Functional Analysis, Sobolev Spaces and Partial Differential Equations

By: Haim Brezis

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Paperback | 10 November 2010 | Edition Number 2011

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616 Pages

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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.

Industry Reviews

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 f¼r 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." (VicenÅ£iu RÄdulescu, Mathematical Reviews, Issue 2012 a)

"The book is the English translation of an 1983 book published in French: Analyse fonctionnelle :th©orie 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)

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Differential Calculus & Equations

Preface	p. vii
The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions	p. 1
The Analytic Form of the Hahn-Banach Theorem: Extension of Linear Functional	p. 1
The Geometric Forms of the Hahn-Banach Theorem: Separation of Convex Sets	p. 4
The Bidual E^**. Orthogonality Relations	p. 8
A Quick Introduction to the Theory of Conjugate Convex Functions	p. 10
	p. 17
	p. 19
The Uniform Boundedness Principle and the Closed Graph Theorem	p. 31
The Baire Category Theorem	p. 31
The Uniform Boundedness Principle	p. 32
The Open Mapping Theorem and the Closed Graph Theorem	p. 34
Complementary Subspaces. Right and Left Invertibility of Linear Operators	p. 37
Orthogonality Revisited	p. 40
An Introduction to Unbounded Linear Operators. Definition of the Adjoint	p. 43
A Characterization of Operators with Closed Range. A Characterization of Surjective Operators	p. 46
	p. 48
	p. 49
Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity	p. 55
The Coarsest Topology for Which a Collection of Maps Becomes Continuous	p. 55
Definition and Elementary Properties of the Weak Topology ¿(E, E^*)	p. 57
Weak Topology, Convex Sets, and Linear Operators	p. 60
The Weak^* Topology ¿(E^* E)	p. 62
Reflexive Spaces	p. 67
Separable Spaces	p. 72
Uniformly Convex Spaces	p. 76
	p. 78
	p. 79
L^p Spaces	p. 89
Some Results about Integration That Everyone Must Know	p. 90
Definition and Elementary Properties of L^p Spaces	p. 91
Reflexivity. Separability. Dual of L^p	p. 95
Convolution and regularization	p. 104
Criterion for Strong Compactness in L^p	p. 111
	p. 114
	p. 118
Hilbert Spaces	p. 131
Definitions and Elementary Properties. Projection onto a Closed Convex Set	p. 131
The Dual Space of a Hilbert Space	p. 135
The Theorems of Stampacchia and Lax-Milgram	p. 138
Hilbert Sums. Orthonormal Bases	p. 141
	p. 144
	p. 146
Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators	p. 157
Definitions. Elementary Properties. Adjoint	p. 157
The Riesz-Fredholm Theory	p. 159
The Spectrum of a Compact Operator	p. 162
Spectral Decomposition of Self-Adjoint Compact Operators	p. 165
	p. 168
	p. 170
The Hille-Yosida Theorem	p. 181
Definition and Elementary Properties of Maximal Monotone Operators	p. 181
Solution of the Evolution Problem du/dt + Au = 0 on (0, +∞), u(0) = u₀. Existence and uniqueness	p. 184
Regularity	p. 191
The Self-Adjoint Case	p. 193
	p. 197
Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension	p. 201
Motivation	p. 201
The Sobolev Space W^1.p (I)	p. 202
The Space &$$$;	p. 217
Some Examples of Boundary Value Problems	p. 220
The Maximum Principle	p. 229
Eigenfunctions and Spectral Decomposition	p. 231
	p. 233
	p. 235
Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions	p. 263
Definition and Elementary Properties of the Sobolev Spaces W^1.p (¿)	p. 263
Extension Operators	p. 272
Sobolev Inequalities	p. 278
The Space &$$$; (¿)	p. 287
Variational Formulation of Some Boundary Value Problems	p. 291
Regularity of Weak Solutions	p. 298
The Maximum Principle	p. 307
Eigenfunctions and Spectral Decomposition	p. 311
	p. 312
Evolution Problems: The Heat Equation and the Wave Equation	p. 325
The Heat Equation: Existence, Uniqueness, and Regularity	p. 325
The Maximum Principle	p. 333
The Wave Equation	p. 335
	p. 340
Miscellaneous Complements	p. 349
Finite-Dimensional and Finite-Codimensional Spaces	p. 349
Quotient Spaces	p. 353
Some Classical Spaces of Sequences	p. 357
Banach Spaces over C: What Is Similar and What Is Different?	p. 361
Solutions of Some Exercises	p. 371
Problems	p. 435
Partial Solutions of the Problems	p. 521
Notation	p. 583
References	p. 585
Index	p. 595
Table of Contents provided by Ingram. All Rights Reserved.

Functional Analysis, Sobolev Spaces and Partial Differential Equations

At a Glance

Paperback

Industry Reviews

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