Functional Analysis

  1. Fundamental Theorems of Functional Analysis
    1. Hahn-Banach Theorem
      1. Analytic Form
        1. Statement for Real Spaces
          1. Statement for Complex Spaces
            1. Proof Outline
            2. Geometric Form
              1. Separation of Convex Sets
                1. Supporting Hyperplanes
                2. Applications
                  1. Extension of Linear Functionals
                    1. Existence of Non-Trivial Functionals
                      1. Norm-Preserving Extensions
                    2. Uniform Boundedness Principle
                      1. Banach-Steinhaus Theorem
                        1. Pointwise vs Uniform Boundedness
                          1. Applications
                            1. Convergence of Operator Sequences
                              1. Unbounded Sequences of Operators
                              2. Corollaries
                              3. Open Mapping Theorem
                                1. Statement and Proof Outline
                                  1. Open Mappings
                                    1. Bounded Inverse Theorem
                                      1. Statement
                                        1. Relationship to Open Mapping
                                      2. Closed Graph Theorem
                                        1. Graph of an Operator
                                          1. Closed Graph Property
                                            1. Statement and Applications
                                              1. Relationship to Boundedness

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                                            5. Linear Operators on Normed Spaces

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                                            7. Duality Theory