Functional analysis 泛函分析

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 func-tional 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, orthogo-hal 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.

Preface
CHAPTER 1.Introduction to Inner Product Spaces
CHAPTER 2.Orthogonal Projections and the Spectral Theorem for Normal Transformations
CHAPTER 3.Normed Spaces and Metric Spaces
CHAPTER 4.Isometries and Completion of a Metric Space
CHAPTER 5.Compactness in Metric Spaces
CHAPTER 6.Category and Separable Spaces
CHAPTER 7.Topological Spaces
CHAPTER 8.Banach Spaces,Equivalent Norms,and Factor Spaces
CHAPTER 9.Commutative Convergence,Hilbert Spaces,and Bessel's Inequality
CHAPTER 10.Complete Orthonormal Sets
CHAPTER 11.The Hahn-Bananch Theorem
CHAPTER 12.Consequences of the Hahn-Banach Theorem
CHAPTER 13.The Conjugate Space of (a,b)
CHAPTER 14.Weak Convergence and Bounded Linear Transformations
CHAPTER 15.Convergence in L(X,Y)and the Principle of Uniform Boundedness
CHAPTER 16.Closed Cransformations and the Glosde Graph Theorem
CHAPTER 17.Closures,Conjugate Sransformations,and Complete Contiunity
CHAPTER 18.Spectral Notions
CHAPTER 19.Introduction to Banach Algebras
CHAPTER 20.Adjoints and Sesquilinear Functionals
CHAPTER 21.Some Spectral Results for Normal and Completely Continuous perators
CHAPTER 22.Orthogonal Projections and Positive Definite Operators
CHAPTER 23.Square Roots and a Spectral Decomposition Theorem
CHAPTER 24.Spectral Theorem for Completely Continuous Normal Operators
CHAPTER 25.Spectral Theorem for Bounded,Self-Adjoint Operators
CHAPTER 26.A Second Approach to the Spectral Theorem for Bounded,Self-Adjoint Operators
CHAPTER 27.A Third Approach to the Spectral Theorem for Bounded,Self-Adjoint Operators and Some Consequences
CHAPTER 28.Spectral Theorem for Bounded,Normal Operators
CHAPTER 29.Spectral Theorem for Unbounded,Self-Adjoin Operators
Bibliography
Index of Symbols
Subject Index
Errata