Book:George Bachman/Functional Analysis
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George Bachman and Lawrence Narici: Functional Analysis
Published $\text {2000}$, Dover
- ISBN 978-0486402512
Subject Matter
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