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Introduction to Mathematical Analysis PDF

759 Pages·1996·19.174 MB·English
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Introduction to Mathematical Analysis Steven A. Douglass A ▼▼ ADDISON-WESLEY PUBLISHING COMPANY Reading, Massachusetts • Menlo Park, California • New York Don Mills, Ontario • Wokingham, England • Amsterdam • Bonn • Sydney Singapore • Tokyo • Madrid • San Juan • Milan • Paris Original edition, entitled Introduction to Mathematical Analysis, 1st Edition, 0201508974 by Douglass, Steven A., published by Pearson Education, Inc., publishing as Addison-Wesley Higher Education, Copyright © 1996. All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage retrieval system, without permission from Pearson Education, Inc. KOREA edition published by PEARSON EDUCATION KOREA LTD, Copyright © 2010 This edition is manufactured in KOREA, and is authorized for sale only in KOREA. ISBN 978-89-450-4022-0 ISBN 89-450-4022-6 To Dr. Anna Penk upon her retirement. An inspiring teacher, steadfast friend, and a woman of integrity. Preface Year after year, when teaching the sequence of courses in undergraduate analysis, I have noticed an ever-widening gulf among the texts available on the market, the expected content of these courses, and the observed capabilities of students in my classes. The classic texts I most admire set a standard rather beyond the reach of many of my students; other texts seem, for various reasons, not quite suitable for my classes. My goal has been to write a book that is mathematically rigorous, that treats those topics generally considered to comprise the central core of analysis at the junior/senior level, and that is accessible to a broad spectrum of these beginning students. I offer this text as my attempt to achieve these goals and to bridge this gap. This text is suitable for use in a one-semester, two-quarter, or year-long course in analysis at the undergraduate level. For students at this stage of development it is rigorous throughout. Mastery of the material presented here provides a solid initial training for students intending to proceed to graduate study in pure or applied mathematics. Students of engineering and the physical sciences, for whom these courses in anal­ ysis are terminal, will receive an essential and pragmatic introduction to classic, nineteenth-century mathematics. I have attempted to soften the hard edge of the mathematics by adopting a somewhat gentle style of presentation, especially in the early chapters. As the student progresses into the text, that style gradually hardens to more nearly approximate the usual terse mode of modem mathematical discourse. Having found that most of my students are intrigued by the historical devel­ opment of mathematics and by the characters in the vast, illustrious cast, I have included brief historical discussions in the text body to serve as motivation. Most students know little of the history of our discipline and often are delighted to realize v vi Preface that mathematics is a purely human endeavor and that all of us have traversed the path from the simple to the complex. An ample number of examples serve to illustrate the standard techniques and maneuvers used in analysis and to guide the beginner in the composition of careful proofs. I have included a rich supply of exercises ranging in degree of difficulty from the straightforward to the challenging. Opinions differ, of course, as to what must be included in an introductory analysis text and how that material ought to be presented. Since no one text can adequately treat this vast subject (and still be portable), inevitable compromises are required. Several features of this text deserve mention. We begin, as we must, with the completeness of R and promptly obtain the Bolzano-Weierstrass theorem. These essential topological concepts form an unfa­ miliar and uncharted region for the beginning student of analysis. Moreover, begin­ ners tend to think almost automatically in terms of sequences rather than in terms of the more subtle concept of the continuum. Therefore we immediately discuss sequences in R and prove that the real numbers are Cauchy complete. In Chapter 2 we generalize these concepts to R" in order to emphasize that the language being developed applies, with only the slightest of adjustments, as well to n-dimensional space as to the real line. This approach prepares the student to learn the basic facts of point set topology in Euclidean spaces and the various equivalent formulations of completeness. Connectedness is treated briefly. Chapter 2 concludes with the crucial concept of compactness. The decision to treat the topology of Euclidean spaces in this early chapter is the first significant compromise. The topology of R must be discussed early on; the same theorems must eventually be obtained for R". Rather than repeat the identical arguments in a later chapter and thereby increase the bulk of the final product, I have chosen to state and prove these topological theorems for R” with the understanding that, by letting n = 1, all the proofs remain valid as written. Instructors who wish to postpone treatment of multivariate analysis can selec­ tively omit much of the material in Chapter 2. In this case, Section 2.1 would be omitted and throughout Section 2.2 R" would be read as R; the notation has been designed so the proofs are valid in either case. Section 2.3 would be omitted except for Cantor’s nested interval theorem (Theorem 2.3.1 and Corollary 2.3.2), which is proved independently for this purpose. Likewise Sections 2.4 and 2.5 can be used by replacing R" by R. The Heine-Borel theorem is proved in R before its n-dimensional version is discussed. Later, when undertaking analysis in R”, instructors can return briefly to Chapter 2 to cover the necessary topological facts. Once the topological preliminaries are established, the way is clear to treat continuity in generality. Chapter 3 deals with continuity in topological terms so that the groundwork is laid for an efficient treatment of functions of a vector variable in Chapters 8 and 9. Again, instructors who wish to postpone the analysis of functions of a vector variable can read R for R" and defer those portions of Chapter 3 applicable only in R". The full import of continuity tends to be downplayed in the teaching of introductory calculus courses in the interest of getting on with more “practical” topics; Chapter 3 is intended to rectify this tendency. Preface Vll The remainder of the text is partitioned into three general areas. Chapters 4 through 7 treat functions of a single real variable. Chapters 8 through 10 discuss functions of a vector variable. Chapters 11 through 14 present the representation of functions by infinite series and integrals. An instructor who so wishes can proceed directly from Chapter 6 to Chapter 11 and return to Chapter 8 at a later time. The results of paramount importance in Chapter 4 are, of course, the Mean Value Theorem and Taylor’s theorem. Chapter 5 treats functions of bounded variation. This material is used in Chapters 6 and 7 and therefore is not optional. These are functions well known to the masters of analysis in the nineteenth century that deserve to be included in the education of every student of mathematics. Functions of bounded variation provide an accessible model of functions more general than the “nice” functions encountered in elementary calculus. As such, they provide a tool to enlarge the student’s field of mathematical vision. Moreover, many of the functions that arise in serious applications of mathematics in physics, chemistry, and engineering are of bounded variation. Many instructors, faced with the inevitable time constraints, will cover the Riemann integral in Chapter 6 but will consider Chapter 7 to be optional. The text has been structured so that this latter chapter can be omitted without disrupting the study of subsequent chapters. It can be used as a resource for self-study by those students who want a deeper understanding of integration. Anyone intending to continue in graduate study in pure or applied mathematics will find Chapter 7 of value. The extension of integration theory to include Lebesgue’s integral seems easy and natural to one already familiar with Stieltjes’s generalization of Riemann’s work. Chapters 8 and 9 present the differential calculus of real and vector-valued functions of a vector variable. These chapters especially require familiarity with linear algebra. Chapter 10 treats double and triple integrals. Since multiple integrals, although important, are used only in a subordinate way in this text, just the basic facts are proved. A classic and accessible proof of the role played by the Jacobian in the transformation of multiple integrals is included. Chapters 11 through 14 comprise the heart of the text; they treat infinite series, improper integrals, and the representation of functions by power series, integrals, and Fourier series. These topics introduce the student to one of the central goals of analysis: the development of tools for the careful analysis of the behavior of functions. Here too a compromise was required. This aspect of analysis is vast. To remain within the usual constraints on time and space, I have chosen to present the central core of basic facts and several classic examples but, reluctantly, to omit some favorite topics. For example, rather than attempt to treat both the Laplace and Fourier transforms (and therefore be forced to treat each somewhat superficially), I have chosen to discuss the Laplace transform in detail, including the convolution product, and have left treatment of the Fourier transform for in a course in complex analysis. Someone who has worked through the details presented here will be able to apply these methods to the study of other transforms. Chapter 14 includes Jordan’s elegant conditions for the pointwise convergence of Fourier series and Fejer’s treatment of the Cesaro summability of these series. viii Preface This book would not have seen the light of day without the continuous encour­ agement of my friends and colleagues to whom I owe a great debt. I especially want to acknowledge the inspiring example of Dr. Anna Penk, who loves mathematics and taught it with passion and who graciously suggested valuable improvements to the various drafts of this text. I must also acknowledge my long-suffering stu­ dents who struggled, sometimes under duress, with the early versions, whose candid responses helped me reshape my thinking and thereby improve the final product. They reminded me, sometimes quite pointedly, that successful teaching requires close attention to the student’s point of view. I appreciate the advice given by those who carefully reviewed the manuscript: Ralph Grimaldi, Rose-Hulman Institute of Technology; Ho Kuen Ng, San Jose State University; Harvey Greenwald, California Polytechnic State University; Ken Johnson, University of Georgia; Jutta Hausen, University of Houston; John D’Angelo, University of Illinois, Urbana-Champaign; Howard Sherwood, University of Central Florida; Mark McConnell, Oklahoma State University; Peter Colwell, Iowa State University; Eric Hayashi, San Francisco State University; John Schiller, Temple University. Typographical errors, like spi­ der mites, seem to thrive and swarm in the process of writing a book; I’ve tried to snag them all. Any that remain and any mathematical errors that persist are my responsibility alone. I owe special thanks to Laurie Rosatone, my editor at Addison-Wesley. She nursed this project from its inception to the present, boosting my morale with her enthusiasm, wit, and commitment and guiding me through the rough parts. Many of the improvements from the original draft to the final product are the result of her wise counsel. Her assistant, Ranjani Srinivasan, kept me smiling with her sprightly good cheer and generous assistance. Kathy Manley, responsible for the production phase, helped enormously with the complex task of transforming my work into polished form. My sincere thanks to you all. SAD

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