Understanding Analysis | Stephen Abbott

Understanding Analysis | Stephen Abbott | A beautifully written introduction to real analysis in 1-dimension.

books:

Understanding Anal...

	Understanding Analysis Stephen Abbott Springer, 2002 - 272 pages average customer review:based on 13 reviews view larger image
	for more information click here

highly recommended

This book outlines an elementary, one-semester course which exposes students 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 of a course in real analysis should be 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.

Too Good To be True

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.

for more information click here

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

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.

for more information click here

Amazing book

When I started reading analysis, I was unfortunately asked to start with Rudin's book. But that book was totally inacessible probably because I come from engineering background. In search of more readable books, I started reading Apostol. It was readable but I wasn't enjoying the subject. Recently, I came across Abbott's book and was totally blow away. It is simply amazing. It makes analysis enjoyable and at the same time you learn a lot.

not bad, but has a lot of gaps left as an exercise

This is not a bad book. However, I dont 't understand how some reviewers claim that this book is ideal for the beginning student. Yes some things are explained very clear, however the reader should be aware that this book contains a lot of gaps left as an exercise for the reader. And I think that most beginning undergraduate students will not be able to complete these gaps without the help of a good teacher.
If you want a very good book for beginning abstract analysis, I would rather recommend "Real Mathematical Analysis" of "Charles Chapman Pugh". Pugh's book is excellent: it is very clear written, motivates the reader by providing the necessary background details that puts everything in the right context (like Abbott does also in an excellent way) but full proofs are present. In Pugh's book, occasionally some proofs contain little gaps left as exercise, but if they do, these gaps are more fair than the gaps in Abbott' s book, if you understand the material you should be able to solve them without a guiding teacher. And indeed Pugh also has very challenging exercises, but het does not mix them up with the theory, wich is more fair to the reader.
An additional plus, in contrast wih Abbott, is that Pugh's book contains more abstract material and is fully n-demensional.

for more information click here

not enough examples!!

I am a beginner in learning Analysis and I took Calculus I and II and got A's. However, I feel that there are not enough examples and proofs for theorems in this book. A lot of proofs are left for exercises. I cannot imagine and don't understand how beginning students can solve the exercise problems with so little examples!!????? I am taking this course right now and I am studying so hard---read books, notes and do whatever I can. However, it takes me more than 20 hours to do homework each week for this single course!! I must have a math tutor to help me solve homework problems. Each time, even with the tutor's help, it takes me a horrible long time to write the proofs down. For beginning students, without examples, how can they create concise proofs by themselves??? I am at least a good student and hard-working. I only recommend this book for complementary reading!! Right now, tonight, I have to stay up for this week's homework again! I am already studying all the time. What happened? Is this book for beginners?

for more information click here

reviews: page 1, 2, 3

products you might be interested in