Book:Haïm Brezis/Functional Analysis, Sobolev Spaces and Partial Differential Equations

Subject Matter

 * Functional Analysis

Contents
Preface


 * 1. The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions


 * 1.1 The Analytic Form of the Hahn-Banach Theorem: Extension of Linear Functionals


 * 1.2 The Geometric Forms of the Hahn-Banach Theorem: Separation of Convex Sets


 * 1.3 The Bidual $E^{**}$. Orthogonality Relations


 * 1.4 A Quick Introduction to the Theory of Conjugate Convex Functions


 * Comments on Chapter 1


 * Exercises for Chapter 1


 * 2. The Uniform Boundedness Principle and the Closed Graph Theorem


 * 2.1 The Baire Category Theorem


 * 2.2 The Uniform Boundedness Principle


 * 2.3 The Open Mapping Theorem and the Closed Graph Theorem


 * 2.4 Complimentary Subspaces. Right and Left Invertability of Linear Operators


 * 2.5 Orthogonality revisited


 * 2.6 An Introduction to Unbounded Linear Operators. Definition of the Adjoint


 * 2.7 A Characterization of Operators with Closed Range. A Characterization of Surjective Operators


 * Comments on Chapter 2


 * Exercises for Chapter 2


 * 3. Weak Topologies, Reflexive Spaces, Separable Spaces, Uniform Convexity


 * 3.1 The Coarsest Topology for Which a Collection of Maps Becomes Continuous


 * 3.2 Definition and Elementary Properties of the Weak Topology $\map \sigma {E,E^*}$


 * 3.3 Weak Topology, Convex Sets and Linear Operators


 * 3.4 The Weak* Topology $\map \sigma {E^*,E}$


 * 3.5 Reflexive Spaces


 * 3.6 Separable Spaces


 * 3.7 Uniformly Convex Spaces


 * Comments on Chapter 3


 * Exercises for Chapter 3


 * 4. $L^p$ Spaces


 * 5. Hilbert Spaces


 * 6. Compact Operators, Spectral Decomposition of Self-Adjoint Compact Operators


 * 7. The Hille-Yosida Problem


 * 8. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension


 * 9. Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in $N$ Dimensions


 * 10. Evolution Problems: the Heat Equation and the Wave Equation


 * 11. Miscellaneous Complements

Solutions of Some Exercises

Problems

Partial Solutions

Notation

References

Index