Functional analysis

Georges Bachman, Lawrence Narici

532 pages, parution le 21/11/2001

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Résumé

This book is meant to serve as an introduction to functional analysis and can be read by anyone with a background in undergraduate mathematics through the junior year. It offers a careful, detailed treatment aimed particularly at advanced undergraduate or beginning graduate mathematics majors as well as to those physics and engineering students who are finding increasing applications of functional analysis in their disciplines.

Topics include an introduction to inner product spaces, normed and metric spaces, isometries and completion of a metric space, topological and Banach spaces, equivalent norms and factor spaces, complete orthonormal sets, the Hahn-Banach theorem and its consequences, weak convergence and bounded linear transformations, spectral notions, adjoints and sesquilinear functionals, orthogonal projections and positive definite operators, and many other related subjects. Each chapter concludes with exercises intended to test and reinforce the reader's understanding of the material covered.

Prerequisites for optimum comprehension of the text are undergraduate courses in linear algebra and advanced calculus; for the convenience of students, a brief glossary of definitions and notations relevant to linear algebra is included. Proofs of all theorems are as detailed as possible. A helpful bibliography and index of symbols complete this clear and well-written introductory volume.

Contents

Introduction to Inner Product Spaces
Orthonormal Projections and the Spectral Theorem for Normal Transformations
Normed Spaces and Metrics Spaces
Isometries and Completion of a Metric Space
Compactness in Metric Spaces
Category and Separable Spaces
Topological Spaces
Banach Spaces, Equivalent Norms, and Factor Spaces
Commutative Convergence, Hilbert Spaces, and Bessel's Inequality
Complete Orthonormal Sets
The Hahn-Banach Theorem
Consequences of the Hahn-Banach Theorem
The Conjugate Space of C[a, b]
Weak Convergence and Bounded Linear Transformations
Convergence in L(X, Y) and the Principle of Uniform Boundedness
Closed Transformations and the Closed Graph Theorem
Closures, Conjugate Transformations, and Complete Continuity
Spectral Notions
Introduction to Banach Algebras
Adjoints and Sesquilinear Functionals
Some Spectral Results for Normal and Completely Continuous Operators
Orthogonal Projections and Positive Definite Operators
Square Roots and a Spectral Decomposition Theorem
Spectral Theorem for Completely Continuous Normal Operators
Spectral Theorem for Bounded, Self-Adjoint Operators
A Second Approach to the Spectral Theorem for Bounded, Self-Adjoint Operators
A Third Approach to the Spectral Theorem for Bounded, Self-Adjoint Operators and some consequences
Spectral Theorem for Bounded, Normal Operators
Spectral Theorem for Unbounded, Self-Adjoint Operators

Bibliography

Index of Symbols

Subject Index

Errata

Caractéristiques techniques

	PAPIER
Éditeur(s)	Dover
Auteur(s)	Georges Bachman, Lawrence Narici
Parution	21/11/2001
Nb. de pages	532
Format	15,5 x 23,3
Couverture	Broché
Poids	666g
Intérieur	Noir et Blanc
EAN13	9780486402512
ISBN13	978-0-486-40251-2