Real and Complex Analysis (Higher Mathematics Series) 3rd Edition

34 customer reviews
ISBN-13: 978-0070542341
ISBN-10: 0070542341
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Product Details

  • Series: Higher Mathematics Series
  • Hardcover: 483 pages
  • Publisher: McGraw-Hill Education; 3 edition (May 1, 1986)
  • Language: English
  • ISBN-10: 0070542341
  • ISBN-13: 978-0070542341
  • Product Dimensions: 6.6 x 1.1 x 9.5 inches
  • Shipping Weight: 1.7 pounds (View shipping rates and policies)
  • Average Customer Review: 4.7 out of 5 stars  See all reviews (34 customer reviews)
  • Amazon Best Sellers Rank: #184,351 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

121 of 123 people found the following review helpful By longhorn24 on June 3, 2003
Format: Hardcover
Rudin's Real and Complex Analysis is an excellent book for several reasons. Most importantly, it manages to encompass a whole range of mathematics in one reasonably-sized volume. Furthermore, its problems are not mere extensions of the proofs given in the text or trivial applications of the results- many of the results are alternate proofs to major theorems or different theorems not proved. With that in mind, this book is not appropriate for a course where the instructor wants students to merely understand the theorems well enough to develop applications- the structure of the book is far better suited for a more theoretical course.
For example, the construction of Lebesgue measure is considered one of the most important topics in graduate analysis courses. After this construction, more abstract measures are developed, and then one proves the Riesz Representation Theorem for positive functionals later.
Conversely, Rudin develops a few basic topological tools, such as Urysohn's Theorem and a finite partition of unity, to construct the Radon measure needed in a sweeping proof of Riesz's Theorem. From this, results about regularity follow clearly, and the construction of Lebesgue measure involves little more than a routine check of its invariance properties.
Another example of where Rudin takes a more theoretical approach to provide a more elegant, yet less intuitive proof, is the Lebesgue-Radon-Nikodym theorem. Other books generally introduce signed measures with several examples, and use this result, along with properties of measures to derive the proof. On the other hand, since the first half of the book contains an intermission on Hilbert Space, Rudin uses the completeless of L^2 and the Riesz Representation Theorem for a more sweeping proof.
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33 of 34 people found the following review helpful By A Customer on June 25, 1999
Format: Hardcover
The first part of this book is a very solid treatment of introductory graduate-level real analysis, covering measure theory, Banach and Hilbert spaces, and Fourier transforms. The second half, equally strong but often more innovative, is a detailed study of single-variable complex analysis, starting with the most basic properties of analytic functions and culminating with chapters on Hp spaces and holomorphic Fourier transforms. What makes this book unique is Rudin's use of 20th-century real analysis in his exposition of "classical" complex analysis; for example, he uses the Hahn-Banach and Riesz Representation theorems in his proof of Runge's theorem on approximation by rational functions. At times, the relationship circles back; for example, he combines work on zeroes of holomorphic functions with measure theory to prove a generalization of the Weierstrass approximation theorem which gives a simple necessary and sufficient condition for a subset S of the natural numbers to have the property that the span of {t^n:n in S} is dense in the space of continuous functions on the interval. All in all, in addition to being a very good standard textbook, Real and Complex Analysis is at times a fascinating journey through the relationships between the branches of analysis.
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30 of 31 people found the following review helpful By Marete on October 8, 2000
Format: Hardcover
This Book of Rudin, Like Principles, rewards perhaps above all else, persistence; a virtue that, if we are to believe some professional Mathematicians, is indispensable for the study of Mathematics.
Its true that it is terse and efficient. However, this "short-coming" is to me not a short-coming at all for the simple reason that Rudin makes up for it. How? The problems. Once you get through the proofs, a TON of challenging questions will be waiting at the other end to hammer out of you any illusions about you depth of understanding. In my opinion, this is the greatest strenghth of Rudin's book. STICK with the problems, attack them relentlessly and at the end of it all, you will have learned, a little perhaps, how to think for yourself in Analysis.
As regards the section on Complex Variables, I found it fruitful to read it while supplementing the problems with those of Ahlfors, which is more computational (E.g. Although Rudin discusses complex int., he scarcely provides any problems for this, and the same goes for expansion in Power Series).
Stick with the book, and soon it will be like a classic novel. (At least it is for me)
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36 of 41 people found the following review helpful By Juan David Gonzalez Cobas on August 15, 2001
Format: Hardcover
This text is a model of mathematical style. The usual Rudin stuff: concise and elegant proofs, great chanllenging exercises and that undefinable sense of quality -mathematical taste- pervading all the book.
The book covers the standard material on 'real variable' (measure theory') in a masterful and compact way; then it goes through the standard complex analysis to a level deeper than usual and showing in a very original way its intertwining with real variable. The final third of the book is devoted to more specialized topics.
Just a warning: the construction of Lebesgue measure is based on Riesz representation theorem, whose lengthy proof is imposed to the reader in chapter 2. It is really tough, and makes this chapter much harder to read than the rest of the book.
If you want to learn REAL mathematics, this is the book for you, you'll learn not only the subject matter, but a great style as well.
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