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Functional Analysis Notes by Prof. Sylvia Serfaty

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Functional Analysis Notes written by Prof. Sylvia Serfaty, Courant Institute of Mathematical Sciences, New York University . This is an other book of mathematics cover the following topics.

  • 1. Hahn-Banach Theorems and Introduction to Convex Conjugation
    1.1 Hahn-Banach Theorem - Analytic Form
    1.1.1 Theorems on Extension of Linear Functionals
    1.1.2 Applications of the Hahn-Banach Theorem
    1.2 Hahn-Banach Theorems - Geometric Versions
    1.2.1 Definitions and Preliminaries
    1.2.2 Separation of a Point and a Convex Set
    1.2.3 Applications (Krein-Milman Theorem)
    1.3 Introduction to the Theory of Convex Conjugate Functions

  • 2. Baire Category Theorem and Its Applications
    2.1 Review
    2.1.1 Reminders on Banach Spaces
    2.1.2 Bounded Linear Transformations
    2.1.3 Duals and Double Duals
    2.2 The Baire Category Theorem
    2.3 The Uniform Boundedness Principle
    2.4 The Open Mapping Theorem and Closed Graph Theorem

  • 3. Weak Topology
    3.1 General Topology
    3.2 Frechet Spaces
    3.3 Weak Topology in Banach Spaces
    3.4 Weak-* Topologies s(X*, X)
    3.5 Reflexive Spaces
    3.6 Separable Spaces
    3.7 Applications
    3.7.1 Lp Spaces
    3.7.2 PDE’s

  • 4. Bounded (Linear) Operators and Spectral Theory
    4.1 Topologies on Bounded Operators
    4.2 Adjoint
    4.3 Spectrum
    4.4 Positive Operators and Polar Decomposition (In a Hilbert Space)

  • 5. Compact and Fredholm Operators
    5.1 Definitions and Basic Properties
    5.2 Riesz-Fredholm Theory
    5.3 Fredholm Operators
    5.4 Spectrum of Compact Operators
    5.5 Spectral Decomposition of Compact, Self-Adjoint Operators in Hilbert Space

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