Applied Functional Analysis

The course introduces those notions and results of functional analysis that are relevant for applications to, e.g., partial differential equations, numerical analysis, signal processing, probability theory, stochastic analysis, and quantum theory. More precisely, the course covers the following subjects:

- Banach spaces: norms, duality, Hahn-Banach theorems.

- Hilbert spaces: best approximation, orthogonality, orthocomplementation, orthonormal bases, Riesz representation theorem.

- C(K)-spaces and L^p-spaces: completeness, duality, compactness, approximation techniques.

- Bounded linear operators: uniform boundedness theorem, open mapping theorem, closed graph theorem, compact operators, adjoint operators, spectrum and resolvent, spectral projections, Riesz-Schauder theory, Fredholm operators.

- Hilbert space operators: orthogonal projections, unitary operators, self-adjoint operators, normal operators, spectral theorem for compact normal operators.

- densely defined operators, closed operators.

- weak derivatives, Sobolev spaces.