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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
Applied Functional Analysis:
An open source version
Thomas Hillen
Department of Mathematical and Statistical Sciences
University of Alberta
thillen@ualberta.ca
Roberto De Leo
Department of Mathematics
Howard University
roberto.deleo@howard.edu
None, Editor
Yet
April 23, 2026
Preface
