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Rating: 
Summary: The Cat's Meow
Review: As stated by prior reveiwers, this books does assume that the reader is Mathematically mature (a saying most young Mathematicians despise), in the sense that he/she must be able to follow the logical development of any given arguement, be able to 'see' where and how topics are related as well as fill in any blanks that may present themsevles in a given definition/proof. Apostol, as compared to Rudin, does a nice job of filling in these blanks by adequately providing all of the necessary details within a proof. This book will provide the willing student with a solid foundation in elementary analysis as well as the confidence to persue higher analysis. The only draw back to Apostols book, aside from cost, is that the constant Theorem - Proof - Theorem format can be overwhelming at times and cause some readers to cover material too quickly. Despite the book's cost I would highly recommend this book over "baby" Rudin (that is, Principles of Mathematical Analysis) since Rudin is notorious for not filling in the blanks within a given proof and instead provides seemingly 'slick proofs'.
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Summary: A cut above the rest...
Review: 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.
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Summary: Best all-around analysis text...
Review: 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.
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Summary: Complete, focused, and well-written.
Review: 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.
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Summary: He is the master,no doubt.
Review: This book is superior in mathematical rigor.Nearly every student have heard about his Calculus,but elegance of this text cannot be surpassed. The author did not matter understandability of the title but completely concentrated on style,elegance and rigor.I have read all the proofs in complete amazement,admiration.The book is very concise and precise,though it is not huge in amount of pages,it says a lot in a few words.Apostol proves every fact that he will need afterwards, and contrary to many analysis books,you cannot learn much about a single chapter,if you haven't read it all from the very beginning. His charming style will soon make you an Apostol fan. You will not only learn analysis but also manner of style and a complete rigor.Makes me remember the words, "Mathematics, rightly viewed, possesses not only truth, but supreme beauty--a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." Bertrand Russell
Rating: 
Summary: Understandable, but overpiced
Review: Well written, about as clear as Kosmala's book "Advanced Calculus, A friendly approach". Very readable, as readable as this material can get, ... If not for the price, I would have given it 5 stars.
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