Functional Analysis

  1. Advanced Topics
    1. Unbounded Operators
      1. Definition and Domain
        1. Closed Operators
          1. Closable Operators
            1. Graph Norm
              1. Core of an Operator
                1. Adjoints of Unbounded Operators
                  1. Symmetric and Self-Adjoint Operators
                    1. Spectral Theory of Unbounded Operators
                    2. Operator Algebras
                      1. Banach Algebras
                        1. Definition and Examples
                          1. Spectrum in Banach Algebras
                            1. Gelfand Theory
                            2. C*-Algebras
                              1. Definition and Basic Properties
                                1. Examples of C*-Algebras
                                  1. Gelfand-Naimark Theorem
                                2. Distributions and Sobolev Spaces
                                  1. Test Functions
                                    1. Distributions as Linear Functionals
                                      1. Operations on Distributions
                                        1. Weak Derivatives
                                          1. Sobolev Spaces
                                            1. Definition and Properties
                                              1. Embedding Theorems
                                                1. Trace Theorems
                                                2. Applications to PDEs
                                                  1. Weak Solutions
                                                    1. Variational Formulations
                                                      1. Existence and Uniqueness Results

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                                                  8. Spectral Theory

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                                                  1. Preliminaries and Foundational Concepts