Book:Alberto Bressan/Lecture Notes on Functional Analysis: With Applications to Linear Partial Differential Equations

Alberto Bressan: Lecture Notes on Functional Analysis: With Applications to Linear Partial Differential Equations

Published $\text {2012}$, American Mathematical Society

ISBN 978-0821887714

Subject Matter

• Functional Analysis
• Partial Differential Equations

Contents

Preface

Chapter 1. Introduction

1.1 Linear equations

1.2 Evolution equations

1.3 Functions spaces

1.4 Compactness

Chapter 2. Banach Spaces

2.1 Basic definitions

2.2 Linear operators

2.3 Finite-dimensional spaces

2.4 Seminorms and Frechet spaces

2.5 Extension theorems

2.6 Separation of convex sets

2.7 Dual spaces and weak convergence

2.8 Problems

Chapter 3. Spaces of Continuous Functions

3.1 Bounded continuous functions

3.2 The Stone-Weierstrass approximation theorem

3.3 Ascoli's compactness theorem

3.4 Spaces of Holder continuous functions

3.5 Problems

Chapter 4. Bounded Linear Operators

4.1 The uniforms boundedness principle

4.2 The open mapping theorem

4.3 The closed graph theorem

4.4 Adjoint operators

4.5 Compact operators

4.6 Problems

Chapter 5. Hilbert Spaces

5.1 Spaces with an inner product

5.2 Orthogonal projections

5.3 Linear functionals on a Hilbert space

5.4 Gram-Schmidt orthogonalization

5.5 Orthonormal sets

5.6 Positive definite operators

5.7 Weak convergence

5.8 Problems

Chapter 6. Compact Operators on a Hilbert Space

6.1 Fredholm theory

6.2 Spectrum of a compact operator

6.3 Selfadjoint operators

6.4 Problems

Chapter 7. Semigroups of Linear Operators

7.1 Ordinary differential equations in a Banach space

7.2 Semigroups of linear operators

7.3 Resolvents

7.4 Generation of a semigroup

7.5 Problems

Chapter 8. Sobolev Spaces

8.1 Distributions and weak derivatives

8.2 Mollifications

8.3 Sobolev spaces

8.4 Approximations of Sobolev functions

8.5 Extension operators

8.6 Embedding theorems

8.7 Compact embeddings

8.8 Differentiability properties

8.9 Problems

Chapter 9. Linear Partial Differential Equations

9.1 Elliptic equations

9.2 Parabolic equations

9.3 Hyperbolic equations

9.4 Problems

Appendix. Background Material

A.1 Partially ordered sets

A.2 Metric and topological spaces

A.3 Review of Lebesgue measure theory

A.4 Integrals of functions taking values in a Banach space

A.5 Mollifications

A.6 Inequalities

A.7 Problems

Summary of Notation

Bibliography

Index