Functional Analysis
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
Reviews / Votes
"This book offers a very readable account, possesses a friendly two-color layout and only asks for minimal prerequisites from linear algebra and topology. "
R. Steinbauer in: Monatshefte für Mathematik 190 (2019), 210
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Content
- Intro
- Contents
- List of Figures
- Basic Notation
- 1 Choice Principles
- 1.1 Axiom of Choice
- 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
