Book:Martin Schechter/Principles of Functional Analysis/Second Edition
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Martin Schechter: Principles of Functional Analysis (2nd Edition)
Published $\text {2002}$, American Mathematical Society
- ISBN 978-0821828953
Subject Matter
Contents
Preface to the Revised Edition
From the Preface to the First Edition
- Chapter 1. Basic Notions
- 1.1 A problem from differential equations
- 1.2 An examination of the results
- 1.3 Examples of Banach spaces
- 1.4 Fourier series
- 1.5 Problems
- Chapter 2. Duality
- 2.1 The Riesz representation theorem
- 2.2 The Hahn-Banach theorem
- 2.3 Consequences of Hahn-Banach theorem
- 2.4 Examples of dual spaces
- 2.5 Problems
- Chapter 3. Linear Operators
- 3.1 Basic properties
- 3.2 The adjoint operator
- 3.3 Annihilators
- 3.4 The inverse operator
- 3.5 Operators with closed ranges
- 3.6 The uniform boundedness principle
- 3.7 The open mapping theorem
- 3.8 Problems
- Chapter 4. The Riesz Theory for Compact Operators
- 4.1 A type of integral equation
- 4.2 Operators of finite rank
- 4.3 Compact operators
- 4.4 The adjoint of a compact operator
- 4.5 Problems
- Chapter 5. Fredholm Operators
- 5.1 Orientation
- 5.2 Further properties
- 5.3 Perturbation theory
- 5.4 The adjoint operator
- 5.5 A special case
- 5.6 Semi-Fredholm operators
- 5.7 Products of operators
- 5.8 Problems
- Chapter 6. Spectral Theory
- 6.1 The spectrum and resolvent sets
- 6.2 The spectral mapping theorem
- 6.3 Operational calculus
- 6.4 Spectral projections
- 6.5 Complexification
- 6.6 The complex Hahn-Banach theorem
- 6.7 A geometric lemma
- 6.8 Problems
- Chapter 7. Unbounded Operators
- 7.1 Unbounded Fredholm operators
- 7.2 Further properties
- 7.3 Operators with closed ranges
- 7.4 Total subsets
- 7.5 The essential spectrum
- 7.6 Unbounded semi-Fredholm operators
- 7.7 The adjoint of a product of operators
- Chapter 8. Reflexive Banach Spaces
- 8.1 Properties of reflexive spaces
- 8.2 Saturated subspaces
- 8.3 Separable spaces
- 8.4 Weak convergence
- 8.5 Examples
- 8.6 Completing a normed vector space
- 8.7 Problems
- Chapter 9. Banach Algebras
- 9.1 Introduction
- 9.2 An example
- 9.3 Commutative algebras
- 9.4 Properties of maximal ideals
- 9.5 Partially ordered sets
- 9.6 Riesz operators
- 9.7 Fredholm perturbations
- 9.8 Semi-Fredholm perturbations
- 9.9 Remarks
- 9.10 Problems
- Chapter 10. Semigroups
- 10.1 A differential equation
- 10.2 Uniqueness
- 10.3 Unbounded operators
- 10.4 The infinitesimal operator
- 10.5 An approximation theorem
- 10.6 Problems
- Chapter 11. Hilbert Space
- 11.1 When is a Banach space a Hilbert space
- 11.2 Normal operators
- 11.3 Approximation by operators of finite rank
- 11.4 Integral operators
- 11.5 Hyponormal operators
- 11.6 Problems
- Chapter 12. Bilinear Forms
- 12.1 The numerical range
- 12.2 The associated operator
- 12.3 Symmetric forms
- 12.4 Closed forms
- 12.5 Closed extensions
- 12.6 Closable operators
- 12.7 Some proofs
- 12.8 Some representation theorems
- 12.9 Dissipative operators
- 12.10 The case of a line or a strip
- 12.11 Selfadjoint extensions
- 12.12 Problems
- Chapter 13. Selfadjoint Operators
- 13.1 Orthogonal projections
- 13.2 Square roots of operators
- 13.3 A decomposition of operators
- 13.4 Spectral resolution
- 13.5 Some consequences
- 13.6 Unbounded selfadjoint operators
- 13.7 Problems
- Chapter 14. Measures of Operators
- 14.1 A seminorm
- 14.2 Perturbation classes
- 14.3 Related measures
- 14.4 Measures of noncompactness
- 14.5 The quotient space
- 14.6 Strictly singular operators
- 14.7 Norm perturbations
- 14.8 Perturbation functions
- 14.9 Factored perturbation functions
- 14.10 Problems
- Chapter 15. Examples and Applications
- 15.1 A few remarks
- 15.2 A differential operator
- 15.3 Does $A$ have a closed extension
- 15.4 The closure of $A$
- 15.5 Another approach
- 15.6 The Fourier transform
- 15.7 Multiplication by a function
- 15.8 More general operators
- 15.9 $B$-Compactness
- 15.10 The adjoint of $\bar A$
- 15.11 An integral operator
- 15.12 Problems
Appendix A. Glossary
Appendix B. Major Theorems
Bibliography
Index