Functional analysis

Syllabus

The first half of the course is a core curriculum; we then offer two options.

Common core: Functional analysis. Main analysis theorems derived from topology: Baire theorem, Banach-Steinhaus theorem, open application and closed graph theorem, Ascoli theorem. L^p spaces: their topology, convolution, density theorems. Hilbert spaces: projection theorem, Riesz representation theorem.

Fundamental option: Duality and elements of spectral theory. Duality in Banach and Hilbert spaces: Hahn-Banach theorem and applications, weak convergence, reflexive spaces, selection theorem. Bounded operators and their spectrum, compact operators.

Applied option: Convex optimization. Convex optimization. Descent algorithms. Krylov's methods. Projection and Uzawa methods for constrained provlems.

Prerequisites