The first part of the course will deal with the spectral theory of bounded and unbounded self-adjoint operators and one-parameter unitary groups (Stone's Theorem), developing both the abstract theory and applications to ordinary and partial differential equations. The second (very short) part will deal with fixed point theorems in infinite-dimensional spaces (especially Schauder's fixed point theorem), and applications to nonlinear PDE's. The third part will pursue the development of the theory of Sobolev spaces, continuing the work done by H. Brezis in Math 507.
No textbook. Brezis' notes will be used especially for the Sobolev spaces part.
Math 507 or something equivalent.