Understanding Analysis

Rating:
Summary: Very Readable
Review: 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!

Rating:
Summary: The best book on Analysis I've ever read
Review: 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?

Rating:
Summary: The best book on Analysis I've ever read
Review: 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?

Rating:
Summary: beware, no solutions
Review: Good book, but no solutions to any exercises. Usually Springer books do have some solutions, so it was a bit disappointing.

Rating:
Summary: the missing star
Review: 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 !!

Rating:
Summary: In Case You Haven't Noticed...
Review: 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.

Rating:
Summary: beware, no solutions
Review: 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.

Rating:
Summary: A Joy to Read
Review: 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.

Rating:
Summary: excellent introduction to mathematics and analysis
Review: 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.

Rating:
Summary: Elementary, my dear Watson.
Review: Upon completion of a multi-semester Calculus course, the encounter with the textbooks for the next level of mathematics comes as a rude awakening to many students. While the books in the Rudin et al. category give a similarly structured, well organized presentation of the field of analysis, they are all based on the assumption that a talented and patient tutor will be there and able to fill in the gaps and to explain the whys and hows. This on the one hand is the source of a lot of frustration, but also fails to open students' eyes to the beauty that mathematics at "the next level" has to offer.

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.