This is the fourth and final volume in the Princeton Lectures in Analysis, a series of textbooks that aim to present, in an integrated manner, the core areas of analysis. Beginning with the basic facts of functional analysis, this volume looks at Banach spaces, *Lp* spaces, and distribution theory, and highlights their roles in harmonic analysis. The authors then use the Baire category theorem to illustrate several points, including the existence of Besicovitch sets. The second half of the book introduces readers to other central topics in analysis, such as probability theory and Brownian motion, which culminates in the solution of Dirichlet’s problem. The concluding chapters explore several complex variables and oscillatory integrals in Fourier analysis, and illustrate applications to such diverse areas as nonlinear dispersion equations and the problem of counting lattice points. Throughout the book, the authors focus on key results in each area and stress the organic unity of the subject.

- A comprehensive and authoritative text that treats some of the main topics of modern analysis

- A look at basic functional analysis and its applications in harmonic analysis, probability theory, and several complex variables

- Key results in each area discussed in relation to other areas of mathematics

- Highlights the organic unity of large areas of analysis traditionally split into subfields

- Interesting exercises and problems illustrate ideas

- Clear proofs provided

**Elias M. Stein** is the Albert Baldwin Dod Professor of Mathematics at Princeton University. **Rami Shakarchi** received his PhD in mathematics from Princeton University. They are the coauthors of *Complex Analysis, Fourier Analysis*, and *Real Analysis* (all Princeton).

"*Functional Analysis* by Elias Stein and Rami Shakarchi is a fast-paced book on functional analysis and related topics. By page 60, you've had a decent course in functional analysis and you've got 360 pages left."—John D. Cook, *Endeavour*

"Characteristically, Stein and Shakarchi reward readers for hard work by making the material pay off."—*Choice*

"This excellent book ends with a proof of the continuity of the averaging operator and applications to the determination of remainder terms in asymptotic formulas for the counting function of lattice points. Reading this book is an enjoyable experience. The reviewer highly recommends it for students and professors interested in a clear exposition of these topics."—Stevan Pilipovic, *MathSciNet*

"This book is accessible for graduate students. Moreover, it plays the role of an instructional book in various branches of mathematical analysis, geometry, probability, and partial differential equations. In most mathematical centers one cannot expect that such lectures will be offered as a semester-long course to students, but both students and teachers have here an excellent guide for learning and teaching the topics presented in this volume. . . . Reading this book is an enjoyable experience. The reviewer highly recommends it for students and professors interested in a clear exposition of these topics."—Stevan Pilipovit, *Mathematical Reviews*

"This book introduces basic functional analysis, probability theory, and most importantly, aspects of modern analysis that have developed over the last half century. It is the first student-oriented textbook where all of these topics are brought together with lots of interesting exercises and problems. This is a valuable addition to the literature."—Gerald B. Folland, University of Washington