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

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## Alberto Bressan:

## 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

### 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**