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

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