Reviews (15)

Reasonable Introduction, Wealth of Exercises
A reasonable introduction to analysis on the real line. The authors repeatedly challenge the student by "leaving the proof as an exercise". But the most crucial results are proved in the text. In addition they provide a wealth of exercises and the coverage is enough to make the reader ready to approach the likes of Rudin (Principles of Mathematical Analysis). Finally, I agree with some of the other reviewers that it is not suitable for learning analysis in the absence of an instructor. But I don't agree with one reviewer who says that the authors leave too much for the student to figure out himself/herself. In my view, this is a good practice rather than bad, for otherwise students wouldn't bother to remember what they read before and couldn't solve more difficult problems by themselves.

Not for the first course in Real Analysis
I cannot recommend this book to beginners. Its hard to understand, and there are very few examples. The only good thing is maybe, that beginners get an intuitive feel of integration, and the underlying ideas, thats all.
Certainly not what an undergraduate needs. Apostol's book is much better.

Reviews (1)

Isbn: 0395959330
Sales Rank: 128948
Subjects:  1. Calculus    2. Mathematics    3. Science/Mathematics   

Reviews (8)

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

the missing star
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 beginnigwith 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 !!

Reviews (11)

My God? What a book!
Godfred Harold Hardy is one of the greatest mathematicians of the 20th century. A Course of Pure Mathematics is an introduction to Analysis, but he only starts Limits on page 110. Until there he talks about real numbers, Dedekind cuts and complex numbers as vectors. I have no words to describe his style: it seems that he is talking with you. If you are looking about a book of Calculus, stop here in buy it.

Best introduction to mathematical analysis
This book is simply beyond any rating whatsoever. Giving 5 stars is to undermine the value of this classic.
The first time I got this book, I was neither aware of it not of its author. I just picked it up randomly from school library. From the contents I figured it was a book on calculus. I immediately searched for the proof that "every continuous function is integrable." This was the first book I encountered which had a rigorous proof of this.
Then I began reading the chapters sequentially thinking that this seems to be a good book on calculus. The book went much beyond my expectation and it satisfied all my mathematical curiousities. All the mysteries of calculus were revelaed. Hardy demystified calculus in the first chapter itself by creating reals out ot rationals.
The Dedekind's construction of reals as presented in this book is the best I have seen. The properties of reals were not stated as axioms (common approach in books on analysis) but rather deduced from those of rationals.
The concepts of functions, limits, continuity, derivative etc. were explained in a prosaic style which has no parallel. This was also my first book on maths which had far more english words than mathematical symbols.
After finishing the entire book I was wondering who was this guy G. H. Hardy who has written such a masterpiece.
Only a few months later I came to know that he was one of the greatest British mathematicians of the century and was responsible for making our Indian Ramanujan famous. After that I read most of his books including "A Mathematician's Apology" and "An Introduction to Theory of Numbers"
Any persons who thinks maths is dull should just read few pages from this book and I bet his old beliefs would be shattered.