Functional Analysis Notes（深入学习泛函分析的好书）

1 Hahn-Banach Theorems and Introduction to Convex Conjugation 1 1.1 Hahn-Banach Theorem - Analytic Form . . . . . . . . . . . . . . 1 1.1.1 Theorems on Extension of Linear Functionals . . . . . . . 1 1.1.2 Applications of the Hahn-Banach Theorem . . . . . . . . 3 1.2 Hahn-Banach Theorems - Geometric Versions . . . . . . . . . . . 5 1.2.1 Definitions and Preliminaries . . . . . . . . . . . . . . . . 5 1.2.2 Separation of a Point and a Convex Set . . . . . . . . . . 6 1.2.3 Applications (Krein-Milman Theorem) . . . . . . . . . . . 8 1.3 Introduction to the Theory of Convex Conjugate Functions . . . 9 2 Baire Category Theorem and Its Applications 13 2.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.1 Reminders on Banach Spaces . . . . . . . . . . . . . . . . 13 2.1.2 Bounded Linear Transformations . . . . . . . . . . . . . . 13 2.1.3 Duals and Double Duals . . . . . . . . . . . . . . . . . . . 15 2.2 The Baire Category Theorem . . . . . . . . . . . . . . . . . . . . 16 2.3 The Uniform Boundedness Principle . . . . . . . . . . . . . . . . 17 2.4 The Open Mapping Theorem and Closed Graph Theorem . . . . 18 3 Weak Topology 21 3.1 General Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Frechet Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3 Weak Topology in Banach Spaces . . . . . . . . . . . . . . . . . . 24 3.4 Weak-* Topologies (X,X) . . . . . . . . . . . . . . . . . . . . 28 3.5 Reflexive Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.6 Separable Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.7.1 Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.7.2 PDE’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4 Bounded (Linear) Operators and Spectral Theory 37 4.1 Topologies on Bounded Operators . . . . . . . . . . . . . . . . . 37 4.2 Adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.3 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.4 Positive Operators and Polar Decomposition (In a Hilbert Space) 46 5 Compact and Fredholm Operators 47 5.1 Definitions and Basic Properties . . . . . . . . . . . . . . . . . . 47 5.2 Riesz-Fredholm Theory . . . . . . . . . . . . . . . . . . . . . . . 49 5.3 Fredholm Operators . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.4 Spectrum of Compact Operators . . . . . . . . . . . . . . . . . . 52 5.5 Spectral Decomposition of Compact, Self-Adjoint Operators in Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 A 57