Functional Analysis, Sobolev Spaces and Partial Differential Equations (Universitext)

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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.

Functional Analysis, Sobolev Spaces and Partial Differential Equations (Universitext) Review

I read the French edition of this book more than 20 years ago at the prompting of my adviser. It was a bit of a cultural shock that shaped my thinking. The author, one of the top experts in partial differential equations, has put together a book that will be extremely useful to any potential user of functional analysis.

Brezis has intelligently chosen several fundamental concepts of functional analysis, and has build the book around them and their applications. For this reason this book is not as comprehensive a source as, e.g., Dunford & Schwartz, Edwards or Yosida's classical texts, but 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. Until the current edition, I could recommend this text only to my students who could read French and I felt very frustrated that the others where denied access to this rewarding intellectual feast. (I am still surprised that it took so long for an English edition to appear since this book has been translated in about a dozen languages.)

One other very pleasing aspect of the book is the novelty, elegance, clarity and efficiency of most proofs. From this point of view this book has things to offer even to more seasoned mathematicians.

The fundamental concepts mentioned above are few and well spelled out: Hahn-Banach separation theorem, open mapping theorem, uniform boundedness principle, the closed range theorem, duality and compactness. I don't know many books that make such a vivid and convincing argument as Brezis's text that these are extremely powerful and versatile tools in knowledgeable hands. The more applied part of the book is a modern approach to partial differential equations. This book is still my favorite source for facts about Sobolev spaces. Brezis's approach to the basic evolution equations (heat and wave equations) takes the road less traveled via the Hille-Yosida theorem. His proofs are designed so they extend with minor conceptual modifications to non-linear situations, namely nonlinear groups generated by maximal monotone operators. This was a subject that underwent a spectacular development in the 70s and it is the subject in which Brezis first made a name for himself.

I think the English edition is even better that the French one due to the inclusion of many interesting exercises and additional comments. (Those circulated for a long time as notes available only to the Parisian students of Professor Brezis.)

A bit of warning. While this book is addressed to newcomers to the subject, it requires a bit of mathematical maturity and some background in measure theory and integration. What else is there to say about this book? Use it! It will open new doors for you.

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