导语

  

内容提要

  

    本书作者擅长写教科书，以选材仔细、论述清晰、实例丰富著称。本书是一部代理科研究生使用的泛函分析教材，读者只需具备积分和测度论的知识即可阅读。全书充分体现了作者的著书风格，以实例先行，从具体到一般，从浅入深，并配有许多精心挑选的例题和习题。

Preface
Preface to the Second Edition
CHAPTER I
  Hilbert Spaces
  1.Elementary Properties and Examples
  2.Orthogonality
  3.The Riesz Representation Theorem
  4.Orthonormal Sets of Vectors and Bases
  5.Isomorphic Hilbert Spaces and the Fourier Transform for the Circle
  6.The Direct Sum of Hilbert Spaces
CHAPTER II
  Operators on Hilbert Space
  1.Elementary Properties and Examples
  2.The Adjoint of an Operator
  3.Projections and Idempotents;Invariant and Reducing Subspaces
  4.Compact Operators
  5.*The Diagonalization of Compact Self-Adjoint Operators
  6.*An Application:Sturm-Liouville Systems
  7.*The Spectral Theorem and Functional Calculus for Compact Normai
  Operators
  8.*Unitary Equivalence for Compact Normai Operators
CHAPTER III
  Banach Spaces
  1.Elementary Properties and Examples
  2.Linear Operators on Normed Spaces
  3.Finite Dimensional Normed Spaces
  4.Quotients and Products of Normed Spaces
  5.Linear Functionals
  6.The Hahn-Banach Theorem
  7.*An Application:Banach Limits
  8.*An Application:Runge's Theorem
  9.*An Application:Ordered Vector Spaces
  10.The Dual of a Quotient Space and a Subspace
  11.Reflexive Spaces
  12.The Open Mapping and Closed Graph Theorems
  13.Complemented Subspaces of a Banach Space
  14.The Principle of Uniform Boundedness
CHAPTER IV
  Locally Convex Spaces
  S1.Elementary Properties and Examples
  2.Metrizable and Normable Locally Convex Spaces
  3.Some Geometric Consequences of the Hahn-Banach Theorem
  4.*Some Examples of the Dual Space of a Locally Convex Space
  5.*Inductive Limits and the Space of Distributions
CHAPTER V
  Weak Topologies
  1.Duality
  2.The Dual of a Subspace and a Quotient Space
  3.Alaoglu's Theorem
  84.Reflexivity Revisited
  5.Separability and Metrizability
  S6.*An Application:The Stone-Cech Compactification
  87.The Krein-Milman Theorem
  8.An Application:The Stone-Weierstrass Theorem
  9.*The Schauder Fixed Point Theorem
  10.*The Ryll-Nardzewski Fixed Point Theorem
  11.*An Application:Haar Measure on a Compact Group
  12.*The Krein-Smulian Theorem
  13.*Weak Compactness
CHAPTER VI
  Linear Operators on a Banach Space
  1.The Adjoint of a Linear Operator
  2.*The Banach-Stone Theorem
  3.Compact Operators
  4.Invariant Subspaces
  5.Weakly Compact Operators
CHAPTER VII
  Banach Algebras and Spectral Theory for
  Operators on a Banach Space
  1.Elementary Properties and Examples
  2.Ideals and Quotients
  3.The Spectrum
  4.The Riesz Functional Calculus
  5.Dependence of the Spectrum on the Aigebra
  6.The Spectrum of a Linear Operator
  7.The Spectral Theory of a Compact Operator
  8.Abelian Banach Algebras
  9.*The Group Algebra of a Locally Compact Abelian Group
CHAPTER VIII
  C*-Algebras
  1.Elementary Properties and Examples
  2.Abelian C*-Algebras and the Functional Calculus in C*-Algebras
  3.The Positive Elements in a C*-Algebra
  4.*Ideals and Quotients of C*-Algebras
  5.*Representations of C*-Algebras and the Gelfand-Naimark-Segal
  Construction
CHAPTER IX
  Normal Operators on Hilbert Space
  1.Spectral Measures and Representations of Abelian C*-Algebras
  2.The Spectral Theorem
  3.Star-Cyclic Normal Operators
  4.Some Applications of the Spectral Theorem
  5.Topologies on (X)
  6.Commuting Operators
  7.Abelian von Neumann Algebras
  8.The Functional Calculus for Normal Operators:
  The Conclusion of the Saga
  Invariant Subspaces for Normal Operators
  9.Multiplicity Theory for Normal Operators:
  10.A Complete Set of Unitary Invariants
CHAPTER X
  Unbounded Operators
  1.Basic Properties and Examples
  2.Symmetric and Self-Adjoint Operators
  3.The Cayley Transform
  4.Unbounded Normal Operators and the Spectral Theorem
  S5.Stone's Theorem
  6.The Fourier Transform and Differentiation
  7.Moments6
CHAPTER XI
  Fredholm Theory
  1.The Spectrum Revisited
  2.Fredholm Operators
  3.The Fredholm Index
  4.The Essential Spectrum
  5.The Components of Sg
  6.A Finer Analysis of the Spectrum
APPENDIX A
  Preliminaries
  1.Linear Algebra
  2.Topology
APENDIX B
  The Dual of LP(u)
  APPENDIX C
  The Dual of Co(X)
Bibliography
List of Symbols
Index