These are notes from a one-semester graduate course in Functional Analysis given by Prof. Sylvia Serfaty at the Courant Institute of Mathematical Sciences, New York University, in the Fall of 2004. Thanks to Atilla Yilmaz, Caroline M¨uller, and Alexey Kuptsov for providing their course notes to help with the preparation of this typed version. The course was largely based on Haim Brezis’ Analyse fonctionnelle : thorie et applications, and Michael Reed’s and Barry Simon’s Methods of modern mathematical physics vol. 1. The geometric versions of the Hahn-Banach Theorems were taken almost entirely out of Brezis and the section on Spectral Theory was based entirely on Reed and Simon.