Book:George Bachman/Functional Analysis

Subject Matter

 * Functional Analysis

Contents
Preface


 * Chapter 1. Introduction to Inner Product Spaces


 * 1.1 Some Prerequisite Material and Conventions


 * 1.2 Inner Product Spaces


 * 1.3 Linear Functionals, the Riesz Representation Theorem, and Adjoints


 * Exercises 1


 * References


 * Chapter 2. Orthogonal Projections and the Spectral Theorem for Normal Transformations


 * 2.1 The Complexification


 * 2.2 Orthogonal Projections and Orthogonal Direct Sums


 * 2.3 Unitary and Orthogonal Transformations


 * Exercises 2


 * References


 * Chapter 3. Normed Spaces and Metric Spaces


 * 3.1 Norms and Normed Linear Spaces


 * 3.2 Metrics and Metrics Spaces


 * 3.3 Topological Notions in Metric Spaces


 * 3.4 Closed and Open Sets, Continuity, and Homeomorphisms


 * Exercises 3


 * Reference


 * Chapter 4. Isometries and Completion of a Metric Space


 * 4.1 Isometries and Homemorphisms


 * 4.2 Cauchy Sequences and Complete Metric Spaces


 * Exercises 4


 * Reference


 * Chapter 5. Compactness in Metric Spaces


 * 5.1 Nested Sequences and Complete Spaces


 * 5.2 Relative Compactness $\epsilon-$Nets and Totally Bounded Sets


 * 5.3 Countable Compactness and Sequential Compactness


 * Exercises 5


 * References


 * Chapter 6. Category and Separable Spaces


 * 6.1 $F_\sigma$ and $G_\delta$ Sets


 * 6.2 Nowhere-Dense Sets and Category


 * 6.3 The Existence of Functions Continuous Everywhere, Differentiable Nowhere


 * 6.4 Separable Spaces


 * Exercises 6


 * References


 * Chapter 7. Topological Spaces


 * 7.1 Definitions and Examples


 * 7.2 Bases


 * 7.3 Weak Topologies


 * 7.4 Separation


 * 7.5 Compactness


 * Exercises 7


 * References


 * Chapter 8. Banach Spaces, Equivalent Norms, and Factor Spaces


 * 8.1 The Hölder and Minkowski Inequalities


 * 8.2 Banach Spaces and Examples


 * 8.3 The Completion of a Normed Linear Space


 * 8.4 Generated Subspaces and Closed Subspaces


 * 8.5 Equivalent Norms and a Theorem of Riesz


 * 8.6 Factor Spaces


 * 8.7 Completeness in the Factor Space


 * 8.8 Convexity


 * Exercises 8


 * References


 * Chapter 9. Commutative Convergence, Hilbert Spaces, and Bessel's Inequality


 * 9.1 Commutative Convergence


 * 9.2 Norms and Inner Products on Cartesian Products of Normed and Inner Product Spaces


 * 9.3 Hilbert Spaces


 * 9.4 A Nonseparable Hilbert Space


 * 9.5 Bessel's Inequality


 * 9.6 Some Results from $\map {L_2} {0, 2\pi}$ and the Riesz-Fischer Theorem


 * 9.7 Complete Orthonormal Sets


 * 9.8 Complete Orthonormal Sets and Parseval's Identity


 * 9.9 A Complete Orthonormal Set for $\map {L_2} {0, 2\pi}$


 * Appendix 9


 * Exercises 9

vReferences


 * Chapter 10. Complete Orthonormal Sets


 * 10.1 Complete Orthonormal Sets and Parseval's Identity


 * 10.2 The Cardinality of Complete Orthonormal Sets


 * 10.3 A Note on the Structure of Hilbert Spaces


 * 10.4 Closed Subspaces and the Projection Theorem for Hilbert Spaces


 * Exercises 10


 * References


 * Chapter 11. The Hahn-Banach Theorem


 * 11.1 The Hahn-Banach Theorem


 * 11.2 Bounded Linear Functionals


 * 11.3 The Conjugate Space


 * Exercises 11


 * Appendix 11. The Problem of Measure and the Hahn-Banach Theorem


 * Exercises 11 Appendix


 * References


 * Chapter 12. Consequences of the Hahn-Banach Theorem


 * 12.1 Some Consequences of the Hahn-Banach Theorem


 * 12.2 The Second Conjugate Space


 * 12.3 The Conjugate Space of $l_p$


 * 12.4 The Riesz Representation Theorem for Linear Functionals on a Hilbert Space


 * 12.5 Reflexivity of Hilbert Spaces


 * Exercises 12


 * References


 * Chapter 13. The Conjugate Space of $C \closedint a b$


 * 13.1 A Representation Theorem for Bounded Linear Functionals on $C \closedint a b$


 * 13.2 A List of Some Spaces and Their Conjugate Spaces


 * Exercises 13


 * References


 * Chapter 14. Weak Convergence and Bounded Linear Transformations


 * 14.1 Weak Convergence


 * 14.2 Bounded Linear Transformations


 * Exercises 14


 * References


 * Chapter 15. Convergence in $\map L {X, Y}$ and the Principle of Uniform Boundedness


 * 15.1 Convergence in $\map L {X, Y}$


 * 15.2 The Principle of Uniform Boundedness


 * 15.3 Consequences of the Principle of Uniform Boundedness


 * Exercises 15


 * References


 * Chapter 16. Closed Transformations and the Closed Graph Theorem


 * 16.1 The Graph of a Mapping


 * 16.2 Closed Linear Transformations and the Bounded Inverse Theorem


 * 16.3 Some Consequences of the Bounded Inverse Theorem


 * Appendix 16. Supplement to Theorem 16.5


 * Exercises 16


 * References


 * Chapter 17. Closures, Conjugate Transformations, and Complete Continuity


 * 17.1 The Closure of a Linear Transformation


 * 17.2 A Class of Linear Transformations that Admit a Closure


 * 17.3 The Conjugate Map of a Bounded Linear Transformation


 * 17.4 Annihilators


 * 17.5 Completely Continuous Operators; Finite-Dimensional Operators


 * 17.6 Further Properties of Completely Continuous Transformations


 * Exercises 17


 * References


 * Chapter 18. Spectral Notions


 * 18.1 Spectra and the Resolvent Set


 * 18.2 The Spectra of Two Particular Transformations


 * 18.3 Approximate Proper Values


 * Exercises 18


 * References


 * Chapter 19. Introduction to Banach Algebras


 * 19.1 Analytic Vector-Valued Functions


 * 19.2 Normed and Banach Algebras


 * 19.3 Banach Algebras with Identity


 * 19.4 An Analytic Function - the Resolvent Operator


 * 19.5 Spectral Radius and the Spectral Mapping Theorem for Polynomials


 * 19.6 The Gelfand Theory


 * 19.7 Weak Topologies and the Gelfand Topology


 * 19.8 Topological Vector Spaces and Operator Topologies


 * Exercises 19


 * References


 * Chapter 20. Adjoints and Sesquilinear Functionals


 * 20.1 The Adjoint Operator


 * 20.2 Adjoints and Closures


 * 20.3 Adjoints of Bounded Linear Transformations in Hilbert Spaces


 * 20.4 Sesquilinear Functionals


 * Exercises 20


 * References


 * Chapter 21. Some Spectral Results for Normal and Completely Continuous Operators


 * 21.1 A New Expression for the Norm of $A \in\map L {X, X}$


 * 21.2 Normal Transformations


 * 21.3 Some Spectral Results for Completely Continuous Operators


 * 21.4 Numerical Range


 * Exercises 21


 * Appendix to Chapter 21. The Fredholm Alternative Theorem and the Spectrum of a Completely Continuous Transformation


 * A.1 Motivation


 * A.2 The Fredholm Alternative Theorem


 * References


 * Chapter 22. Orthogonal Projections and Positive Definite Operators


 * 22.1 Properties of Orthogonal Projections


 * 22.2 Products of Projections


 * 22.3 Positive Operators


 * 22.4 Sums and Differences of Orthogonal Projections


 * 22.5 The Product of Positive Operators


 * Exercises 22


 * References


 * Chapter 23. Square Roots and a Spectral Decomposition Theorem


 * 23.1 Square Root of Positive Operators


 * 23.2 Spectral Theorem for Bounded, Normal, Finite-Dimensional Operators


 * Exercises 23


 * References


 * Chapter 24. Spectral Theorem for Completely Continuous Normal Operators


 * 24.1 Infinite Orthogonal Direct Sums: Infinite Series of Transformations


 * 24.2 Spectral Decomposition Theorem for Completely Continuous Normal Operators


 * Exercises 24


 * References


 * Chapter 25. Spectral Theorem for Bounded, Self-Adjoint Operators


 * 25.1 A Special Case - the Self-Adjoint, Completely Continuous Operator


 * 25.2 Further Properties of the Spectrum of Bounded, Self-Adjoint Transformations


 * 25.3 Spectral Theorem for Bounded, Self-Adjoint Operators


 * Exercises 25


 * References


 * Chapter 26. A Second Approach to the Spectral Theorem for Bounded, Self-Adjoint Operators


 * 26.1 A Second Approach to the Spectral Theorem for Bounded, Self-Adjoint Operators


 * Exercises 26


 * References


 * Chapter 27. A Third Approach to the Spectral Theorem for Bounded, Self-Adjoint Operators and Some Consequences


 * 27.1 A Third Approach to the Spectral Theorem for Bounded, Self-Adjoint Operators


 * 27.2 Two Consequences of the Spectral Theorem


 * Exercises 27


 * References


 * Chapter 28. Spectral Theorem for Bounded, Normal Operators


 * 28.1 The Spectral Theorem for Bounded, Normal Operators on a Hilbert Space


 * 28.2 Spectral Measures; Unitary Transformations


 * Exercises 28


 * References


 * Chapter 29. Spectral Theorem for Unbounded, Self-Adjoint Operators


 * 29.1 Permutativity


 * 29.2 The Spectral Theorem for Unbounded, Self-Adjoint Operators


 * 29.3 A Proof of the Spectral Theorem Using the Cayley Transform


 * 29.4 A Note on the Spectral Theorem for Unbounded Normal Operators


 * Exercises 29


 * References

Bibliography

Index of Symbols

Subject Index

Errata