Functional Analysis : A Terse Introduction

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 * Modularly structured into teaching units. * Bridges the gap between elementary and advanced topics in analysis for advanced undergraduate and graduate students in mathematics and physics. 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 This graduate textbook on functional analysis offers a concise introduction into the field. The authors established a modern and modular structure which allows easy assimilation of content often displayed in much more elaborate ways. It is thus not only suitable for graduate students but also for advanced undergraduates.