Elements of Functional Analysis

This book arose from a course taught for several years at the Univer- sity of Evry-Val d'Essonne. It is meant primarily for graduate students in mathematics. To make it into a useful tool, appropriate to their knowl- edge level, prerequisites have been reduced to a minimum: essentially, basic concepts of topology of metric spaces and in particular of normed spaces (convergence of sequences, continuity, compactness, completeness), of "ab- stract" integration theory with respect to a measure (especially Lebesgue measure), and of differential calculus in several variables. The book may also help more advanced students and researchers perfect their knowledge of certain topics. The index and the relative independence of the chapters should make this type of usage easy. The important role played by exercises is one of the distinguishing fea- tures of this work. The exercises are very numerous and written in detail, with hints that should allow the reader to overcome any difficulty. Answers that do not appear in the statements are collected at the end of the volume. There are also many simple application exercises to test the reader's understanding of the text, and exercises containing examples and coun- terexamples, applications of the main results from the text, or digressions to introduce new concepts and present important applications. Thus the text and the exercises are intimately connected and complement each other.