This is a softcover reprint of the English translation of 1987 of the second edition of Bourbaki's Espaces Vectoriels Topologiques (1981).

This Äsecond editionÜ is a brand new book and completely supersedes the original version of nearly 30 years ago. But a lot of the material has been rearranged, rewritten, or replaced by a more up-to-date exposition, and a good deal of new material has been incorporated in this book, all reflecting the progress made in the field during the last three decades.

Table of Contents.

Chapter I: Topological vector spaces over a valued field.

Chapter II: Convex sets and locally convex spaces.

Chapter III: Spaces of continuous linear mappings.

Chapter IV: Duality in topological vector spaces.

Chapter V: Hilbert spaces (elementary theory).

Finally, there are the usual "historical note", bibliography, index of notation, index of terminology, and a list of some important properties of Banach spaces.

in E x R. For every x E X, the hyperplane H separates (x, f(x») strictly from (x, b), and therefore meets the line {x} x R in a single point w(x); thus H is the graph of a continuous PROPOSITION affine function whose restriction w to X is a member of L, that is a lower bound for u and v and that satisfies the inequality w(x) > f(x) for all x E X. This proves that the set L' is decreasing directed. Prop. 5 of II, p. 39, applied to - f shows that f is the lower envelope of L'. Let f be a

points (x, 1;) of R" where x E B, Y E R and fl (x) 0( 1; 0( - fz(x). 37) Lct E be a vector space; in order that a convex set F of E x R should be formed of pairs 0( 1; (resp. f(x) < 1;) for a convex function I defined over a convex subset X of E, it is necessary and sufficient that the projection of F on E should be identical with X and that, for all x E X, the set F(x) of F that projects onto x should be a closed (resp. open) interval unlimited to the right (i.e. not bounded above). (x, 1;)

subspace. 'IT 26) Let E be an infinite dimensional Frechet space of enumerable type. a) Show that there exists a sequence (a.) of linearly independent elements of E such that each sequence (a 2 .) and (a 2n + 1) is total (use exerc. 24, c». b) Let F be the vector subspace of E generated by the a 2n + 1 (n EN). For every n > 0 let Mn be the subspace generated by the a 2k with k ~ n. For each n, let

set of \E is contained in a set of \E 1 . The intersection of a family of bomologies on E is a bomology; consequently for every subset 6 of I.]3(E), there exists a smallest bomology containing 6; this bomology is said to be generated by 6 and admits as a base the set of finite unions of sets of 6. If E and E' are two sets, and \E (resp. \E') a bomology on E (resp. E'), TVS 111.2 SPACES OF CONTINUOUS LINEAR MAPPINGS §I the product bomology is the bomology on E x E' which admits the sets M x

K is a nondiscrete valued division ring (1 I, p. 7, prop. 4) is not valid for all topological vector spaces over K as the preceding example shows. 4. Locally compact topological vector spaces 3. Let K be a complete non-discrete valued division ring. If E is a Hausdorff topological vector space over K, which is such that some neighbourhood V of 0 in E is precompact, then E is of finite dimension. If E =1= { 0 }, then both K and E are locally compact. In proving the first assertion, we need