# Book:George Bachman/Functional Analysis

## George Bachman and Lawrence Narici: Functional Analysis

Published $\text {2000}$, Dover

ISBN 978-0486402512.

### 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
20.3 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