Functional Analysis ：A Terse Introduction ( De Gruyter Textbook )

Publication subTitle ：A Terse Introduction

Publication series ：De Gruyter Textbook

Author： Chacón Gerardo;Rafeiro Humberto;Vallejo Juan Camilo

Publisher： De Gruyter‎

Publication year： 2016

E-ISBN: 9783110441925

P-ISBN(Paperback): 9783110441918

Subject： O151.2 Linear Algebra

Keyword：泛函分析

Language： ENG

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Description

This textbook on functional analysis offers a short and concise introduction to the subject. The book is designed in such a way as to provide a smooth transition between elementary and advanced topics and its modular structure allows for an easy assimilation of the content. Starting from a dedicated chapter on the axiom of choice, subsequent chapters cover Hilbert spaces, linear operators, functionals and duality, Fourier series, Fourier transform, the fixed point theorem, Baire categories, the uniform bounded principle, the open mapping theorem, the closed graph theorem, the Hahn–Banach theorem, adjoint operators, weak topologies and reflexivity, operators in Hilbert spaces, spectral theory of operators in Hilbert spaces, and compactness. Each chapter ends with workable problems.
The book is suitable for graduate students, but also for advanced undergraduates, in mathematics and physics.

Contents:
List of Figures
Basic Notation
Choice Principles
Hilbert Spaces
Completeness, Completion and Dimension
Linear Operators
Functionals and Dual Spaces
Fourier Series
Fourier Transform
Fixed Point Theorem
Baire Category Theorem
Uniform Boundedness Principle
Open Mapping Theorem
Closed Graph Theorem
Hahn–Banach Theorem
The Adjoint Operator
Weak Topologies and Reflexivity
Operators in Hilbert Spaces
Spectral Theory of Operators on Hilbert Spaces
Compactness
Bibliography
Index

Chapter

1.2 Some Applications

1.3 Problems

2 Hilbert Spaces

2.1 Norms

2.2 Inner Products and Hilbert Spaces

2.3 Some Geometric Properties

2.4 Orthogonality

2.5 Orthogonal Sequences

2.6 Problems

3 Completeness, Completion and Dimension

3.1 Banach Spaces

3.2 Completion and Dimension

3.3 Separability

3.4 Problems

4 Linear Operators

4.1 Linear Transformations

4.2 Back to Matrices

4.3 Boundedness

4.4 Problems

5 Functionals and Dual Spaces

5.1 A Special Type of Linear Operators

5.2 Dual Spaces

5.3 The Bra-ket Notation

5.4 Problems

6 Fourier Series

6.1 The Space L2[–0, 0]

6.2 Convergence Conditions for the Fourier Series

6.2.1 Sufficient Convergence Conditions for the Fourier Series in a Point

6.2.2 Conditions for Uniform Convergence for the Fourier Series

6.3 Problems

7 Fourier Transform

7.1 Convolution

7.2 L1 Theory

7.3 L2 Theory

7.4 Schwartz Class

7.5 Problems

8 Fixed Point Theorem

8.1 Some Applications

8.1.1 Neumann Series

8.1.2 Differential Equations

8.1.3 Integral Equations

8.1.4 Fractals

8.2 Problems

9 Baire Category Theorem

9.1 Baire Categories

9.2 Baire Category Theorem

9.3 Problems

10 Uniform Boundedness Principle

10.1 Problems

11 Open Mapping Theorem

11.1 Problems

12 Closed Graph Theorem

12.1 Problems

13 Hahn–Banach Theorem

13.1 Extension Theorems

13.2 Minkowski Functional

13.3 Separation Theorem

13.4 Applications of the Hahn–Banach Theorem

13.5 Problems

14 The Adjoint Operator

14.1 Hilbert Spaces

14.2 Banach Spaces

14.3 Problems

15 Weak Topologies and Reflexivity

15.1 Weak* Topology

15.2 Reflexive Spaces

15.3 Problems

16 Operators in Hilbert Spaces

16.1 Compact Operators

16.2 Normal and Self-Adjoint Operators

16.3 Problems

17 Spectral Theory of Operators on Hilbert Spaces

17.1 A Quick Review of Spectral Theory in Finite Dimensions

17.2 The Spectral Theorem for Compact Self-Adjoint Operators

17.3 Problems

18 Compactness

18.1 Metric Spaces

18.2 Compactness in Some Function Spaces

18.2.1 Space l2

18.2.2 Space of Continuous Functions

18.2.3 Lebesgue Spaces

18.3 Problems

Bibliography

Index

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