Book:Joseph Muscat/Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras

Subject Matter

 * Functional Analysis

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