Princeton Lectures in Analysis - UC Davis Mathematics

1 Princeton Lecturesin Analysisby Elias M. Stein andRami Shakarchi A Book ReviewReviewed by Charles Fefferman and Robert Feffermanwith contributions from Paul Hagelstein,Nata a Pavlovi c, and Lillian PierceComments by Charles Fefferman and RobertFeffermanFor the last ten years, Eli Stein and Rami Shakarchihave undertaken a labor of love, producing a se-quence of intensive undergraduate Analysis coursesand an accompanying set of four books, called thePrinceton Lectures in Analysis . The individual titlesare: Fourier Analysis : An Introduction Complex Analysis Real Analysis : Measure Theory, Integration,and Hilbert Spacesand Functional Analysis : Introduction to FurtherTopics in four books are now available; all four booksbear the unmistakable imprint of Eli mathematician knows Stein as an analystof unsurpassed originality and impact.

2 A few dozenof us have had the privilege of writing a the-sis under his supervision. We know firsthand howEli conveys the essential unity of many seeminglydisparate ideas. To him, Analysis is always an or-ganic whole. Even more remarkably, his own enthusi-asm for the subject instills great optimism in all wholearn from him. First-rate math is right in front ofCharles Fefferman is the Herbert Jones University Pro-fessor of Mathematics at Princeton University. His emailaddress Fefferman is the Max Mason Distinguished ServiceProfessor of Mathematics at the University of Chicago.

3 Hisemail address , ready to be discovered. Eli turns us into researchmathematicians by encouraging us to become activeparticipants in the who have not enjoyed the privilege of study-ing under Stein or collaborating with him have nev-ertheless benefited greatly by studying from his clas-sic booksSingular Integrals,Introduction to FourierAnalysis in Euclidean Spaces(with Guido Weiss), andHarmonic Analysis . People far removed from Prince-ton have been able to read those books and then goon to do significant work. Eli succeeded in puttingon the printed page the kernel of what he conveyedas a teacher and research proceeding further, we should say a fewwords about Rami Shakarchi.

4 Shakarchi is a remark-able man in his own right. He is, among other things,a passionate and accomplished pilot. He is now anactive worker in the financial sector. As a graduatestudent, Rami volunteered to help Eli to plan thesequence of courses and to write the four collaboration was a great success. Eli andRami got along famously and communicated per-fectly. Rami earned his under one of us (CF)and took a demanding finance job in London. Evenwhen his firm, Lehman Brothers, declared bank-ruptcy, he stayed with the Stein project and saw itthrough to the Stein-Shakarchi books constitute an extra-ordinary achievement.

5 They are accessible (witha lot of work) to any math student who has hada rigorous one-variable calculus course and a lit-tle linear algebra, yet they cover an astonishingrange of material, including (in alphabetical order)Brownian motion, the Brunn-Minkowski inequality,May2012 Notices of the AMS641 Dirichlet s principle, Dirichlet s theorem on primesin an arithmetic progression, elliptic functions, theergodic theorem (maximal, mean, and pointwise),the gamma function, the Hardy-Littlewood maximaltheorem, Hausdorff dimension, the isoperimetricinequality, the Kakeya problem, the partition func-tion, the prime number theorem, representationsof positive integers as sums of two squares andas sums of four squares, the Riemann zeta func-tion, the Runge approximation theorem, Stirling sformula, theta functions, and a lot , these topics ap-pear, not as a zoo of isolatedwonders, but quite naturally aspart of a unified picture.

6 Forinstance, results from analyticnumber theory and probabilitygive the books a relevance thatreaches beyond Analysis to otherbranches of Mathematics material is explained withthe perfect clarity, focus on es-sentials, and stress upon theinterconnection of ideas that oneexpects from Eli Stein. Numerous challengingproblems encourage active audience working the problems, the student earns anunderstanding of key remarkablefeatureof the Stein-Shakarchibooks is that they take very seriously the historicaldevelopment of Analysis .

7 The authors boldly presentFourier Analysis before passing from the Riemannto the Lebesgue integral. This makes it possible forthe student to start doing interesting mathematicsright away, without getting bogged down in unmo-tivated technicalities. When the time comes to startdiscussing measure and integration, the subject isintroduced with the following words:Starting in about 1870, a revolutionary changein the conceptual framework of Analysis beganto take shape, one that ultimately led to a vasttransformation and generalization of the un-derstanding of such basic objects as functions,and such notions as continuity, differentiabil-ity and earlier view that relevant functionsin Analysis were given by formulas or other analytic expressions, that these functionswere by their nature continuous (or nearly so)

8 ,that by necessity such functions had deriva-tives at most points, and moreover thesewere integrable by the accepted methods ofintegration all of these ideas began to giveway under the weight of various examplesand problems that arose in the subject, whichcould not be ignored and required new con-cepts to be understood. Parallel with thesedevelopments came new insights that wereat once more geometric and more abstract: aclearer understanding of the nature of curves,their rectifiability and their extent; also thebeginnings of the theory of sets, starting withsubsets of the line, the plane, etc.

9 , and the measure that could be assigned to sees here the vast and penetrating scope ofStein s view of Analysis and how he is able to weaveit into his question remains, of course, how such anambitious set of books can work in practice. Toanswer this query, we asked Paul Hagelstein, Nata aPavlovi c, and Lillian Pierce to comment on theirown experiences. Paul taught the Stein-Shakarchicourses as a VIGRE postdoc. Nata a taught ComplexVariables la Stein-Shakarchi at Princeton . Lilliantook the Stein-Shakarchi courses as an undergradu-ate, then went on to serve as a TA for Nata a s are grateful to Paul, Nata a, and Lillian for theirthoughtful comments, which appear remains for us only to add that, in our view, theStein-Shakarchi books will be immensely valuablefor any undergraduate or graduate math student,for a wide audience of working mathematicians, andfor many science or engineering students and re-searchers with a mathematical bent.

10 Even those fewmathematicians who thoroughly know the contentsof all four books will find pleasure in the beauty oftheir unified presentation of a vast by Lillian B. PierceThe CoursesIn the fall of 1999 a murmur spread through thecommunity of math students at Princeton : a newcourse would be offered in the spring, the first offour intensive courses in Analysis , taught by Profes-sor Elias M. Stein. Professor Stein had already ac-quired status in our eyes, as it happened that mostof my cohort had taken his Introduction to SingleVariable Real Analysis course in our first semesterat Princeton .