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

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

Published $\text {2014}$, Springer

ISBN 978-3319067278

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