exercise preceded by a dagger symbol t is cited at some point in the text.