Functional Analysis

  • Instructor: Dávid KUNSZENTI-KOVÁCS
  • Contact: mathkukac at gmail com
  • Prerequisites: Linear Algebra: Euclidean spaces, matrices, eigenvalues and eigenvectors, diagonalisation of symmetric matrices;
    Analysis/Calculus: convergence/limits in Euclidean spaces, uniform continuity, Taylor/power series, uniform convergence and its connection to integrals and derivatives of limit functions.
    Familiarity with metric spaces/topology is not required, the required basics will be covered in class.
  • Text: A Course in Functional Analysis, by John Conway, Springer 2007

Course description: The course provides an introduction to functional analysis. Only some knowledge of calculus and linear algebra is assumed. As time permits at the end of the semester, we shall have a look at unbounded linear operators and applications to PDE theory.

Topics covered:

  • Normed spaces, Banach spaces: standard examples of function spaces, bounded linear operators, linear functionals, dual spaces and weak topologies
  • Hilbert spaces: inner products, orthogonal complements, representation of linear functionals, adjoint operator, self-adjoint, unitary and normal operators.
  • Fundamental theorems of functional analysis: Hahn-Banach theorem, Uniform Boundedness Theorem, Open Mapping Theorem, Closed Graph Theorem.
  • Spectral theory: resolvent and spectrum, bounded self-adjoint operators, compact operators.
  • Unbounded linear operators and applications to PDE theory.

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