75 of 76 people found the following review helpful
on 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.
54 of 55 people found the following review helpful
on February 18, 2007
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.
40 of 44 people found the following review helpful
on February 22, 2001
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.
18 of 20 people found the following review helpful
on March 18, 2007
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.
13 of 14 people found the following review helpful
on February 12, 2003
Understanding Analysis is an excellent textbook for anyone wanting to learn more about mathematics beyond the high school and calculus levels. It shows how mathematics is more than simply multiplying numbers or solving integrals. Elementary analysis rigorously builds the foundations of calculus starting from first principles. If you ever had trouble understanding limits, sequences, series, derivatives, or integrals and want to really learn them, I strongly recommend this book.
The book provides a lucid introduction to proof writing and non-computational mathematics best suited to students who have just completed calculus. In the author's own words, "The proofs in Understanding Analysis are written with the introductory student firmly in mind. Brevity and other stylistic concerns are postponed in favor of including a significant level of detail." When contrasted with many other mathematics books that are terse presentations of theorems, the textbook is remarkably readable, focusing on teaching material and developing students.
17 of 20 people found the following review helpful
on August 7, 2006
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.
27 of 34 people found the following review helpful
on August 16, 2003
Stephen Abbott is with no doubt a very talented writer in mathematics. The book is a fun to read because
of its style : each chapter starts describing a basic mathematical question that challenged the human mind
in history. This always makes you curious to read further to discover the great constructs made by the creative thinkers who solved these problems. Also, each chapter ends with adescription of related topics and some historical notes . I really like this style ....
However I did not give this book a five star rating for the following reasons :
-Some proofs contain gaps that are left as an exercise to the reader. Not all of these exercises are
staightforward however. It sometimes took me several hours to find a solution for these exercises ...
This is OK for real exercises, though it is no fun to have to spend this time filling up some basic proofs..
Sometimes I also had the impression that the hints were misleading. For example, I completed the proof on the double summation bit did not at all understood why we needed the hint prooved in exercise 2.8.4. Also when I tried to complete the
proof of the sequential criterium for nonuniform continuity (theorem 4.4.6), I did not see why we would need the hint
to take values 1/n for epsilon...
-Some explanations are missing, (maybe this will be solved in second edition). For instance :
a)please give a clear definition of what an interval is before using the name interval throughout the book.
b)In baires theorem, the author claims that every open set is either a finite or countable union of open intervals .... Please explain why ...
-Especially the 'more advanced' topics like baire 's theorem, fourier analysis, metric spaces, ... are rather presented as one big exercise. If you want to learn these topics, there are better books, providing you with much more information....
-This book only covers a limited range of topics. All the analysis is done for real variables in one dimension.
I think we need a broader scope, even for an introductionary course. My opinion is that modern analysis should start from the beginnig with n-dimensional metric spaces, conveying your mind to the beautifull theories of normed linear spaces and banach spaces.
-Since the book is targeted to the beginning student of abstract math, it would be good idea to include some pages
(appendix) on logic reasoning like second order predicate calculus, and some basic set theory ....
So, no five stars for this edition (maybe for a next edition ??)...
Giving this beautifull book less then four stars however would be unfair, since it definitely has it strengths : the things that are explained are explained very clear and the narrative style of the author always keeps the reader interested!!! Nobody could have done that better !!
13 of 15 people found the following review helpful
on December 15, 2002
As an engineering student trying to specialize in sigal processing, I found my mathematics background filled with holes. I embarked upon a journey to try to fill these holes and discovered what disservice my engineering education had done to me in terms of dealing with abstract notions like probability theory, etc. This book has been indispensible in re-engaging not only my anlytical skills but also continue on my path to a better understanding of the mathematics required to succeed professionally. Thank you, Prof. Abbott!
4 of 4 people found the following review helpful
on April 17, 2013
Format: HardcoverVerified Purchase
I got this book for an "Advanced Calculus 2" course which did integration and beginners measure theory. This book covers everything from functional limits, continuity, differentiation, integration, fourier series, weierstrass approximation theory, and so much more. I waited a long time to review this book and there aren't enough stars in the sky to convey how I would rate this book. It is an absolute gem and worth reading from cover to cover and there is so much to learn from every problem. Shroeder-bernstein theorem is walked through, cantor sets, convergence of functions (uniform and pointwise), completion of R, Baire category theorem, and so so so much more. The amazing thing is all of these really advanced concepts are conveyed through the language of that taught to people who are learning analysis for the first time and so going back and reading this, even as a graduate student, I always find a new deeper understanding in things that I learn now (functional analysis, measure theory, etc.). This book has given me so much through the many years I've had it, it makes no sense to hold off on reviewing it any more. If you are serious about being successful in analysis then it is never too late to pick this book up and start reading. It is worth it for the exposition the author gives alone, explaining concepts in words and plain english sentences instead of line after line of definition and proof. Pass this gem along to your friends too, this book will be remembered as a classic when we are all old and can't remember which way the triangle inequality works anymore
4 of 5 people found the following review helpful
on May 7, 2009
As a student of an introductory analysis course which used this book, I would like to share my perspective to all those who might find it helpful. No doubt, this is a great book which encouraged me to read back through all the project sections. Abbott manages to do two things which I have not seen in many other analysis books: firstly, he manages to be very thorough with every topic he presents, thus there are no confusing interludes, nor does he jump from easy topics to hard topics within the length of a page. Secondly, the problems are all very well thought out, I can guarantee that there is no "busy work" in this book. Each problem is worth looking at and so encourages students to test themselves outside of any assigned work.
As has been said before, this book is an investigation into the reasoning behind this reworking, which enables students to formulate correct definitions even when they have no prior experience with a concept. Abbott's book is a very effective approach to introductory analysis, encouraging true thought and understanding over busy work or memorization, which, as a student I am very thankful for. Again, read correctly, this book will create a lasting love for the subject, and will encourage any diligent student to think beyond the scope of the book.