Understanding Analysis [Hardcover]

Stephen Abbott (Author)

Book Description

ISBN-10: 0387950605 | ISBN-13: 978-0387950600 | Publication Date: January 12, 2001

This elementary presentation exposes readers to both the process of rigor and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable. The aim is to challenge and improve mathematical intuition rather than to verify it. The philosophy of this book is to focus attention on questions which give analysis its inherent fascination. Each chapter begins with the discussion of some motivating examples and concludes with a series of questions.

Editorial Reviews

Review

From the reviews: "The author has written an interesting book designed as a text for a one-semester course ... . He also expects the students to do their part by leaving to them (with copious hints) the details of many of the steps in the proofs. This enables him to cover a surprising amount of material in an attractive and stimulating manner. Students who work their way through this text will have seen a lot of interesting analysis and have developed a good understanding of the material." (R.G. Bartle, Mathematical Reviews, 2001 m) "This is a dangerous book. Understanding Analysis is so well-written and the development of the theory so well-motivated that exposing students to it could well lead them to expect such excellence in all their textbooks. ... Understanding Analysis is perfectly titled; if your students read it that’s what’s going to happen. This terrific book will become the text of choice for the single-variable introductory analysis course; take a look at it next time you’re preparing that class." (Steve Kennedy, The Mathematical Association of America, 2001) "This is a well-constructed text on single variable real analysis. ... students are meeting many new concepts for the first time so, from a pedagogical point of view, it is essential that the material be motivated. Abbott makes a concerted and laudable effort to do so. ... This book lives up to its title by helping the readers to develop an understanding of analysis. I wish I could say I wrote it!" (Steve Abbott, The Mathematical Gazette, 87:509, 2003) "Understanding Analysis outlines an elementary, one-semester course designed to expose students to the rich rewards inherent in taking a mathematically rigorous approach to the study of functions of a real variable ... The philosophy of this book is to focus attention on the questions that give analysis its inherent fascination ... In giving these topics center stage, the hard work of a rigorous study is justified by the fact that they are inaccessible without it." (L'enseignement mathématique, 48:3-4, 2002) "In writing Understanding Analysis, Stephen Abbott appreciates that modern readers may be unfamiliar with axiomatic arguments. Thus, many courses in analysis have been made easier by making them less interesting. The alternative is to make advanced topics accessible and focus on questions that give real analysis its value ... Prerequisite knowledge has been kept to a minimum: an understanding of single variable calculus. The text is written with introductory students in mind and will certainly be persuasive in establishing the value of analysis." (Nigel Steele, Times Higher Education Supplement, November 2002) "Each chapter begins with a discussion section and ends with an epilogue. The discussion serves to motivate the content of the chapter while the epilogue points tantalisingly to more advanced topics. ... I wish I had written this book! The development of the subject follows the tried-and-true path, but the presentation is engaging and challenging. Abbott focuses attention immediately on the topics which make analysis fascinating ... and makes them accessible to an inexperienced audience." (Scott Sciffer, The Australian Mathematical Society Gazette, 29:3, 2002) "In this book the author presents fundamental facts connected with the real functions of a real variable. ... In my opinion the didactic conception is interesting. The structure of the chapters is the following: First: discussion section (motivation, some questions or examples). Second: the fundamental text (also with some open questions) and a lot of exercises ... . Each chapter is closed by an Epilogue." (Ryszard Pawlak, Zentralblatt MATH, 966, 2001) "This book aims to build up the elements of analysis properly – that is, with a good level of understanding ... The author’s intention was ‘to restore intellectual liveliness of the course ...’ In other words, he tries to present relatively advanced topics in an accessible way, and thus make the course interesting and worthy of effort ... This book will be useful for gifted students for supplementary reading to a basic analysis course or for students not specialising in mathematics who wish to put this material on a solid basis." (European Mathematical Society Newsletter, 42, December 2001) "The main goal of the author is to write an elementary one-semester book that exposes students to the rich rewards inherent in taking a mathematically rigorous approach to the study of functions of a real variable. This goal is excellently completed in this textbook. ... Each chapter begins with the discussion of some motivating examples and open questions. Each section contains many exercises ... . The book is warmly recommended to students, instructors interested in the study of introductory analysis." (József Németh, Acta Scientiarum Mathematicarum, 67, 2001)

Product Details

• Hardcover: 272 pages
• Publisher: Springer (January 12, 2001)
• Language: English
• ISBN-10: 0387950605
• ISBN-13: 978-0387950600
• Product Dimensions: 9.3 x 6.2 x 0.7 inches
• Shipping Weight: 1.2 pounds (View shipping rates and policies)
• Average Customer Review: 4.3 out of 5 stars  See all reviews (25 customer reviews)
• Amazon Best Sellers Rank: #53,967 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

37 of 38 people found the following review helpful:

5.0 out of 5 stars Too Good To be True, July 2, 2006

By 

Benjamin Balsam - See all my reviews
(REAL NAME)   

This review is from: Understanding Analysis (Hardcover)

Once in a while, a book comes along that is so wonderfully written, the reader reflexively searches for other books by its author. Understanding Analysis is a prime example of this rare breed (Unfortunately, this is Abbott's only book as far as I know: write more!).

Undergraduates often begin analysis courses with dread and finish in a state of utter confusion,knowing the definitions of key phrases, and sometimes even being able to supply proofs for some elementary results, but having no intution as to why the main theorems are pertinent.

But it does not have to be so. 'Understanding Analysis' has the distinction of being so readable, it is sometimes difficult to pry oneself away from its pages and attempt the exercises. On multiple occasions I found myself skimming through the book and reading the various 'special topics' (e.g. Cantor Sets, Integration, Fourier Series) interspersed throughout the book to pique the readers' interest. But most importantly, a reader will come away with an understanding of many theorems in analysis. He or she will begin to develop a vocabulary of results that make sense both mathematically and intuitively, be able to use the results to complete the exercises (which are by no means simple 'plug-and-chug' problems), and be excellently prepared for study at a more advanced level.

Bottom line: Abbott's book may not be encyclopedaic in content, but it, without a doubt covers a sufficient amount of material to warrant its use for a one-semester course in analysis. My only concern is that after such a fantasticly lucid treatment, students may have difficulty adapting to the vast selection of more advanced, less pedagogical texts available. I sincerely hope Abbott writes a sequel.

30 of 31 people found the following review helpful:

5.0 out of 5 stars A beautifully written introduction to real analysis in 1-dimension., February 18, 2007

By 

Joshua E. Hill "math grad student" (Irvine, CA USA) - See all my reviews
(REAL NAME)   

This review is from: Understanding Analysis (Hardcover)

If you're attempting to learn real analysis in one dimension, Abbott's Understanding Analysis is a great place to start. It is everything that a math textbook used for instruction should be: it has clean, concise prose, it assumes modest jumps in understanding, and it includes a good selection of exercises. Additionally, Abbott's book maintains a conversational tone without watering down the formality at the center of the mathematics while managing to convey the feeling of seeing "the big picture". Yes, there are more complete treatments (Rudin, Bartle, Browder, etc), but none of them are nearly as accessible, and frankly they aren't as good at providing an introduction to the subject.

This last statement may cause cries of anguish from mathies everywhere, as I've just suggested that there are some ways in which this book is better than Rudin's Principles of Mathematical Analysis. Rudin's texts (and most other upper division and graduate math texts that I've read) seem to fall into the same pedagogical trap: they assume that the student is already familiar with the material, but they may need a quick reminder of the particulars. This is, of course, not generally the case, so the student is left to fill in whatever gaps exist, hopefully with the aid of an instructor. Indeed, there is a sort of book for which this strategy is ideal: a reference. For this use, Rudin is spectacular. For actually learning the material for the first time, it is useful to have a bit of guidance, a bit of context, and a bit of direction. It is as if many math authors have forgotten a time where they didn't thoroughly understand the material, or worse, that they have somehow conflated the pain that they experienced as students while trudging through poorly realized texts with learning the material! Abbott does not fall into this trap, and for this, he deserves more praise than I can manage. The quality of the exposition in this book has re-awakened my dissatisfaction with most other math texts.

The only negative comments that I can make about this book come as a direct consequence of the material that Abbott chose not to cover. The chapters are as follows: the real numbers, sequences and series, basic topology on the reals, functional limits and continuity, the derivative, sequences and series of functions, the Riemann integral and additional topics, which include the generalized Riemann integral (a.k.a the gauge integral), metric spaces and the Baire category theorem, Fourier series and a construction of the reals from the rationals. All of these topics are done with respect to the real line, and there is no move toward generalizing the results to multiple dimensions.

I desperately want to see this book in general use, but for this to happen I think that it needs to cover sufficient material for a year long sequence. If Abbott were to include material on real analysis in n-dimensions (including vector valued functions), more information on metric spaces, and an introduction to function spaces, that should do it.

To summarize: if you're trying to learn the material presented in this book, buy it, but beware: the quality of the exposition of this book will spoil you and make you dissatisfied with other texts.

31 of 33 people found the following review helpful:

4.0 out of 5 stars In Case You Haven't Noticed..., February 21, 2001

By 

"another_anonymous" (Dallas, Texas United States) - See all my reviews

This review is from: Understanding Analysis (Hardcover)

The book is aimed at introductory students. The problems are interesting and often challenging (as they should be). Abbott spends some time explaining the topics and providing examples (and pictures). Each chapter ends with a summary containing a bit of the historical aspects of what was learned and some of the implications of the more important results, and each chapter begins with a discussion to pique interest in the material (the chapter on functional limits & continuity begins with Dirichilet & Thomae and the chapter on the basic topology of R begins with a construction of the Cantor set). At the end is a wonderful chapter on more advanced topics like the Generalized Riemann Integral and Metric Spaces & the Baire Category Theorem. Also, the causal dialogue in this book may make it reasonable for self study (the only prerequisite is a good understanding of single variable calc). I can't do this book justice with my review, you may want to check it out for yourself.