Mathematical Analysis, Second Edition [Paperback]

Tom M. Apostol (Author)

Book Description

ISBN-10: 0201002884 | ISBN-13: 978-0201002881 | Publication Date: January 11, 1974 | Edition: 2nd

It provides a transition from elementary calculus to advanced courses in real and complex function theory and introduces the reader to some of the abstract thinking that pervades modern analysis.

Product Details

• Paperback: 492 pages
• Publisher: Addison Wesley; 2nd edition (January 11, 1974)
• Language: English
• ISBN-10: 0201002884
• ISBN-13: 978-0201002881
• Product Dimensions: 9.5 x 6.5 x 1.2 inches
• Shipping Weight: 1.6 pounds (View shipping rates and policies)
• Average Customer Review: 4.8 out of 5 stars  See all reviews (19 customer reviews)
• Amazon Best Sellers Rank: #479,442 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

121 of 126 people found the following review helpful:

5.0 out of 5 stars Excellent, May 27, 2001

By 

Dougabug "dougabug" (Orangevale, CA United States) - See all my reviews

This review is from: Mathematical Analysis, Second Edition (Paperback)

I stole...er, borrowed this book from my friend back in my Berkeley days. If I see him again, I'll give it back, and then ask if I can borrow it for another dozen years or so. It was dog eared when you lent it to me, Max, honest! What a great book. A great combo to learn introductory analysis from (advanced calculus to schools stuck in a 19th century time warp) is Rosenlicht's "Intro. to Analysis", Apostol's "Mathematical Analysis", and Rudin's "Principles of Mathematical Analysis". Rosenlicht is dirt cheap (one the few reasonably modern Dover books) and the combo is likely to be no more than a single elementary calculus book at today's inflated prices. At any rate, it's maybe 5-10% of the cost of a university course so it's a bargain, given that between the three you get just about any material that's likely to be presented in an undergraduate analysis course and then some. Bear in mind that the leap from introductory calculus to real analysis is a fairly mind blowing leap of both rigor and abstraction. You really would do yourself to acquire these books *before* you took the class, and preferrably study at least one of them over the summer prior.

Rosenlicht is short, a fast read. Concise, yet still rigorous. However, it's almost certain that after a first exposure, many logical subtlety's will have slipped passed you if you've never studied analysis. The relentless attention to detail in Apostol, and its definition-theorem-proof structure make it difficult to miss something without realizing it. However, Rosenlicht's more expository style lends itself to bedtime reading, where Apostol can become quickly very dry if you don't read it carefully and process every line. Apostol is not for the lazy reader, although in reality no book is, since the lazy reader will have gone away learning little, but fail to realize it. Rudin, you can save for the last, since it is in some sense the most mathematically elegant, especially once you already grasp the ideas. However, it is not as comprehensive as Apostol, and takes a certain mathematical maturity for granted, a maturity that working through and understanding Apostol helps build. The great thing about having all three books is that in addition to the better coverage (ie it's unlikely that an important concept like absolute continuity or a lipshitz condition has gone undefined), you get to see several equivalent definitions and presentation/proofs of the same ideas. And you'll need them, because, you don't know sh*t until you at least reached this level. Even then, you're still at the base of a huge mountain.

32 of 32 people found the following review helpful:

5.0 out of 5 stars One of the best I own..., March 10, 2005

By 

me "me" (here) - See all my reviews

This review is from: Mathematical Analysis, Second Edition (Paperback)

I own books on mathematical analysis by Browder (0387946144), Douglas S Bridges (0387982396
), Haaser Sullivan (0486665097), Pfaffenberger(0486421740), Dudley (0521007542),Abbot(0387950605) and Apostol.

All books cover abstract multivariable spaces, except Abbott who limits himself to the real line.
None of these books are perfect, but of all these books Apostol is the one I prefer for the following reasons :

1. The contents : I think a beginning analysis course should serve two aims :
a. teach basic techniques that can be used in other theoretical oriented courses like physics,economics,...
b. at the same time let the students discover the beauty of abstract and rigorous math.

In this context Apostol has reached the ideal mix between abstraction and usability. He covers practical topics , used as a basis in a lot of other courses, but he does this by making the needed level of abstraction in order to proof everything in a rigorous way.

Each book is self contained, though none of these books give a good introduction into basic mathematical logic. However an introduction to set theory is explained well in all books.
Dudley 's beautifull book is the most abstract but requires the highest level of mathematical maturity.

2 Layout : The books of Haaser Sullivan , Pfaffenberger cover excellent material in a very clear way but they are cheap Dover editions, putting as much text as possible on one page. Browder 's contents I like most (and contains really excellent explanations), but his layout is also very dense and not always comfortable to read. The layout of Apostol is the best of all these books, its pages are well filled, but the difficult proofs contain enough whitspace for a confortable read.

3.Completeness and rigor : Apostol and all these books, except Abbott and Douglas S Bridges, proof everything they mention (exceptionally, they leaf a proof as an exercise, but then the proof is relatively easy enough if you understand the material). This is an approach I like : present the complete theory and then (like all of them do) create challenging exercises seperate from the basic theory.
In contrast, the book of Douglas S Bridges represents all material as one big exercise.This is nice if you have anough time, but most of us do not have that much time,I am afraid. Also Abbott has a lot of difficult proofs left as an exercise to the reader. But at the same time, Abbott is the best in motivating the reader. Abbott often provides excellent background in order to motivate the reader and sharpen the readers mathematical intuition.

While Apostol is not best on all the criteria mentioned above, Apostol scores good on all off them and as a consequence he has the best total average. This being said, I must omit that reading Apostol requires patience. Yes his explanations are clear, but can be very terse (especially his examples). Though, in principle everything is explained without gaps. This book requires reading every word carefully and take the time to reflect, but maybe that is the only way to learn advanced math.

Finally a remark about the price, I bought this book in Europe where it is much cheaper (check amazon.co.uk)

So compared with the others this a very good book.

30 of 30 people found the following review helpful:

5.0 out of 5 stars Excellent Intermediate Real Analysis Text, February 12, 2007

By 

Richard T. McCoy - See all my reviews
(REAL NAME)   

This review is from: Mathematical Analysis, Second Edition (Paperback)

"Mathematical Analysis (2nd Ed.)," by Tom Apostol, does an excellent job of bridging the gap between standard introductory calculus texts and full-fledged treatments of topics in analysis. Apostol's book covers significantly more material than the gold standard of such texts, "Principles of Mathematical Analysis" by Rudin, and does so in a very different style. Where Rudin is brief and elegant, Apostol is thorough, detailed and friendly. Both Apostol's and Rudin's books have been around a long time, for very good reasons.

Unlike some intermediate texts, Apostol's book spends little time restating the particular results of elementary calculus (e.g., the derivative of sin x or x^n) in the new language of a more theoretical approach. Unlike Rudin and similar texts, Apostol *does* give detailed proofs, with thorough explanations. As a result of this approach, Apostol's book is not particularly well-suited to serve as a reference work for use by more advanced students or by professionals -- it is strictly a vehicle, and a very good vehicle indeed, for moving from elementary calculus to an introductory careful theoretical treatment of the material. Apostol does a particularly good job of presenting the "backbone ideas" of limits and continuity in a brief but very clear chapter (Chapter 4).

Apostol's problems are excellent and should be considered an important part of his presentation of the material. (This is one area in which Apostol perhaps surpasses Rudin, although MIT's online materials contain answers to so many of Rudin's problems that they now must be viewed as "worked-out examples!") Students find Apostol's tone, and the hints given in connection with the problems, to be helpful and engaging.

I suspect that the final few chapters of Apostol's book are used only rarely, due to the typical two-semester structure of real analysis courses (with a third semester being devoted to complex analysis). If true, this is a shame, because Apostol does a nice job of moving from a fairly standard treatment of the Lebesgue integral to Fourier integrals, multiple Riemann integrals and multiple Lebesgue integrals.

I should mention, as a minor point, that students can become confused, at least momentarily and episodically, by Apostol's parallel system of numbering (i) subsections and (ii) theorems and definitions. For example, the first line of page 166 reads "7.23 RIEMANN-STIELTJES INTEGRALS DEPENDING ON A PARAMETER" and the very next line reads (in italics) "Theorem 7.38 Let f be continuous at each point (x,y) of a rectangle . . . " Although the fonts differentiate these two parallel numbering systems, confusion can occur.