Università Roma Tre

Broad program:

Banach and Hilbert spaces, general properties, projections in Hilbert spaces, orthonormal systems.
Hahn-Banach theorem, analytical and geometric form, consequences.
First and second category spaces, Baire theorem, Banach-Steinhaus theorem, open map and closed graph, applications.
Weak, closed and convex topologies, Banach-Alaoglu theorem, separability and reflexivity.
Sobolev spaces in one dimension, Immersion theorems, Poincaré inequality, application to variational problems.
Spectral theory, Fredholm alternative, spectral theorem for compact and self-adjoint operators, application to variational problems.

H. Brezis - Analisi Funzionale - Liguori (1986)
H. Brezis - Functional Analysis, Sobolev Spaces and Partial Differential Equations - Springer (2010)
W. Rudin - Functional Analysis - McGraw-Hill (1991)