These notes are an expanded version of a set written for a course given to final-yearundergraduates at the University of Oxford.A thorough understanding of the Oxford third-year b4 analysis course (an introductionto Banach and Hilbert spaces) or its equivalent is a prerequisite for this material. Weuse [24] as a compendium of results from that series of lectures. (Numbers in squarebrackets refer to items in the bibliography.)The author acknowledges his debt to all those from whom he has learnt functionalanalysis, especially Professor D. A. Edwards, Dr G. R. Allen and Dr J. M. Lindsay. Thestudents attending the course were very helpful, especially Mr A. Evseev, Mr L. Taitzand Ms P. Iley.This document was typeset using LATEX2ε with Peter Wilson’s memoir class and theAMS-LATEX and XY-pic packages. The index was produced with the aid of the MakeIndexprogram.This edition contains a few additional exercises and the electronic version is equippedwith hyperlinks, thanks to the hyperref package of Sebastian Rahtz and Heiko Oberdiek.