# Book:George Bachman/Functional Analysis

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## George Bachman and Lawrence Narici:

## George Bachman and Lawrence Narici: *Functional Analysis*

Published $2000$, **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**