Book:Joseph Muscat/Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras
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Joseph Muscat: Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras
Published $\text {2014}$, Springer
- ISBN 978-3319067278
Subject Matter
Contents
Introduction
- Part I: Metric Spaces
- 2. Distance
- 2.1 Balls and Open Sets
- 2.2 Closed Sets
- 3. Convergence and Continuity
- 3.1 Convergence
- 3.2 Continuity
- 4. Completeness and Separability
- 4.1 Completeness
- 4.2 Uniformly Continuous Maps
- 4.3 Separable Spaces
- 5. Connectedness
- 5.1 Connected Sets
- 5.2 Components
- 6. Compactness
- 6.1 Bounded Sets
- 6.2 Totally Bounded Sets
- 6.3 Compact Sets
- 6.4 The Space $\map C {X, Y}$
- Part II: Banach and Hilbert Spaces
- 7. Normed Spaces
- 7.1 Vector Spaces
- 7.2 Norms
- 7.3 Metric and Vector Properties
- 7.4 Complete and Separable Normed Vector Spaces
- 7.5 Series
- 8. Continuous Linear Maps
- 8.1 Operators
- 8.2 Quotient Spaces
- 8.3 $\R^N$ and Totally Bounded Sets
- 9. Main Examples
- 9.1 Sequence Spaces
- 9.2 Functions Spaces
- 10. Hilbert Spaces
- 10.1 Inner Products
- 10.2 Least Squares Approximation
- 10.3 Duality $H^* \approx H$
- 10.4 The Adjoint Map $T^*$
- 10.5 Inverse Problems
- 10.6 Orthonormal Bases
- 11. Banach Spaces
- 11.1 The Open Mapping Theorem
- 11.2 Compact Operators
- 11.3 The Dual Space $X^*$
- 11.4 The Adjoint $T^T$
- 11.5 Pointwise and Weak Convergence
- 12. Differentiation and Integration
- 12.1 Differentiation
- 12.2 Integration for Vector-Valued Functions
- 12.3 Complex Differentiation and Integration
- Part III: Banach Algebras
- 13. Banach Algebras
- 13.1 Introduction
- 13.2 Power Series
- 13.3 The Group of Invertible Elements
- 13.4 Analytic Functions
- 14. Spectral Theory
- 14.1 The Spectral Radius
- 14.2 The Spectrum of an Operator
- 14.3 Spectra of Compact Operators
- 14.4 The Functional Calculus
- 14.5 The Gelfand Transform
- 15. $C^*$-Algebras
- 15.1 Normal Elements
- 15.2 Norml Operators in $\map B H$
- 15.3 The Spectral Theorem for Compact Normal Operators
- 15.4 Representation Theorems
Hints to Selected Problems
Glossary of Symbols
Further Reading
References
Index