The basic structures of linear functional analysis: Banach spaces, Hilbert spaces, locally convex topological vector spaces. Examples of spaces of functions and function-like objects: Lebesgue spaces, Sobolev spaces, spaces of holomorphic functions, spaces of distributions. Bounded linear operators. The standard classical theorems: Hahn-Banach, open mapping, closed graph, Banach-Steinhaus, Bourbaki-Alaoglu, nonemptyness of the spectrum, elementary properties of the resolvent. Spectral theory of compact operators. Applications to eigenfunction expansions. The Schauder fixed point theorem. Extreme points of convex sets in locally convex spaces and the Krein-Milman theorem. Elementary facts about Hilbert spaces.
"Functional Analysis, Sobolev Spaces and Partial Differential Equations"
(Universitext) by Haim Brezis, Springer, ISBN-13: 978-0387709130, ISBN-10: 0387709134
640:502 or equivalent