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This is an advanced text for the one- or two-semester course in analysis taught primarily to math, science, computer science, and electrical engineering majors at the junior, senior or graduate level.
Book Description
This is an advanced text for the one- or two-semester course in analysis taught primarily to math, science, computer science, and electrical engineering majors at the junior, senior or graduate level. The basic techniques and theorems of analysis are presented in such a way that the intimate connections between its various branches are strongly emphasized. The traditionally separate subjects of 'real analysis' and 'complex analysis' are thus united in one volume. Some of the basic ideas from functional analysis are also included. This is the only book to take this unique approach. The third edition includes a new chapter on differentiation. Proofs of theorems presented in the book are concise and complete and many challenging exercises appear at the end of each chapter. The book is arranged so that each chapter builds upon the other, giving students a gradual understanding of the subject.
This text is part of the Walter Rudin Student Series in Advanced Mathematics. --This text refers to the Hardcover edition.
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Most Helpful Customer Reviews
11 of 11 people found the following review helpful
Format:Paperback
Rudin's book is a classic in analysis, and deservedly so. Unlike many analysis texts, he maintains as much generality as possible, working in locally compact, Hausdorff spaces and forgoing the reals when possible. He develops the standard measure and integration theory (Fubini's theorem, Lebesgue Dominated Convergence, etc) in these general cases. The proofs are basically the same (although Rudin likes slick proofs), and there is a massive gain in applicability. This book is inappropriate for a (US-level) undergraduate analysis course, but after a first introduction to the Riemann integral, I think this is an excellent way to explore measure theory and the Lebesgue integral. His treatment of complex analysis is also excellent and quite standard (Max Mod, Riemann Mapping, Mittag-Leffler, etc). In both parts, he proves nearly all his statements, but his proofs are quick enough to allow some detail-checking calisthenics for the interested reader. His exercises are generally excellent--they truly test understanding of the material, and even present some variant proofs (like an alternate proof of Riemann Mapping Theorem). At the very least, this is an excellent reference, but it also makes a very good text if you are ready for it.
9 of 9 people found the following review helpful
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.
6 of 6 people found the following review helpful
Format:Hardcover
This year we have been using Walter Rudin's treatise as the main text for a standard first-year graduate sequence on real analysis, backed up by Wheeden/Zygmund's book on Measure and Integral, and the two seem to complement each other quite nicely. Rudin writes in a very user-friendly yet concise manner, and at the same time he masterfully manages to maintain the high level of formality required from a graduate mathematics text. To be totally honest, a few years ago my very first attempt at learning graduate-level real analysis in a classroom setting (via Folland's book) was unsuccessful, as I found the exposition in Folland very dense and rigid, and the homework problems too difficult to do. Rudin's book however, is a lot more accessible for the beginning graduate students who may not have had any more than some basic exposure to measure theory in their upper division analysis classes. One point to keep in mind is that, Rudin developes the measure in the more formal axiomatic way, instead of the more concrete (constructive) approach. In the constructive approach, one first introduces the "subadditive" outer measure as a set function which is defined on the power set P(X) of a nonempty set X. One then proceeds by showing that the restriction of the domain of the outer measure to a smaller class of subsets of X (a sigma algebra M), obtained via applying the Caratheodory's criterion, results in a "countably additive" set function that is called a measure on (X,M). (The latter is the approach taken in both H.L. Royden and Wheeden/Zygmund). The formal axiomatic approach is not very intuitive and is less natural for the readers who have not yet developed a certain level of mathematical maturity. Also, Rudin does not discuss some of the more advanced or interdisciplinary topics such as distribution theory (Sobolev spaces, weak derivatives, etc.), or applications of measure theory to the probability theory, both explored in the book by Folland. (Please also note that contrary to the common practice, Folland gives many end-of-chapter notes outlining the historical development of the topics, as well as a good few references and suggestions for further study). All said, I can recommend taking up Royden as your very first approach to measure theory, then based on how well you think you have learned the first course, move on to either Rudin or Folland for a more advanced treatment. Rudin also does a great job on the complex analysis part, a subject not discussed in the other books mentioned above. There are however a few other equally well-written complex analysis books to pick from, for example try John B. Conway's and L.V. Ahlfors's classics, to name just a couple.
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Most Recent Customer Reviews
A start in mathematical analysis.
I am a fan of Rudin's books. This one "Real and Complex Analysis" has served as a standard textbook in the first graduate course in analysis at lots of universities in the US, and... Read more
Published on 4 Dec 2004 by Palle E T Jorgensen
Joining real analysis and complex analysis
This book presents a large part of the classical real and complex analysis. The good thing is that the two diciplins is joint.
Published on 26 Sep 1998
An Excellent Read
A terrific book for all Mathematicians, Rudin's books sometimes feel like they have plot, if you know what I mean. Maybe thats because often his proofs rely on trickery. Read more
Published on 3 Sep 1998
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