Customer Reviews

Mathematical Analysis, Second Edition

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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.

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.

"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.

I am currently studying from Apostol's book, completeing a year-long course with his treatment of the Lebesgue integral. While my experience with comperable analysis texts is not exhaustive, I am familiar with the more notable: "Baby" Rudin, Marsden,... So, I can confidently say that Apostol's text is among best covering the subject. His treatment is well modivated with examples, and his proofs, while not as not as "elegant" as those of Rudin, are surely more pedagogical in nature. Apostol has included a large amount of exercises that range througout the gamut of difficulty, and the material is peppered with a treatment of complex varaibles. Also, the readability is something to be attained by all authors of mathematics texts.

One drawback to the text is a too abstract approach to the Implict and Inverse Function Theorems. I found these to be the most challenging in the text, and I was forced to return to my copy of Stewart's Calculus text to re-acquiant myself with each concept. Also, at times Apostol falls into the pattern of Definition, Theorem, Definition, Theorem,..., but this seems to be only in the cases when ample preparation is needed to provide noteworthy examples; eg. Lebesgue integration.

So, in spite of the cost, I highly recommend this text for the study of real analysis (even for self study), although at [this price] there are bound to be others that have a higher value to cost ratio. Having completed the text (almost), I feel prepared to begin a more abstract study of analysis.

I own analysis texts by Apostol, Rudin, Bear, Fulks, Protter, and Kosmala. This one by Apostol gets my vote as the best all-around text on the subject. It's rigorous, elegant, readable, and has just the right amount of explanatory text. This would be my first choice as an undergraduate textbook, a self-study text, or as a supplemental reference to another text. I also recommend Bear for his elegance and witty style, and Kosmala for his thorough explanations. But if you are going to buy only one, make it this one.

If you're the type of person who likes crisp and clear proofs but don't want to have the proofs be as skinny as Rudin's then this is the perfect book. Apostol's writing style is not only accessible and clear but the organization of the text is excellent too. There are plenty of problems with a good mix of difficulty levels. He also throws in an example here and there to give you firm footing on some difficult topics. If I had to recommend one analysis text this would be it.

This book constitutes one of the best expositions of multidimensional calculus I've ever read. I'm writing my Ph.D. dissertation on hypercomplex analysis and it is still useful for me! The author shows his high skills by presenting a well-organized text with the right amount of abstraction and rigor, as required in modern teaching. The topics covered are more than enough for undergraduate courses and the exercises have the right level of difficulty.

I find this book suitable for most advanced calculus courses. It even includes some material on elementary complex analysis.

Its contents are: Real and Comlex Number Systems; Fundamental Notions of Set Theory; Elements of Point-Set Theory; The Concepts of Limit and Continuity; Differentiation of Real-Valued Functions; Differentiation of Functions of Several Variables; Applications of Partial Differentiation; Functions of Bounded Variation, Rectifiable Curves, and Connected Sets; Riemann-Stieltjes Integration Theory; Multiple Integrals and Line Integrals; Vector Analysis; Infinite Series and Products; Sequences of Functions; Improper Riemann-Stieltjes Integrals; Fourier Series and Fourier Integrals; Cauchy's Theorem and Calculus of Residues.

The lists of references for each chapter are somewhat short and could be updated, but they're O.K. The only one complaint I have is the price.

This is an outstanding textbook that is also one of the more comprehensive books as advanced calculus and introductory analysis texts go. It makes an excellent reference because it is quite comprehensive, covering a number of topics that don't make it into most introductory analysis books.

Other reviewers have said enough about the quality of this book; I just want to add a few comments. The second edition of this book is very different from the first--it cuts out much of the material on vector calculus, but it adds material on Lebesgue integration, which it presents without the use of measure theory.

Anyone who finds this text a little too difficult might want to look at the book "Advanced Calculus" by Taylor & Mann. It moves a little bit slower than this book, is a little bit less abstract, and covers less material. This book is in some ways a logical "next step" after that book. I strongly prefer this book to the "baby" Rudin, both as a learning text and a reference. This book is more detailed, and the dependency of the material is less strict--it's easier to open this book to a specific topic and understand it without having to cross-reference earlier theorems.

Apostol's book well deserves the label "classic" because of its conciseness, elegance, and completeness, among other features.

I think this book is one of the best introductions to multivariate calculus availble. It includes everything that should be covered in such a course, from the basic topology of n-dimensional euclidean space up to a thorough exposition of the calculus itself, and even some complex analysis.

Apostol is a long-lasting companion in the matematicians' libraries.

I don't know about the spanish guy but the other reviews are misleading. This is the best book at the level not for it's elegance, rigour (which every analysis book tends to have) or sophistication, but for being eminently readable. Apostol is known for being pedagogical without compromising standards. Though he doesn't have anything like spivak's or munkres talent for teaching, thiry years as calculus drill seargent at caltech have given him enough experience to write fairly high quality books. For true elegance and style one must go to rudin, deudonne and upwards. But tommy is a good place to start. One note, most of the problems are easy butthe hard ones tend to be pointless and irritating. Apostol teh number theorist comes through here.