Infinite-dimensional linear spaces, Banach and Hilbert spaces.

Geometry of Hilbert spaces, lemma of Riesz for Banach spaces.

Linear functionals, Hahn-Banach theorem, weak convergence, uniform boundedness principle.

Linear operators, open mapping theorem and closed graph theorem for Hilbert spaces.

Compact operators, Fredholm theory.

Spectral theory for compact self-adjoint operators and applications to ODEs.

Introduction to Sobolev spaces