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Applied Functional Analysis:
An open source version
Thomas Hillen, Roberto De Leo, None (Editor)
Contents
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Contents
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Front Matter
Preface
1
Introduction
Applied mathematics
Partial Differential Equations
Spectrum
Compactness
Optimization
Fixed Point Methods
Outline
Recommended Literature
References and Suggested Readings
2
Basic space structures
Topology
Topological Vector Spaces
Metric spaces
Complete Metric Spaces
Norms
Scalar Products
Exercises
References and Suggested Readings
3
Basic Functional Analysis Spaces
Banach spaces
Hilbert spaces
Mollifiers
Differentiable functions
Inequalities
Integrable functions
Fourier series
Exercises
References and Suggested Readings
4
Linear Operators
Introduction
Bounded Operators
Compact Operators
Three fundamental results on mappings
Closed Operators
Closed operators in Hilbert spaces
A Glance Ahead to Spectral Theory
Fractional Powers
Exercises
References and Suggested Readings
5
Dual Spaces
Bounded linear functionals
The Hahn-Banach Theorem
The adjoint map
Locally Convex spaces
Dual Spaces
Weak Topology
Weak\(^*\) Topology
Exercises
6
Sobolev Spaces
Distributional and Weak Derivatives
Sobolev Spaces
Embeddings
Exercises
7
Fixed Point Theorems
The Banach Fixed-Point Theorem
The Brouwer and Schauder fixed-point theorems
The Leray-Schauder Principle
The Lax Milgram Lemma
References and Suggested Readings
8
Spectral Theory
Spectrum and resolvent of an operator
Adjoint Operators
Self-Adjoint Operators
Spectral Decomposition
Fredholm Alternatives
Summary of Spectral Theory
Exercises
References and Suggested Readings
9
Semiflows
Introduction
Banach-Space Valued Functions
The Infinitesimal Generator
Solutions to Abstract ODEs
Flow of an ODE
The Hille-Yosida and Stone Theorems for self-adjoint operators
The Hille-Yosida Theorem
The Lumer-Phillips Theorem
Application to PDEs
Bounded Perturbations
Analytic Semigroups
Supplemental Material
Semigroup Summary
Exercises
References and Suggested Readings
Authored in PreTeXt
Chapter
2
Basic space structures
We review some basic concepts that we will use throughout the book.
2.1
Topology
2.2
Topological Vector Spaces
2.3
Metric spaces
2.4
Complete Metric Spaces
2.5
Norms
2.6
Scalar Products
2.7
Exercises
2.8
References and Suggested Readings