Uniform Convergence and the Hahn-Schur Theorem

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 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 multiplier convergent with respect to w(E, F) if for each t = {tj} G ,the series Xj=! tj Xj is w(E,F) convergent in E. For multiplier spaces satisfying certain gliding hump properties we establish the following uniform convergence result: Suppose j XX ij is multiplier convergent with respect to w(E, F) for each i G N and for each t G 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 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 ,y G F. Applications of these abstract results are given to spaces of linear operators, vector spaces in duality, spaces of continuous functions and spaces with Schauder bases.