Sunday, 17 May 2020

Walter Rudin: Real and Complex Analysis

[Reviewed by 
Allen Stenger
, on 
 in MAA Reviews]
This book is full of interesting things, mostly proofs. The chapter on Banach algebras is a gem; this subject combines algebra, analysis, and topology, and the exposition shows clearly how the three areas work together. Walter Rudin (1921–2010) wrote the book in 1966 to show that real and complex analysis should be studied together rather than as two subjects, and to give a a modern treatment. Fifty years later it is still modern.
The first third of the book is devoted to measure and integration. The presentation is based on measures on abstract spaces with σ-algebras. It includes brief introductions to Hilbert space and Banach spaces, with material that will be used in the complex-variables proofs later. This beginning section is the only part of the book that deals with spaces more general than the real line and the complex plane, however it’s not any harder than it would be if we stuck to the real line. This includes a chapter on differentiation (of measures) and a chapter on product spaces (i.e., the Fubini theorem). The rest of the book is about analysis on the complex plane. It starts with a short chapter on Fourier transforms, then presents a course in complex variables that is traditional in terms of the theorems proved, but has very slick proofs using what has gone before. The traditional part ends with the little Picard theorem. The last quarter of the book consists of several short chapters on advanced topics in complex analysis; these include Hp spaces, Banach algebras, holomorphic Fourier transforms, and a characterization of functions that are the uniform limit of polynomials (Mergelyan’s theorem).
The approach is not very concrete; there are very few worked examples (many of the exercises do deal with specific functions). The book does not have the detailed chapters that we are used to on evaluating series and integrals and on special functions. But it is also not very abstract; it truly is mostly complex analysis, not general spaces. The proofs are informed by the more general viewpoint, and there is a strong functional-analysis flavor. For example, much use is made of the Hahn-Banach Theorem and some use of the Urysohn lemma and Tietze extension theorem.
The book has been widely criticized for lacking motivation, and this criticism is accurate. You don’t absolutely need a lot of background to read the book, but it is a collection of beautiful proofs without much context. For example, the discussion of spectra comes out of nowhere and is very mysterious unless you are well-acquainted with linear algebra and eigenvalues.
The book was aimed at first-year graduates and has been used successfully in many first-year graduate courses, and I think that is still about the right level for it. Undergraduates interested in the subject matter would be better served, before they tackle this book, by a more traditional complex analysis book such as Bak & Newman’s Complex Analysis or Ahlfors’s more advanced Complex Analysis, and by one of the many good introductions to Lebesgue integration (I like Boas’s A Primer of Real Functions).

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis

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