Book:Barbara D. MacCluer/Elementary Functional Analysis

Subject Matter

 * Functional Analysis

Contents
Preface


 * 1. Hilbert Space Preliminaries


 * 1.1 Normed Linear Spaces


 * 1.2 Orthogonality


 * 1.3 Hilbert Space Geometry


 * 1.4 Linear Functionals


 * 1.5 Orthonormal Bases


 * 1.6 Exercises


 * 2. Operator Theory Basics


 * 2.1 Bounded Linear Operators


 * 2.2 Adjoints of Hilbert Space Operators


 * 2.3 Adjoints of Banach Space Operators


 * 2.4 Exercises


 * 3. The Big Three


 * 3.1 The Hahn-Banach Theorem


 * 3.2 Principle of Uniform Boundedness


 * 3.3 Open Mapping and Closed Graph Theorems


 * 3.4 Quotient Spaces


 * 3.5 Banach and the Scottish Café


 * 3.6 Exercises


 * 4. Compact Operators


 * 4.1 Finite-Dimensional Spaces


 * 4.2 Compact Operators


 * 4.3 A Preliminary Spectral Theorem


 * 4.4 The Invariant Subspace Problem


 * 4.5 Introduction to the Spectrum


 * 4.6 The Fredholm Alternative


 * 4.7 Exercises


 * 5. Banach and $C^*$-Algebras


 * 5.1 First Examples


 * 5.2 Results on Spectra


 * 5.3 Ideals and Homomorphisms


 * 5.4 Commutative Banach Algebras


 * 5.5 Weak Topologies


 * 5.6 The Gelfand Transform


 * 5.7 The Continuous Functional Calculus


 * 5.8 Fredholm Operators


 * 5.9 Exercises


 * 6. The Spectral Theorem


 * 6.1 Normal Operators Are Multiplication Operators


 * 6.2 Spectral Measures


 * 6.3 Exercises


 * Appendix A: Real Analysis Topics


 * A.1 Measures


 * A.2 Integration


 * A.3 $L^p$ Spaces


 * A.4 The Stone-Weierstrass Theorem


 * A.5 Positive Linear Functionals on $\map C X$

References

Index