Princeton Lectures in Analysis by Elias M. Stein and Rami Shakarchi—A Book Review Reviewed by Charles Fefferman and Robert Fefferman with contributions from Paul Hagelstein, Nataša Pavlovi´c, and Lillian Pierce Comments by Charles Fefferman and Robert us, ready to be discovered. Eli turns us into research Fefferman mathematicians by encouraging us to become active For the last ten years, Eli Stein and Rami Shakarchi participants in the process. have undertaken a labor of love, producing a se- Many who have not enjoyed the privilege of study- quence of intensive undergraduate analysis courses ing under Stein or collaborating with him have nev- and an accompanying set of four books, called the ertheless benefited greatly by studying from his clas- Princeton Lectures in Analysis. The individual titles sic books Singular Integrals, Introduction to Fourier are: Analysis in Euclidean Spaces (with Guido Weiss), and • Fourier Analysis: An Introduction Harmonic Analysis. People far removed from Prince- • Complex Analysis ton have been able to read those books and then go • Real Analysis: Measure Theory, Integration, on to do significant work. Eli succeeded in putting and Hilbert Spaces on the printed page the kernel of what he conveyed and as a teacher and research collaborator. • Functional Analysis: Introduction to Further Before proceeding further, we should say a few Topics in Analysis. words about Rami Shakarchi. Shakarchi is a remark- All four books are now available; all four books able man in his own right. He is, among other things, bear the unmistakable imprint of Eli Stein. a passionate and accomplished pilot. He is now an Every mathematician knows Stein as an analyst active worker in the financial sector. As a graduate of unsurpassed originality and impact. A few dozen student, Rami volunteered to help Eli to plan the of us have had the privilege of writing a Ph.D. the- sequence of courses and to write the four books. sis under his supervision. We know firsthand how The collaboration was a great success. Eli and Eli conveys the essential unity of many seemingly Rami got along famously and communicated per- disparate ideas. To him, analysis is always an or- fectly. Rami earned his Ph.D. under one of us (CF) ganic whole. Even more remarkably, his own enthusi- and took a demanding finance job in London. Even asm for the subject instills great optimism in all who when his firm, Lehman Brothers, declared bank- learn from him. First-rate math is right in front of ruptcy, he stayed with the Stein project and saw it through to the end. Charles Fefferman is the Herbert Jones University Pro- The Stein-Shakarchi books constitute an extra- fessor of Mathematics at Princeton University. His email ordinary achievement. They are accessible (with address is [email protected]. a lot of work) to any math student who has had Robert Fefferman is the Max Mason Distinguished Service a rigorous one-variable calculus course and a lit- Professor of Mathematics at the University of Chicago. His tle linear algebra, yet they cover an astonishing email address is [email protected]. range of material, including (in alphabetical order) DOI: http://dx.doi.org/10.1090/noti832 Brownian motion, the Brunn-Minkowski inequality, May 2012 Notices of the AMS 641 Dirichlet’s principle, Dirichlet’s theorem on primes at once more geometric and more abstract: a in an arithmetic progression, elliptic functions, the clearer understanding of the nature of curves, ergodic theorem (maximal, mean, and pointwise), their rectifiability and their extent; also the the gamma function, the Hardy-Littlewood maximal beginnings of the theory of sets, starting with theorem, Hausdorff dimension, the isoperimetric subsets of the line, the plane, etc., and the inequality, the Kakeya problem, the partition func- “measure” that could be assigned to each. tion, the prime number theorem, representations One sees here the vast and penetrating scope of of positive integers as sums of two squares and Stein’s view of analysis and how he is able to weave as sums of four squares, the Riemann zeta func- it into his teaching. tion, the Runge approximation theorem, Stirling’s The question remains, of course, how such an formula, theta functions, and a lot more. ambitious set of books can work in practice. To Moreover, these topics ap- answer this query, we asked Paul Hagelstein, Nataša pear, not as a zoo of isolated Pavlovi´c, and Lillian Pierce to comment on their wonders, but quite naturally as own experiences. Paul taught the Stein-Shakarchi part of a unified picture. For courses as a VIGRE postdoc. Nataša taught Complex instance, results from analytic Variables à la Stein-Shakarchi at Princeton. Lillian number theory and probability took the Stein-Shakarchi courses as an undergradu- give the books a relevance that ate, then went on to serve as a TA for Nataša’s class. reaches beyond analysis to other We are grateful to Paul, Nataša, and Lillian for their branches of mathematics and thoughtful comments, which appear below. science. It remains for us only to add that, in our view, the The material is explained with Stein-Shakarchi books will be immensely valuable the perfect clarity, focus on es- for any undergraduate or graduate math student, sentials, and stress upon the for a wide audience of working mathematicians, and interconnection of ideas that one for many science or engineering students and re- expects from Eli Stein. Numerous challenging searchers with a mathematical bent. Even those few problems encourage active audience participation. mathematicians who thoroughly know the contents By working the problems, the student earns an of all four books will find pleasure in the beauty of understanding of key ideas. their unified presentation of a vast subject. Another remarkable feature of the Stein-Shakarchi books is that they take very seriously the historical Comments by Lillian B. Pierce development of analysis. The authors boldly present The Courses Fourier analysis before passing from the Riemann to the Lebesgue integral. This makes it possible for In the fall of 1999 a murmur spread through the the student to start doing interesting mathematics community of math students at Princeton: a new right away, without getting bogged down in unmo- course would be offered in the spring, the first of tivated technicalities. When the time comes to start four intensive courses in analysis, taught by Profes- discussing measure and integration, the subject is sor Elias M. Stein. Professor Stein had already ac- introduced with the following words: quired status in our eyes, as it happened that most of my cohort had taken his Introduction to Single Starting in about 1870, a revolutionary change Variable Real Analysis course in our first semester in the conceptual framework of analysis began at Princeton. So as the rumor of the new course rum- to take shape, one that ultimately led to a vast bled through our ranks, it was clear that we’d all sign transformation and generalization of the un- up for it. It was also clear from the first lectures that derstanding of such basic objects as functions, we were in for something unlike anything we’d ever and such notions as continuity, differentiabil- experienced before. ity and integrability. At the beginning of each of the four courses, Pro- The earlier view that relevant functions fessor Stein started by saying that he would work in analysis were given by formulas or other very hard on the course and that the course would in “analytic” expressions, that these functions turn require us to work very hard. Then he expressed were by their nature continuous (or nearly so), his gratitude, ahead of time, for all our ensuing hard that by necessity such functions had deriva- work. This was a striking attitude of enfranchise- tives at most points, and moreover these ment, which I certainly never encountered in any were integrable by the accepted methods of other course. integration—all of these ideas began to give Indeed, we did have to work hard. Of course, way under the weight of various examples substantial courses in mathematics are often de- and problems that arose in the subject, which scribed as “hard”, and this superficial descriptor could not be ignored and required new con- can conceal a multitude of sins or blessings. Are cepts to be understood. Parallel with these the lectures simply unclear, the assigned problems developments came new insights that were ill chosen? Or are the lectures deeply insightful, 642 Notices of the AMS Volume 59, Number 5 the assigned problems a methodology for building of individual mathematicians (while avoiding a a versatile toolbox? In this analysis sequence, the digression into blow-by-blow timelines). lectures were fluid and painstakingly prepared; they Looking back, I realize that we students did not possessed Stein’s trademark of logical progressions feel that we were merely inputting stale mathemat- of big ideas blended with clarifying close-up exam- ics into our heads; the beautiful presentations in the inations. In particular, he was careful to lift each lectures and the intense work on the problem sets concept up from first principles, so that we could made us feel like we were part of a creative force, start from our existing knowledge (as second-year (re)discovering mathematics. I am sure we had this undergraduates) and follow him deeply into the sense of creation partly because material. It was as though he was successfully accel- the courses were being devel- erating a car full of passengers, having first checked oped into books, but I think that all our seatbelts were fastened! that it also came from the re- The problem sets were memorably substantial spect given to the work we were (one group of students estimated that they some- doing, conveyed by the high stan- times spent about thirty hours per week on the dards and high expectations of courses).