Syllabus:

1. Review of linear algebra (with Fredholm theory and perturbation theory in mind);
2. Topological vector spaces;
3. The Hahn-Banach theorem with applications such as the Müntz-Szász theorem and Runge's approximation theorem;
4. The open mapping, closed graph and uniform boundedness theorems with applications to partial differential equations (such as Lewy's non-solvability);
5. Fredholm theory; Noether's index theorem for Toeplitz operators;
6. The Banach-Alaoglu theorem; Malgrange's theorem on the range of PDE with constant coefficients (as an application);
7. Symmetric and self-adjoint operators on Hilbert spaces;
8. The spectral theorem; spectral decomposition of ordinary differential operators and of Schrödinger operators.

Grading: