Real Analysis: A Comprehensive Course in Analysis, Part 1

This review covers the entire five-volume set. Follow the links for the other volumes.

What a treat! Barry Simon, the IBM Professor of Mathematics and Theoretical Physics at Caltech, has launched a titanic set of graduate level analysis textbooks. They cover the full spectrum, and then some — says Simon in the Preface to the Series (available online and in a very welcome and useful Companion booklet): “This five-volume set began and, to some extent, remains a set of texts for a basic graduate analysis course. In part it reflects Caltech’s three-terms-per-year schedule and the actual courses I have taught in the past.” The five volumes are titled, respectively, Real Analysis (Part 1), Basic Complex Analysis and Advanced Complex Analysis (Parts 2A and 2B), Harmonic Analysis (Part 3), and Operator Theory (Part 4). Simon indicates that “[w]hile there is, of course, overlap between these books and other texts, there are some places where we differ, at least from many …” and the reader should really start with this list of six differences, presumably from “Green Rudin” or Royden (or am I dating myself?), both to get his bearings and in order to maximize the effect of Simon’s engaging prose: see pp.2–3 of this Preface. However, in the interest of time and space I’ll cite six corresponding highlights:

1. “… a unified approach to … real and complex analysis [enables us] to use notions like contour integrals as Stieltjes integrals …”
2. “… while discussing maximal functions, I present Garcia’s proof of the maximal (and so Birkhoff) ergodic theorem”
3. “These books are written to be keepers — the idea is that for many students, this may be the last analysis course they take … [so] I’ve included ‘bonus’ chapters and sections … [e.g.] for an instructor who can’t imagine a complex analysis course without a proof of the prime number theorem …”
4. “… I have long collected ‘best’ proofs …[e.g.] von Neumann’s proof of the Lebesgue decomposition and Radon-Nikodym theorems, the Hermite expansion treatment of Fourier transform, … and Newman’s proof of the prime number theorem …”
5. “… all chapters, except those on preliminaries, have a listing of ‘Big Notions and Theorems’ at their start …too many textbooks are monotone listings of theorems and proofs.”
6. “I’ve included copious ‘Notes and Historical Remarks’ at the end of each section. These notes illuminate and extend, and they (and the Problems) allow us to cover more material than would otherwise be possible …”

Simon goes on to talk a little about the problems in his book to the effect that there is “a wide variety … in line with a multiplicity of uses. The serious reader should at least skim them since there is often interesting supplementary material covered there.” There is also a much larger bibliography (good) and “a little more focus on special functions than is standard” (excellent).

I could go on at this point to discuss the contents of the five books in great detail, but permit me instead to refer to the respective Tables of Contents for this information. Let me just note that Real Analysis goes from metric space topology quickly to Hilbert space and Fourier series, then to a lot of material on measure theory, then to convexity and Banach spaces, and finally tempered distributions and the Fourier transform. Well, not quite “finally,” as there is a trio of bonus chapters on, respectively, probability, Hausdorff measure and dimension, and inductive limits and ordinary distributions.

Basic Complex Analysis (Part 2A) does Cauchy theory with a proper vengeance, but with a focus on homologous chains and what Simon calls “Dixon’s proof of the ultimate C[auchy] I[ntegral] T[heorem]” — cf. p. 140: a locally analytic function (understood in the usual way) integrates to zero over a chain homologous to zero. This is followed by the ultimate Cauchy Integral Formula, and, down the line, ultimate Laurent splitting, the ultimate argument principle, the ultimate residue theorem, and the ultimate Rouché theorem. Pretty cool.

Of course there’s much more, even before we get to Part 2B, including stuff on conformal mapping, linear fractional transformations, infinite products, elliptic functions, and a number of big classical theorems with very famous names. In the latter connection, Simon does mention early on that he subscribes to Arnol’d’s Principle, which asserts that “If a notion bears a personal name then that name is not the name of the discoverer (and the related Berry’s Principle: The Arnol’d principle is applicable to itself) …” However, I guess it’s safe to argue that Blaschke’s Products are Blaschke’s and Jensen’s formula is Jensen’s. In Part 2A we also encounter Paley and Wiener as well as Picard, and here, too, I guess credit has been given where it was due. Simon does not disagree (cf. e.g. p. 573). Finally, Part 2A ends with a “first glance at compact Riemann surfaces” — than which little more important can be thought, of course, at least in Mathematics.

And so it is that Riemann is all over the first part of Part 2B: after hitting Riemannian metrics and related things, Simon does a (to me very welcome) chapter on analytic number theory. Then he goes at ODEs, asymptotic methods (including the all-important method of stationary phase and the WKB approximation), univalent functions, and finally Nevanlinna Theory.

We are pretty far along in our travels at this point: we get Part 3, Harmonic Analysis, and Part 4, Operator Theory, and we encounter some rather sophisticated stuff here. For example, regarding harmonic analysis Simon does not shrink back from using martingales and microlocal analysis (even if it’s in the form of “a first glimpse”). We also find, e.g., a specific discussion of the uncertainty principle in this connection: not only is this very welcome, but it is in phase, so to speak, with Simon’s status as a premier expert on the mathematics of quantum theory. Part 3 is capped off with a bonus section of “more inequalities,” where we meet Hardy, Littlewood, Sobolev, Stein, Weiss, Calderon, and Zygmund. The last section deals with Tauberian theorems.

Speaking of Simon’s reputation as a quantum mechanic (consider, e.g. Reed-Simon, or Simon’s definitive Functional Integration and Quantum Physics), after all of this there’s even more: Part 4, aptly opening with Dorothy’s phrase, “Toto, we’re not in Kansas anymore,” from The Wizard of Oz, is a truly wonderful text on operator theory, covering (well, you pick the pun:) pretty much the entire spectrum or most of the good stuff (I’m tapping into the connection with another Caltech QM (and QED) figure). Let’s just say that once the basics are taken care of, compact operators appear in the third chapter, the spectral theorem (with bells on) appears in the fifth chapter, and chapter six is devoted to Banach algebras. This is especially good stuff, as Simon adds bonus sections on, among other things, the representation theory of L(ocally) C(ompact) A(belian) groups and Fourier analysis on LCAs: these things are pretty hard to find in the literature — as far as I know just about the only text explicitly devoted to this very important subject is the LMS monograph by Alain Robert. Simon has exquisite taste. Indeed, he closes section 6 with nothing less than the prime number theorem revisited, using Tauberian theorems. But arguably the most noteworthy fact about Part 3 is that Simon adds a pretty sizable bonus chapter devoted entirely to unbounded self-adjoint operators: yes, indeed, quantum mechanics vindicatus, and I guess this is all fitting and proper with Simon’s locale — at Caltech the mathematics department is housed in the Alfred P. Sloan Laboratory of Mathematics and Physics.

So there you have it: a brief tour of Barry Simon’s Comprehensive Course in Analysis. There is no need to belabor the point that this is a fabulous set of texts, and will be a smash hit in graduate programs with good taste and good students, willing to work hard, and ready for exposure to mathematical culture of a wonderful sort: in addition to being a fine mathematician and teacher of mathematics, Simon is something of a raconteur (in the best sense of the word) with a strong interest in history. The books are peppered with historical asides and human interest material, and this feature adds to their readability. Indeed, they are beautifully and clearly written and certainly make for a major contribution to the literature at the intended level and beyond. When I learned that the AMS (which is to be congratulated with this publication) was launching this series by Barry Simon, I of course expected a great deal — I was by no means disappointed: these books are terrific.

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Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.