Customer Reviews
A start in mathematical analysis. - By: Palle E T Jorgensen, 05 Dec 2004 
I am a fan of Rudin's books. This one "Real & Complex Analysis" has served as a standard textbookin the first graduate coursein analysis at lots of universitiesin the US, & around the world.
The book is dividedin the two main parts, real & complex analysis. Butin addition, it contains a good amount of functional & harmonic analysis; & a little operator theory.
I loved it when I was a student, & since then I have taught from it many times. It has stood the test of time over almost three decades, & it is still my favorite. I have to admit that it is not the favorite of everyone I know.
What I like is that it is concise, & that the material is systematically built upin a way that is both effective & exciting.
Some of the exercises are notoriously hard, but I think that is good: It simply means that they serve as work-projects when the students use the book. And this approach probably is more pedagogical as well.
After surviving some of the hard exercisesin Rudin's Real & Complex, I think we learn things that stay with us for life; you will be "marked for life!"
Review by Palle Jorgensen, December 2004.
Welcome to the self-service analysis center! - By: Farshid Arjomandi, 05 Feb 2004 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 & Integral, & the two seem to complement each other quite nicely. Rudin writesin a very user-friendly yet concise manner, & 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 analysisin a classroom setting (via Folland's book) was unsuccessful, as I found the expositionin Folland very dense & rigid, & 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 theoryin their upper division analysis classes. One point to keepin mind is that, Rudin developes the measurein 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, resultsin a "countably additive" set function that is called a measure on (X,M). (The latter is the approach takenin both H.L. Royden & Wheeden/Zygmund). The formal axiomatic approach is not very intuitive & 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 exploredin 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 & 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 discussedin 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 & L.V. Ahlfors's classics, to name just a couple.
An excellent text and reference - By: , 11 Jun 2001 Rudin's book is a classicin analysis, & deservedly so. Unlike many analysis texts, he maintains as much generality as possible, workingin locally compact, Hausdorff spaces & forgoing the reals when possible. He develops the standard measure & integration theory (Fubini's theorem, Lebesgue Dominated Convergence, etc)in these general cases. The proofs are basically the same (although Rudin likes slick proofs), & there is a massive gainin 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 & the Lebesgue integral. His treatment of complex analysis is also excellent & 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, & 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.
Excellent, often intriguing treatment of the subject - By: , 26 Jun 1999 The first part of this book is a very solid treatment of introductory graduate-level real analysis, covering measure theory, Banach & Hilbert spaces, & 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 & culminating with chapters on Hp spaces & holomorphic Fourier transforms. What makes this book unique is Rudin's use of 20th-century real analysisin his exposition of "classical" complex analysis; for example, he uses the Hahn-Banach & Riesz Representation theoremsin 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 & sufficient condition for a subset S of the natural numbers to have the property that the span of {t^n:nin S} is densein the space of continuous functions on the interval. Allin all,in addition to being a very good standard textbook, Real & Complex Analysis is at times a fascinating journey through the relationships between the branches of analysis.
Joining real analysis and complex analysis - By: , 26 Sep 1998 This book presents a large part of the classical real & complex analysis. The good thing is that the two diciplins is joint.
