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

Preface

Chapter 1  Introduction
     1.1. Linear equations
     1.2. Evolution equations
     1.3. Function spaces
     1.4. Compactness

Chapter 2  Banach Spaces
     2.1. Basic definitions
          2.1.1. Examples of normed and Banach spaces.
     2.2. Linear operators
          2.2.1. Examples of 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
          2.7.1. Weak convergence.
          2.7.2. Weak-star convergence
     2.8. Problems

Chapter 3  Spaces of Continuous Functions
     3.1. Bounded continuous functions
     3.2. The Stone-Weierstrass approximation theorem
          3.2.1. Complex-valued functions
     3.3. Ascoli's compactness theorem
     3.4. Spaces of Holder continuous functions
     3.5. Problems

Chapter 4  Bounded Linear Operators
     4.1. The uniform boundedness principle
     4.2. The open mapping theorem
     4.3. The closed graph theorem
     4.4. Adjoint operators
     4.5. Compact operators
          4.5.1. Integral 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
          5.5.1. Fourier series.
     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
          7.1.1. Linear homogeneous ODEs.
     7.2. Semigroups of linear operators
          7.2.1. Definition and basic properties of semigroups
     7.3. Resolvents
     7.4. Generation of a semigroup
     7.5. Problems

Chapter 8  Sobolev Spaces
     8.1. Distributions and weak derivatives
          8.1.1. Distributions.
          8.1.2. Weak derivatives.
     8.2. Mollificat ions
     8.3. Sobolev spaces
     8.4. Approximations of Sobolev functions
     8.5. Extension operators
     8.6. Embedding theorems
          8.6.1. Morrey's inequality
          8.6.2. The Gagliardo-Nirenberg inequality
          8.6.3. High-order Sobolev estimates.
     8.7. Compact embeddings
     8.8. Differentiability properties
     8.9. Problems

Chapter 9  Linear Partial Differential Equations
     9.1. Elliptic equations
          9.1.1. Physical interpretation.
          9.1.2. Classical and weak solutions
          9.1.3. Homogeneous second-order elliptic operators
          9.1.4. Representation of solutions in terms of eigenfunctions
          9.1.5. More general linear elliptic operators.
     9.2. Parabolic equations
          9.2.1. Representation of solutions in terms of eigenfunctions
          9.2.2. More general operators
     9.3. Hyperbolic equations
     9.4 Problems

Appendix  Background Material
     A.1. Partially ordered sets
     A.2. Metric and topological spaces
          A.2.1. Fixed points of contractive maps
          A.2.2. The Baire category theorem
     A.3. Review of Lebesgue measure theory
          A.3.1. Measurable sets
          A.3.2. Lebesgue integration
     A.4. Integrals of functions taking values in a Banach space
     A.5. Mollifications
          A.5.1. Partitions of unity.
     A.6. Inequalities
          A.6.1. Convex sets and convex functions
          A.6.2. Basic inequalities
          A.6.3. A differential inequality
     A.7. Problems

Summary of Notation

Bibliography

Index