This textbook is a completely revised, updated, and expanded English edition of the important Analyse fonctionnelle (1983). In addition, it contains a wealth of problems and exercises (with solutions) to guide the reader. Uniquely, this book presents in a coherent, concise and unified way the main results from functional analysis together with the main results from the theory of partial differential equations (PDEs). Although there are many books on functional analysis and many on PDEs, this is the first to cover both of these closely connected topics. Since the French book was first published, it has been translated into Spanish, Italian, Japanese, Korean, Romanian, Greek and Chinese. The English edition makes a welcome addition to this list.

Table of Contents

1. The Hahn–Banach Theorems. Introduction to the Theory of Conjugate Convex Functions

2. The Uniform Boundedness Principle and the Closed Graph Theorem

3. Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity

4. Lp Spaces

5. Hilbert Spaces

6. Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators

7. The Hille–Yosida Theorem

8. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension

9. Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions

10. Evolution Problems: The Heat Equation and the Wave Equation

11. Miscellaneous Complements

Review

From the reviews:

“Brezis has intelligently chosen several fundamental concepts of functional analysis, and has build the book around them and their applications. … for a newcomer who intends to become a user of functional analysis this book is an ideal place to start. In fact I would recommend this over any other source to any beginning graduate student. … the book also has all the basic tools for a beginer PDE researcher … . Its a bible for the field of research.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, October, 2013)

“This textbook has its origin in the French version Analyse fonctionnelle published in 1985, which has become a standard reference and was translated into several languages. … At the end of each chapter the reader will find comments with further information, references, and historic remarks. … In summary, the present textbook provides an excellent basis for a course on functional analysis plus a follow-up course on partial differential equations. It is well-written and I can wholeheartedly recommend it to both students and teachers.” (G. Teschl, Monatshefte für Mathematik, Vol. 165 (3-4), March, 2012)

“This book is a tour de force by the author, who is a master of modern nonlinear functional analysis and who has contributed extensively to the development of the theory of partial differential equations. … The writing is lively, the material is diverse and maintains a strong unity. … the book is a very useful contribution to the growing literature on this circle of ideas. I wholeheartedly recommend this book both as a textbook, as well as for independent study.” (Vicenţiu Rădulescu, Mathematical Reviews, Issue 2012 a)

“The book is the English translation of an 1983 book published in French: Analyse fonctionnelle :théorie et applications … . It has seen translations into numerous languages and the Springer edition was especially anticipated, as it announced a number of practice exercises following each chapter. I can honestly say that it was well worth the wait. … The text is a pleasure to read. … I wholeheartedly recommend this book as both a textbook as well as for independent study.” (Florin Catrina, The Mathematical Association of America, June, 2011)

About the author

Haim Brezis is a French mathematician, who mainly works in functional analysis and partial differential equations.