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.