Understanding Analysis (Undergraduate Texts in Mathematics) 1st ed. 2001. Corr. 2nd printing 2002 Edition

by Stephen Abbott (Author)

77 customer reviews

ISBN-13: 978-0387950600

ISBN-10: 0387950605

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Understanding Analysis (Undergraduate Texts in Mathematics)

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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.

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Product Details

Series: Undergraduate Texts in Mathematics
Hardcover: 260 pages
Publisher: Springer; 1st ed. 2001. Corr. 2nd printing 2002 edition (July 12, 2002)
Language: English
ISBN-10: 0387950605
ISBN-13: 978-0387950600
Product Dimensions: 6.1 x 0.7 x 9.2 inches
Shipping Weight: 1.2 pounds
Average Customer Review: 4.5 out of 5 stars See all reviews (77 customer reviews)
Amazon Best Sellers Rank: #433,889 in Books (See Top 100 in Books)
- #69 in Books > Science & Math > Mathematics > Pure Mathematics > Functional Analysis
- #275 in Books > Science & Math > Mathematics > Mathematical Analysis
- #397 in Books > Textbooks > Science & Mathematics > Mathematics > Calculus

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Top Customer Reviews

Too Good To be True

ByBenjamin Balsamon July 2, 2006

Format: HardcoverVerified Purchase

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.

44 comments 88 people found this helpful. Was this review helpful to you?YesNo

A beautifully written introduction to real analysis in 1-dimension.

ByJoshua E. Hillon February 18, 2007

Format: 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.

11 comment 61 people found this helpful. Was this review helpful to you?YesNo

In Case You Haven't Noticed...

ByAmazon Customeron February 21, 2001

Format: 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.

0Comment 42 people found this helpful. Was this review helpful to you?YesNo

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By far the best introductory analysis book, and one of the best introductory math books around. One of the few math texts books that is so well written it is easy for even a...Read more

Published 1 month ago by A. Chen

Great book that will challenge you!

Published 4 months ago by Sunny M.

Actually readable, while still conveying the core ideas of analysis

We've used this as a textbook for over a decade for first-semester real analysis, targeted at an audience that includes math majors, minors, and math/econ concentrators.Read more

Published 5 months ago by R. Manning

Published 5 months ago by NINI

Published 6 months ago by Xiaoxiong

Good even if this is you first exposure to analysis.

Fantastic book. Even if you have never had real analysis you can teach yourself from this book.

Published 6 months ago by Cogsbox6.28

This book is an excellent introduction to real analysis. It is worth at least two reads.

Published 7 months ago by Spencer H

Published 7 months ago by Robert

Unfinished proofs.

I have two complaints about this book. First, there are no answers to any problem. None at all. Second, proofs are not complete.Read more

Published 8 months ago by M. Rumore

Best analysis book ever!!

Published 8 months ago by DLR2006

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