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Essential Results of Functional Analysis : Chicago Lectures in Mathematics - Robert J. Zimmer

Essential Results of Functional Analysis

Chicago Lectures in Mathematics

Paperback Published: 1st January 1990
ISBN: 9780226983387
Number Of Pages: 168

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Functional analysis is a broad mathematical area with strong connections to many domains within mathematics and physics. This book, based on a first-year graduate course taught by Robert J. Zimmer at the University of Chicago, is a complete, concise presentation of fundamental ideas and theorems of functional analysis. It introduces essential notions and results from many areas of mathematics to which functional analysis makes important contributions, and it demonstrates the unity of perspective and technique made possible by the functional analytic approach.

Zimmer provides an introductory chapter summarizing measure theory and the elementary theory of Banach and Hilbert spaces, followed by a discussion of various examples of topological vector spaces, seminorms defining them, and natural classes of linear operators. He then presents basic results for a wide range of topics: convexity and fixed point theorems, compact operators, compact groups and their representations, spectral theory of bounded operators, ergodic theory, commutative C*-algebras, Fourier transforms, Sobolev embedding theorems, distributions, and elliptic differential operators. In treating all of these topics, Zimmer's emphasis is not on the development of all related machinery or on encyclopedic coverage but rather on the direct, complete presentation of central theorems and the structural framework and examples needed to understand them. Sets of exercises are included at the end of each chapter.

For graduate students and researchers in mathematics who have mastered elementary analysis, this book is an entrée and reference to the full range of theory and applications in which functional analysis plays a part. For physics students and researchers interested in these topics, the lectures supply a thorough mathematical grounding.

Preface 0
Background 0
Review of basic functional analysis 0
Some special properties of integration in Rn
Topological vector spaces and operators
Examples of spaces
Examples of operators
Operator topologies and groups of operators
Convexity and fixed point theorems
Kakutani-Markov fixed point theorem
Haar measure for compact groups
Krein-Millman theorem
Compact operators
Compact operators and Hilbert-Schmidt operators
Spectral theorem for compact normal operators
Peter-Weyl theorem for compact groups
General spectral theory
Spectrum of an operator
Spectral theorem for self-adjoint operators
Gelfand's theory of commutative C*-algebras
Mean ergodic theorem
Fourier transforms and Sobolev embedding theorems
Basic properties of the Fourier transform and the Plancherel theorem
Sobolev and Rellich embedding theorems
Distributions and elliptic operators
Basic properties of distributions
Distributions and Sobolev spaces
Regularity for elliptic operators
Appendix to
6.3
A spectral theorem for elliptic operators
Index
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780226983387
ISBN-10: 0226983382
Series: Chicago Lectures in Mathematics
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 168
Published: 1st January 1990
Publisher: The University of Chicago Press
Country of Publication: US
Dimensions (cm): 20.4 x 13.2  x 0.64
Weight (kg): 0.19