Real and Complex Analysis (McGraw-Hill International Editions: Mathematics Series)

This text is part of the Walter Rudin Student Series in Advanced Mathematics.

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.

In the real analysis section, Rudin covers advanced topics generally not covered in a first course on measure theory. The chapters on differentiation and Fourier analysis are key examples of this. Rudin uses maximal functions to develop the Lebesgue Point theorem and results from complex analysis, and provides an incredibly thorough proof of the change-of-variables theorem. The ninth chapter, on Fourier transforms, relies heavily on convolutions, which are developed as a product of Fubini's theorem. This, in turn, is used to prove Plancherel's theorem and the uniqueness of Fourier transforms as a character homomorphism.

The tenth chapter, on basic complex analysis, essentially covers an entire undergraduate course on the subject, with added results based on a solid knowledge of topology on the plane. Once a solid foundation on the topic is laid, Rudin can develop more advanced topics from Harmonic analysis using general results from real analysis like the Hahn-Banach theorem and the Lebesgue Point theorem (for Poisson integrals).

Most of the basic results from the power series perspective are covered in the text, but while the geometric view is examined, it is still done in a very analytic, formula-based way that does not allow the reader to gain too much intuition. Nonetheless, all the basic results are covered, and Rudin uses these to develop the main theorems, such as the Mittag-Leffler and Weierstrass theorems on meromorphic functions, and the Monodromy Theorem and a modular function used to prove Picard's Little Theorem.

As an introductory text, even for advanced students, Rudin should probably be accompanied by more descriptive texts to develop better intuition. In fact, I would recommend Folland's Real Analysis and Ahlfors' Complex Analysis for self-study, because the problems are easier and one can learn better through those. With a good instructor, though, Rudin's text is concise and elegant enough to be both useful and enjoyable. It is also a good test to see how well one REALLY knows the subject.


A start in math.   September 21, 2004
Palle E T Jorgensen (Iowa City, Iowa United States)
29 out of 31 found this review helpful

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 around the world.

The book is divided in the two main parts, real and complex analysis. But in addition, it contains a good amount of functional and harmonic analysis; and a little operator theory.

I loved it when I was a student, and since then I have taught from it many times. It has stood the test of time over almost three decades, and 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, and that the material is systematically built up in a way that is both effective and 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 exercises in Rudin's Real and Complex, I think we learn things that stay with us for life; you will be "marked for life!"

Review by Palle Jorgensen, September 2004.


Welcome to the self-service analysis center!   February 1, 2004
Farshid Arjomandi (California, USA)
14 out of 15 found this review helpful

This year we have been using the 1987's third edition of 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 not successful, 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. (I will inevitably be making a few comparisons between the two texts in the following.) One point to keep in mind though, is that Rudin developes the measure in a more formal axiomatic direction, instead of in 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 which is called a measure on (X, M) (The latter is the approach also taken in both H.L. Royden and Wheeden/Zygmund). The formal approach is not very intuitive and is less natural for a beginning graduate student who might not have developed some level of mathematical maturity yet.

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. Last but not least, it's worth noting that contrary to the common practice, Folland includes many end-of-chapter notes where he outlines some important historical aspects of the development of the topics, and also gives a few references for further study. For example, in the notes section at the end of the chapter on Lebesgue integration, he mentions --and briefly outlines-- the basics of the theory of "gauge integration" (aka Henstock-Kurzweil theory) which serves to construct a more powerful integral than that of the Lebesgue's. As another instance, having already defined and used "nets" within the chapter on topology, in the end-notes Folland also introduces "filters" and "ultrafilters". These are all machineries which have been developed to play the role of the metric space sequences in general (locally Hausdorff) topological spaces, but for some historical reasons, ultrafilters have nowadays taken a backseat to the nets (the older general topology books usually prove the Tychonoff theorem using ultrafilters). 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. Please note that the other books I have mentioned above do not discuss complex analysis, a subject which is also masterfully presented in Rudin. There are however a few other equally well-written complex analysis books to pick from, for example John B. Conway's classic from the Springer-Verlag graduate text series, or L.V. Ahlfors's wonderful monograph, to name just a couple.


The "Bible"   February 15, 2002
Jason Schorn (Spokane, WA)
6 out of 10 found this review helpful

This book is no frills, but the pay off is insight into not only a beautiful subject but the mind of one of the masters in the field. This book is worth every penny and should be read by anyone who is serious about Analysis. A big word of caution to prospective buyers... You must have considerable mathematical maturity inorder to benefit from this book. Rudin's book is not a fly by the seat of your pants, I just want to be published book. This is the real deal and if I could I would give this book 8 stars.


Best (math) book ever written   August 15, 2001
Juan David Gonzalez Cobas (Gijon, Asturias Spain)
22 out of 22 found this review helpful

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.