Mathematical Analysis: A Modern Approach to Advanced Calculus

"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.

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.

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 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.