Understanding Analysis

While I was familiar with the contents of this book, I am deeply impressed by Abbott's presentation of elementary analysis. In many respects this book is the next chapter after Spivak's Calculus, both in content and style. From his explanation and exploration of Euclid's proof on the infinity of the number of primes -check out the sample pages that Amazon provides- at the very beginning of the book, until the last page, Abbott reduces the material to its essentials and explains them with the utmost clarity. In this process the author does not "dumb down" for an instant, since elementary does not imply simple, when it comes to the subject matter. Yet Abbot's clarity in explaining the problems and providing proofs and explanations results in a book that delivers on its title.

Both for those who need guidance in studying elementary analysis and equally for their Profs/TAs, studying this inspiring book will be an eye opening experience.


Too Good To be True   July 2, 2006
Benjamin Balsam
18 out of 18 found this review helpful

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.


A beautifully written introduction to real analysis in 1-dimension.   February 18, 2007
Joshua E. Hill (San Luis Obispo, CA USA)
9 out of 9 found this review helpful

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.


The best book on Analysis I've ever read   August 19, 2001
10 out of 14 found this review helpful

Finally, someone took the time to write a book on Real Analysis that connects the topics in a way that it makes sense. I really hope to see more books by Abbott. How about a book on Topology, Steve?


A Joy to Read   March 26, 2003
Erik Lundberg (Gainesville, FL United States)
7 out of 12 found this review helpful

This is my first analysis book. So, I have no basis for camparing it as an analysis book. But, as a math book, it is honestly the most readable and enjoyable I have ever read.