Topological Vectors Spaces: Elements of Mathematics

This book is the English translation of the new and expanded version of Bourbaki's "Espaces vectoriels topologiques". Chapters 1 and 2 contain the general definitions and a thorough study of convexity; they are organized around the basic theorems (closed graph, Hahn-Banach and Krein-Milman), and differ only by minor changes from those of older editions. Chapter 3 and 4 have been substantially rewritten; the order of exposition has been modified and a number of notions and results have been inserted, whose importance emerged in the last twenty years. Bornological spaces are introduced together with barrelled ones; almost every space of practical use today belongs in fact to these two categories, which have good stability properties, and in which the basic theorems (the Banach-Steinhaus theorem for example) apply. Recent results on the completion of a dual space (Grothendieck theorem) or on the continuity of linear maps with measurable graphs are treated. An important place is devoted to properties of Fréchet spaces and of their dual spaces, to compactness criteria (Eberlein-Smullian) and to the existence of fixed points for groups of linear maps. Chapter 5 is devoted to Hilbert spaces; it includes in particular the spectral decomposition of Hilbert-Schmidt operators and the construction of symmetric and exterior powers of Hilbert spaces, whose applications are of growing importance. At the end, an appendix restates the principal results obtained in the case of normed spaces, providing convenient references. The book addresses all mathematicians and physicists interested in a structural presentation of contemporary mathematics.