Book:John B. Conway/A Course in Functional Analysis/Second Edition

John B. Conway: A Course in Functional Analysis (2nd Edition)

Published $\text {1990}$, Springer-Verlag

ISBN 0-387-97245-5

Subject Matter

• Functional Analysis

Contents

CHAPTER I: Hilbert Spaces

$\S 1.$ Elementary Properties and Examples

$\S 2.$ Orthogonality

$\S 3.$ The Riesz Representation Theorem

$\S 4.$ Orthonormal Sets of Vectors and Bases

$\S 5.$ Isomorphic Hilbert Spaces and the Fourier Transform for the Circle

$\S 6.$ The Direct Sum of Hilbert Spaces

CHAPTER II: Operators on Hilbert Space

$\S 1.$ Elementary Properties and Examples

$\S 2.$ The Adjoint of an Operator

$\S 3.$ Projections and Idempotents; Invariant and Reducing Subspaces

$\S 4.$ Compact Operators

$\S 5.$* The Diagonalization of Compact Self-Adjoint Operators

$\S 6.$* An Application: Sturm-Liouville Systems

$\S 7.$* The Spectral Theorem and Functional Calculus for Compact Normal Operators

$\S 8.$* Unitary Equivalence for Compact Normal Operators

CHAPTER III: Banach Spaces

$\S 1.$ Elementary Properties and Examples

$\S 2.$ Linear Operators on Normed Spaces

$\S 3.$ Finite Dimensional Normed Spaces

$\S 4.$ Quotients and Products of Normed Spaces

$\S 5.$ Linear Functionals

$\S 6.$ The Hahn-Banach Theorem

$\S 7.$* An Application: Banach Limits

$\S 8.$* An Application: Runge's Theorem

$\S 9.$* An Application: Ordered Vector Spaces

$\S 10.$ The Dual of a Quotient Space and a Subspace

$\S 11.$ Reflexive Spaces

$\S 12.$ The Open Mapping and Closed Graph Theorems

$\S 13.$ Complemented Subspaces of a Banach Space

$\S 14.$ The Principle of Uniform Boundedness

CHAPTER IV: Locally Convex Spaces

$\S 1.$ Elementary Properties and Examples

$\S 2.$ Metrizable and Normable Locally Convex Spaces

$\S 3.$ Some Geometric Consequences of the Hahn-Banach Theorem

$\S 4.$* Some Examples of the Dual Space of a Locally Convex Space

$\S 5.$* Inductive Limits and the Space of Distributions

CHAPTER V: Weak Topologies

$\S 1.$ Duality

$\S 2.$ The Dual of a Subspace and a Quotient Space

$\S 3.$ Alaoglu's Theorem

$\S 4.$ Reflexivity Revisited

$\S 5.$ Separability and Metrizability

$\S 6.$* An Application: The Stone-Cech Compactification

$\S 7.$ The Krein-Milman Theorem

$\S 8.$ An Application: The Stone-Weierstrass Theorem

$\S 9.$* The Schauder Fixed Point Theorem

$\S 10.$* The Ryll-Nardzewski Fixed Point Theorem

$\S 11.$* An Application: Haar Measure on a Compact Group

$\S 12.$* The Krein-Smulian Theorem

$\S 13.$* Weak Compactness

CHAPTER VI: Linear Operators on a Banach Space

$\S 1.$ The Adjoint of a Linear Operator

$\S 2.$* The Banach-Stone Theorem

$\S 3.$ Compact Operators

$\S 4.$ Invariant Subspaces

$\S 5.$ Weakly Compact Operators

CHAPTER VII: Banach Algebras and Spectral Theory for Operators on a Banach Space

$\S 1.$ Elementary Properties and Examples

$\S 2.$ Ideals and Quotients

$\S 3.$ The Spectrum

$\S 4.$ The Riesz Functional Calculus

$\S 5.$ Dependence of the Spectrum on the Algebra

$\S 6.$ The Spectrum of a Linear Operator

$\S 7.$ The Spectral Theory of a Compact Operator

$\S 8.$ Abelian Banach Algebras

$\S 9.$* The Group Algebra of a Locally Compact Abelian Group

CHAPTER VIII: $C^*$-Algebras

$\S 1.$ Elementary Properties and Examples

$\S 2.$ Abelian $C^*$-Algebras and the Functional Calculus in $C^*$-Algebras

$\S 3.$ The Positive Elements in a $C^*$-Algebra

$\S 4.$* Ideals and Quotients of $C^*$-Algebras

$\S 5.$* Representations of $C^*$-Algebras and the Gelfand-Naimark-Segal Construction

CHAPTER IX: Normal Operators on Hilbert Space

$\S 1.$ Spectral Measures and Representations of Abelian $C^*$-Algebras

$\S 2.$ The Spectral Theorem

$\S 3.$ Star-Cyclic Normal Operators

$\S 4.$ Some Applications of the Spectral Theorem

$\S 5.$ Topologies on $\map \BB \HH$

$\S 6.$ Commuting Operators

$\S 7.$ Abelian von Neumann Algebras

$\S 8.$ The Functional Calculus for Normal Operators: The Conclusion of the Saga

$\S 9.$ Invariant Subspaces for Normal Operators

$\S 10.$ Multiplicity Theory for Normal Operators: A Complete Set of Unitary Invariants

CHAPTER X: Unbounded Operators

$\S 1.$ Basic Properties and Examples

$\S 2.$ Symmetric and Self-Adjoint Operators

$\S 3.$ The Cayley Transform

$\S 4.$ Unbounded Normal Operators and the Spectral Theorem

$\S 5.$ Stone's Theorem

$\S 6.$ The Fourier Transform and Differentiation

CHAPTER XI: Fredholm Theory

$\S 1.$ The Spectrum Revisited

$\S 2.$ Fredholm Operators

$\S 3.$ The Fredholm Index

$\S 4.$ The Essential Spectrum

$\S 5.$ The Components of $\LL \FF$

$\S 6.$ A Finer Analysis of the Spectrum

APPENDIX A: Preliminaries

$\S 1.$ Linear Algebra

$\S 2.$ Topology

APPENDIX B: The Dual of $L^p(\mu)$

APPENDIX C: The Dual of $C_0(X)$

Bibliography

List of Symbols

Index