(Where relevant, I attempted in these notes to formulate the course's results in the setting of topological vector spaces) - Topological duals, quotients, complemented subspaces; - Hahn–Banach (extension & separation); - reflexivity; - weak & weak* topology; - Banach–Alaoglu; - Goldstine; - Schur property; - Compact & weakly compact operators; - Schauder’s theorem; - Closed range & Fredholm theory (index, perturbation, Fredholm alternative); - Spectral theory of compact (self-adjoint) operators; - Schauder bases; examples in classical Banach spaces.