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Title: Understanding Analysis by Stephen Abbott ISBN: 0-387-95060-5 Publisher: Springer-Verlag Pub. Date: 12 January, 2001 Format: Hardcover Volumes: 1 List Price(USD): $49.95 |
Average Customer Rating: 4.5 (8 reviews)
Rating: 3
Summary: beware, no solutions
Comment: Good book, but no solutions to any exercises. Usually Springer books do have some solutions, so it was a bit disappointing.
Rating: 4
Summary: the missing star
Comment: 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: 5
Summary: A Joy to Read
Comment: 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.
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Title: Counterexamples in Analysis by Bernard R. Gelbaum, John M. H. Olmsted ISBN: 0486428753 Publisher: Dover Publications Pub. Date: 01 June, 2003 List Price(USD): $14.95 |
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Title: Introductory Real Analysis by A. N. Kolmogorov, S. V. Fomin ISBN: 0486612260 Publisher: Dover Publications Pub. Date: 01 June, 1975 List Price(USD): $15.95 |
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Title: Yet Another Introduction to Analysis by Victor Bryant ISBN: 052138835X Publisher: Cambridge University Press Pub. Date: 01 November, 1990 List Price(USD): $33.00 |
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Title: Problems and Solutions for Undergraduate Analysis by Rami Shakarchi, Serge Lang ISBN: 0387982353 Publisher: Springer-Verlag Pub. Date: 01 January, 1998 List Price(USD): $49.95 |
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Title: A Primer of Real Functions by Ralph Philip Boas, Harold P. Boas, Ralph P. Boas ISBN: 088385029X Publisher: Mathematical Assn of America Pub. Date: 01 October, 1996 List Price(USD): $41.95 |
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