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Tom M. Apostol "Mathematical Analysis, 2 Ed
Addison Wesley Publishing Company | 1974-01 | ISBN: 0201002884 | 492 pages | Djvu | 10 MB
Reader's review:
Summary: Excellent Intermediate Real Analysis Text
Rating: 5
"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.
