Resumen de Uniform Convergence and the Hahn-Schur Theorem

Charles Swartz

• Let E be a vector space, F aset, G be a locally convex space, b : E X F — G a map such that ò(-,y): E — G is linear for every y G F; we write b(x, y) = x · y for brevity. Let λ be a scalar sequence space and w(E,F) the weakest topology on E such that the linear maps b(-,y): E — G are continuous for all y G F .A series Xj in X is λ multiplier convergent with respect to w(E, F) if for each t = {tj} G λ ,the series Xj=! tj Xj is w(E,F) convergent in E. For multiplier spaces λ satisfying certain gliding hump properties we establish the following uniform convergence result: Suppose j XX ij is λ multiplier convergent with respect to w(E, F) for each i G N and for each t G λ the set {Xj=! tj Xj : i} is uniformly bounded on any subset B C F such that {x · y : y G B} is bounded for x G E.Then for each t G λ the series ^jjLi tj xj · y converge uniformly for y G B,i G N. This result is used to prove a Hahn-Schur Theorem for series such that lim¿ Xj=! tj xj · y exists for t G λ,y G F.