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.