Functional Analysis

  1. Duality Theory
    1. Dual Spaces
      1. Definition of Dual Space X*
        1. Dual Norm
          1. Examples of Dual Spaces
            1. Dual of ℓₚ Spaces
              1. Dual of Lₚ Spaces
                1. Dual of C[a,b]
                  1. Dual of c₀
                2. Second Dual and Reflexivity
                  1. Second Dual Space X**
                    1. Canonical Embedding
                      1. Reflexive Spaces
                        1. Examples and Non-Examples
                          1. Properties of Reflexive Spaces
                          2. Weak Topologies
                            1. Weak Topology σ(X,X*)
                              1. Weak-* Topology σ(X*,X)
                                1. Basic Neighborhoods
                                  1. Relationship to Norm Topology
                                  2. Weak Convergence
                                    1. Weak Convergence in X
                                      1. Weak-* Convergence in X*
                                        1. Characterizations
                                          1. Examples and Counterexamples
                                            1. Relationship to Norm Convergence
                                            2. Compactness in Weak Topologies
                                              1. Banach-Alaoglu Theorem
                                                1. Weak Compactness
                                                  1. Eberlein-Šmulian Theorem

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                                                6. Fundamental Theorems of Functional Analysis

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                                                8. Spectral Theory