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Real Analysis with Real Applications Hardcover – December 30, 2001

ISBN-13: 978-0130416476 ISBN-10: 0130416479

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Product Details

  • Hardcover: 624 pages
  • Publisher: Prentice Hall (December 30, 2001)
  • Language: English
  • ISBN-10: 0130416479
  • ISBN-13: 978-0130416476
  • Product Dimensions: 9.3 x 7.2 x 1.1 inches
  • Shipping Weight: 2.5 pounds
  • Average Customer Review: 2.7 out of 5 stars  See all reviews (7 customer reviews)
  • Amazon Best Sellers Rank: #2,034,316 in Books (See Top 100 in Books)

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From the Back Cover

Using a progressive but flexible format, this book contains a series of independent chapters that show how the principles and theory of real analysis can be applied in a variety of settings—in subjects ranging from Fourier series and polynomial approximation to discrete dynamical systems and nonlinear optimization. Users will be prepared for more intensive work in each topic through these applications and their accompanying exercises. Chapter topics under the abstract analysis heading include: the real numbers, series, the topology of R^n, functions, normed vector spaces, differentiation and integration, and limits of functions. Applications cover approximation by polynomials, discrete dynamical systems, differential equations, Fourier series and physics, Fourier series and approximation, wavelets, and convexity and optimization. For math enthusiasts with a prior knowledge of both calculus and linear algebra.

Excerpt. © Reprinted by permission. All rights reserved.

This book provides an introduction both to real analysis and to a range of important applications that require this material. More than half the book is a series of essentially independent chapters covering topics from Fourier series and polynomial approximation to discrete dynamical systems and convex optimization. Studying these applications can, we believe, both improve understanding of real analysis and prepare for more intensive work in each topic. There is enough material to allow a choice of applications and to support courses at a variety of levels.

The first part of the book covers the basic machinery of real analysis, focusing on that part needed to treat the applications. This material is organized to allow a streamlined approach that gets to the applications quickly, or a more wide-ranging introduction. To this end, certain sections have been marked as enrichment topics or as advanced topics to suggest that they might be omitted. It is our intent that the instructor will choose topics judiciously in order to leave sufficient time for material in the second part of the book.

A quick look at the table of contents should convince the reader that applications are more than a passing fancy in this book. Material has been chosen from both classical and modern topics of interest in applied mathematics and related fields. Our goal is to discuss the theoretical underpinnings of these applied areas concentrating on the role of fundamental principles of analysis. This is not a methods course, although some familiarity with the computational or methods-oriented aspects of these topics may help the student appreciate how the topics are developed. In each application, we have attempted to get to a number of substantial results and to show how these results depend on the fundamental ideas of real analysis. In particular, the notions of limit and approximation are two sides of the same coin, and this interplay is central to the whole book.

We emphasize the role of normed vector spaces in analysis, as they provide a natural framework for most of the applications. This begins early with a separate treatment of Rn. Normed vector spaces are introduced to study completeness and limits of functions. There is a separate chapter on metric spaces that we use as an opportunity to put in a few more sophisticated ideas. This format allows its omission, if need be.

The basic ideas of calculus are covered carefully, as this level of rigour is not generally possible in a first calculus course. One could spend a whole semester doing this material, which forms the basis of many standard analysis courses today. When we have taught a course from these notes, however, we have often chosen to omit topics such as the basics of differentiation and integration on tile grounds that these topics have been covered adequately for many students. The goal of getting further into the applications chapters may make it worth cutting here.

We have treated only tangentially some topics commonly covered in real analysis texts, such as multivariate calculus or a brief development of the Lebesgue integral. To cover this material in an accessible way would have left no time, even in a one-year course, for the real goal of the book. Nevertheless, we deal throughout with functions on domains in Rn, and we do manage to deal with issues of higher dimensions without differentiability. For example, the chapter on convexity and optimization yields some deep results on "nonsmooth" analysis that contain the standard differentiable results such as Lagrange multipliers. This is possible because the subject is based on directional derivatives, an essentially one-variable idea. Ideas from multivariate calculus appear once or twice in the advanced sections, such as the use of Green's Theorem in the section on the isoperimetric inequality.

Not covering measure theory was another conscious decision to keep the material accessible and to keep the size of the book under control. True, we do make use of the L2 norm and do mention the LP spaces because these are important ideas. We feel, however, that the basics of Fourier series, approximation theory, and even wavelets can be developed while keeping measure theory to a minimum. Of course, this does not mean we think that the subject is unimportant. Rather we wished to aim the book at an undergraduate audience. To deal partially with some of the issues that arise here, we have included a section on metric space completion. This allows a treatment of LP spaces as complete spaces of bona fide functions, by means of the Daniell jntegral. This is certainly an enrichment topic, which can be used to motivate the need for measure theory and to satisfy curious students.

This book began in 1984 when the first author wrote a short set of course notes (120 pages) for a real analysis class at the University of Waterloo designed for students who came primarily from applied math and computer science. The idea was to get to the basic results of analysis quickly and then illustrate their role in a variety of applications. At that time, the applications were limited to polynomial approximation, Newton's method, differential equations, and Fourier series.

A plan evolved to expand these notes into a textbook suitable for one semester or a year-long course. We expanded both the theoretical section and the choice of applications in order to make the text more flexible. As a consequence, the text is not uniformly difficult. The material is arranged by topic, and generally each chapter gets more difficult as one progresses through it. The instructor can choose to omit some more difficult topics in the chapters on abstract analysis if they will not be needed later. We provide a flow chart indicating the topics in abstract analysis required for each part of the applications chapters. For example, the chapter on limits of functions begins with the basic notion of uniform convergence and the fundamental result that the uniform limit of continuous functions is continuous. It ends with much more difficult material, such as the Arzela-Ascoli Theorem. Even if one plans to do the chapter on differential equations, it is possible to stop before the last section on Peano's Theorem, where the Arzela-Ascoli Theorem is needed. So both topics can be conveniently omitted. Although one cannot proceed linearly through the text, we hope there is some compensation in demonstrating that, even at a high level, there is a continued interplay between theory and application.

The background assumed for using this text is decent courses in both calculus and linear algebra. What we expect is outlined in the background chapter. A student should have a reasonable working knowledge of differential and integral calculus. Multivariable calculus is an asset because of the increased level of sophistication and the incorporation of linear algebra; it is not essential. We certainly expect that the student is used to working with exponentials, logarithms, and trigonometric functions. Linear algebra is needed because we treat Rn, C(X), and L2 (-π, π) as vector spaces. We develop the notion of norms on vector spaces as an important tool for measuring convergence. As such, the reader should be comfortable with the notion of a basis in finite-dimensional spaces. Familiarity with linear transformations is also sometimes useful. A course that introduces the student to proofs would also be an asset. Although we have attempted to address this in the background chapter (Chapter 1), we have no illusions that this text would be easy for a student having no prior experience with writing proofs.

While this background is in principle enough for the whole book, sections marked with a • require additional mathematical maturity or are not central to the main development, and sections marked with a * are more difficult yet. By and large, the various applications are independent of each other. However, there are references to material in other chapters. For example, in the wavelets chapter (Chapter 15), it seems essential to make comparisons with the classical approximation results for Fourier series and for polynomials.

It is also possible to use an application chapter on its own for a student seminar or other topics course. We have included several modern topics of interest in addition to the classical subjects of applied mathematics. The chapter on discrete dynamical systems (Chapter 11) introduces the notions of chaos and fractals and develops a number of examples. The chapter on wavelets (Chapter 15) illustrates the ideas with the Haar wavelet. It continues with a construction of wavelets of compact support, and gives a complete treatment of a somewhat easier continuous wavelet. In the final chapter (Chapter 16), we study convex optimization and convex programming. Both of these latter chapters require more linear algebra than the others.

We would like to thank various people who worked with early versions of this book for their helpful comments; in particular, Robert Andre, John Baker, Brian Forrest, John Holbrook, David Seigel, and Frank Zorzitto. We also thank various people who offered us assistance in various ways, including Jon Borwein, Stephen Krantz, Justin Peters, and Ed Vrscay. We also thank our student Masoud Kamgarpour for working through parts of the book. We would particularly like to thank the students in various classes, at the University of Waterloo and at the University of Nebraska, where early versions of the text were used.

We welcome comments on this book.

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Most Helpful Customer Reviews

4 of 4 people found the following review helpful By Stephanie Jakus on July 15, 2008
Format: Hardcover Verified Purchase
I'm now almost a third year in graduate school and searched Amazon for this book, as it was my favorite analysis textbook as an undergraduate, and I thought of buying it for reference to go with the solutions that I wrote (and saved) as an undergraduate.

I was shocked to find such poor reviews of such a well written text. I found the book very readable, the examples helpful, and most of all, the exercises very interesting and fun to solve.

Do not be put off by the previous critiques. This is an excellent book and it is the first book in analysis that enjoyed learning from.
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2 of 2 people found the following review helpful By Anthony on December 21, 2005
Format: Hardcover
I am currently a graduate student, and we are using this book in my first-year graduate course in analysis. To be quite honest, I find this book utterly useless, except for looking up homework problems that are so hard you are forced to look elsewhere just to learn how to solve them! The authors spend way too little time building up the theory and just expect their readers to be able to follow what they're doing with very few examples (or ones too complicated to really illustrate what's going on), and then give problems where even the easier ones can seem near impossible. This book makes more sense as a graduate text, certainly, especially if you've already had analysis; in that case, then you may only need to see the major theorems as a refresher and then you can start right on the challenging problems.

However, if you're an undergrad and this is your first exposure to analysis, go elsewhere, please! My fellow grad students and I have gotten so frustrated over this book and its problems, and we've all had analysis before! If you've never had analysis before, I would suggest Bartle/Sherbert's Intro. to Real Analysis; they spend a good amount of time with examples and what I call "warm-up" homework problems to get you used to the concepts, followed by some doozies (and, yes; selected answers and hints are in the back!). If you're very strong in math, then perhaps Rudin's "Principles of Mathematical Analysis" may be more up your alley (aka Baby Rudin). Best of luck to you!
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5 of 7 people found the following review helpful By X. Zhang on April 22, 2005
Format: Hardcover
I've been using this text for two semesters. I have to say this book is too advanced for starters, especially after chapter 6. My biggest complaint is the author does not provide enough examples to illustrate the theorems. A majority of the sections usually go like: 1. proposition of a theorem 2. proof 3. major theorem. 4. proof 5. corollary 6. proof 7. tons of hard problems left to homework.

I would suggest the author give more examples when showing off those hard theorems. It could be better if the author also provides solutions to (at least) half of the exercises at the end of each section. Remember your readers are not academic conference colleagues, but first-time undergrad students. We learn things from examples!
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8 of 12 people found the following review helpful By Palle E T Jorgensen VINE VOICE on August 14, 2002
Format: Hardcover
A well balanced book! The first solid analysis course, with proofs, is central in the offerings of any math.-dept.;-- and yet, the new books that hit the market don't always hit the mark: The balance between theory and applications, --between technical profs and intuitive ideas,--between classical and modern subjects, and between real life exercises vs. the ones that drill a new concept. The Davidson-Donsig book is outstanding, and it does hit the mark. The writing is both systematic and engaged.- Refreshing! Novel: includes wavelets, approximation theory, discrete dynamics, differential equations,
Fourier analysis, and wave mechanics.
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