Resumen de Familles sommables dans les espaces vectoriels topologiques.

Michel Mazan

• Let E and F be two vector spaces in separating duality. Let us consider T0, the uniform convergence topology on E on the partial sums of families of F which are weakly summable to 0 in F; then, if (E',T'0) is the completion of (E,T0), the finest locally convex topology T on F for which all the weakly summable families in F are also T-summable, is the uniform convergence topology on the T'0-compact subsets of E'. If F is a Banach space and E its dual space F', every weakly summable family in F is summable in F if and only if the closed unit ball of F' is a T0-compact set. If TZ is further the uniform convergence topology on F on the partial sums of absolutely summable and Ts(F)-summable to 0 sequences of F', F is reflexive if and only if F is Tz-complete. We give also in appendix a generalization of the Orlicz-Pettis theorem.