Functional Analysis

Author

John Robin Inston

Published

September 25, 2026

1 Functional Analysis

1.1 Foundations

  • Metric Spaces
    • Normed Spaces
  • [[banarch-spaces]]
    • Hahn-Banach Theorem
    • Bounded Linear Operators
    • Banach-Steinhaus
      • Principle of Uniform Boundedness
    • Open Mapping and Closed Graph Theorems
  • Hilbert Spaces
    • Inner Product Spaces
    • Orthonormal Bases
      • Gram-Schmidt Process
    • Riesz Representation Theorem
    • Projection Theorem
    • Parseval’s Identity
      • Bessel’s Inequality
  • [[functional]]

1.2 Operator Theory

  • Bounded Linear Operators
    • Operator norm, continuity, dual operators
    • Compact Operators
    • Spectral theory of compact operators
  • Unbounded Operators
    • Closable and closed operators
    • Domains and graphs
    • Symmetric and self-adjoint operators
  • Spectral Theory (for Banach and Hilbert Spaces)
    • Spectrum, resolvent set
    • Spectral Theorem for compact self-adjoint operators
    • Functional calculus for self-adjoint operators

1.3 Duality and Weak Topologies

  • Dual Spaces
    • Reflexivity
    • Weak and weak-* convergence
    • Alaoglu’s Theorem
    • Goldstine’s and Mazur’s Theorems
  • Topological Vector Spaces
    • Locally convex spaces
    • Fréchet and LF spaces
    • Seminorms and Minkowski functionals

1.4 Distributions and Function Spaces

  • Schwartz Space and Tempered Distributions

    • Test functions, duality

    • Fourier transform on Schwartz space

    • Sobolev spaces Wk,pWk,p, Sobolev Embedding Theorems

  • Interpolation and Compactness

    • Rellich-Kondrachov Theorem

    • Lions–Magenes theory (for PDEs)

1.4.1 V. Spectral Theory and Functional Calculus

  • Spectral Theorem for Unbounded Self-Adjoint Operators

  • Functional Calculus (Borel, holomorphic)

  • Stone’s Theorem on One-Parameter Unitary Groups

1.4.2 VI. Applications to PDEs, Quantum Mechanics, and Optimization

  • Weak Formulations and Variational Methods

    • Lax-Milgram Theorem

    • Galerkin method

  • Semigroups of Operators

    • Hille–Yosida Theorem

    • Generation of C0C0​-semigroups

    • Applications to linear evolution equations (heat, Schrödinger, wave)

  • Quantum Mechanics

    • Rigged Hilbert spaces

    • Self-adjointness and physical observables

    • Spectral decomposition of Hamiltonians

1.4.3 VII. Nonlinear Functional Analysis

  • Fixed Point Theorems

    • Banach, Schauder, Brouwer fixed point theorems

    • Applications to nonlinear PDEs

  • Monotone Operator Theory

    • Minty-Browder Theorem

    • Maximal monotone operators

  • Degree Theory, Variational Inequalities

1.4.4 VIII. Modern and Research-Level Topics

  • Operator Algebras

    • C∗C∗-algebras and von Neumann algebras

    • Gelfand-Naimark Theorem

    • Noncommutative geometry

  • Functional Analytic Methods in Stochastic Processes

    • Dirichlet forms

    • Infinite-dimensional Gaussian measures

  • Banach Space Geometry

    • Type and cotype

    • Radon–Nikodym property

    • Asymptotic theory of Banach spaces (e.g., Ribe program)

  • Advanced Spectral and Scattering Theory

    • Mourre theory

    • Resonances and scattering poles

  • Category-Theoretic and Topos-Theoretic Approaches

    • Derived functional analysis

    • Nonlinear topological vector spaces

1.5 Backlinks

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