## Description

Reed–Simon2 starts with “Mathematics has its roots in numerology, ge-ometry, and physics.” This puts into context the division of mathematics into algebra, geometry/topology, and analysis. There are, of course, other areas of mathematics, and a division between parts of mathematics can be artiﬁcial. But almost universally, we require our graduate students to take courses in these three areas.

This ﬁve-volume series 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’ve taught in the past. Much of the contents of Parts 1 and 2 (Part 2 is in two volumes, Part 2A and Part 2B) are common to virtually all such courses: point set topology, measure spaces, Hilbert and Banach spaces, distribution theory, and the Fourier transform, complex analysis including the Riemann mapping and Hadamard product theorems. Parts 3 and 4 are made up of material that you’ll ﬁnd in some, but not all, courses—on the one hand, Part 3 on maximal functions and Hp-spaces; on the other hand, Part 4 on the spectral theorem for bounded self-adjoint operators on a Hilbert space and det and trace, again for Hilbert space operators. Parts 3 and 4 reﬂect the two halves of the third term of Caltech’s course.

While there is, of course, overlap between these books and other texts, there are some places where we diﬀer, at least from many:

(a) By having a uniﬁed approach to both real and complex analysis, we are able to use notions like contour integrals as Stietljes integrals that cross the barrier.

(b) We include some topics that are not standard, although I am sur-prised they are not. For example, while discussing maximal functions, I present Garcia’s proof of the maximal (and so, Birkhoﬀ) ergodic the-orem.

(c) These books are written to be keepers—the idea is that, for many stu-dents, this may be the last analysis course they take, so I’ve tried to write in a way that these books will be useful as a reference. For this reason, I’ve included “bonus” chapters and sections—material that I do not expect to be included in the course. This has several advantages.

First, in a slightly longer course, the instructor has an option of extra topics to include. Second, there is some ﬂexibility—for an instructor who can’t imagine a complex analysis course without a proof of the prime number theorem, it is possible to replace all or part of the (non-bonus) chapter on elliptic functions with the last four sections of the bonus chapter on analytic number theory. Third, it is certainly possible to take all the material in, say, Part 2, to turn it into a two-term course. Most importantly, the bonus material is there for the reader to peruse long after the formal course is over.

(d) I have long collected “best” proofs and over the years learned a num-ber of ones that are not the standard textbook proofs. In this re-gard, modern technology has been a boon. Thanks to Google books and the Caltech library, I’ve been able to discover some proofs that I hadn’t learned before. Examples of things that I’m especially fond of are Bernstein polynomials to get the classical Weierstrass approxi-mation theorem, von Neumann’s proof of the Lebesgue decomposition and Radon–Nikodym theorems, the Hermite expansion treatment of Fourier transform, Landau’s proof of the Hadamard factorization theo-rem, Wielandt’s theorem on the functional equation for Γ(z), and New-man’s proof of the prime number theorem. Each of these appears in at least some monographs, but they are not nearly as widespread as they deserve to be.

(e) I’ve tried to distinguish between central results and interesting asides and to indicate when an interesting aside is going to come up again later. In particular, all chapters, except those on preliminaries, have a listing of “Big Notions and Theorems” at their start. I wish that this attempt to diﬀerentiate between the essential and the less essential didn’t make this book diﬀerent, but alas, too many texts are monotone listings of theorems and proofs.

(f) 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. The history is there to enliven the discussion and to emphasize to students that mathematicians are real people and that “may you live in interesting times” is truly a curse. Any discussion of the history of real analysis is depressing because of the number of lives ended by the Nazis. Any discussion of nineteenth-century mathematics makes one appreciate medical progress, contemplating Abel, Riemann, and Stieltjes. I feel knowing that Picard was Hermite’s son-in-law spices up the study of his theorem.

On the subject of history, there are three cautions. First, I am not a professional historian and almost none of the history discussed here is based on original sources. I have relied at times—horrors!—on information on the Internet. I have tried for accuracy but I’m sure there are errors, some that would make a real historian wince.

A second caution concerns looking at the history assuming the mathe-matics we now know. Especially when concepts are new, they may be poorly understood or viewed from a perspective quite diﬀerent from the one here. Looking at the wonderful history of nineteenth-century complex analysis by Bottazzini–Grey3 will illustrate this more clearly than these brief notes can.

The third caution concerns naming theorems. Here, the reader needs to bear in mind Arnol’d’s principle:4 If a notion bears a personal name, then that name is not the name of the discoverer (and the related Berry principle: The Arnol’d principle is applicable to itself ). To see the applica-bility of Berry’s principle, I note that in the wider world, Arnol’d’s principle is called “Stigler’s law of eponymy.” Stigler5 named this in 1980, pointing out it was really discovered by Merton. In 1972, Kennedy6 named Boyer’s law Mathematical formulas and theorems are usually not named after their original discoverers after Boyer’s book.7 Already in 1956, Newman8 quoted the early twentieth-century philosopher and logician A. N. Whitehead as saying: “Everything of importance has been said before by somebody who