Book:Martin Schechter/Principles of Functional Analysis/Second Edition

Subject Matter

 * Functional Analysis

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