%0 Journal Article
%T Uniform Convergence and the Hahn-Schur Theorem
%A Swartz
%A Charles
%J Proyecciones (Antofagasta)
%D 2012
%I Universidad Cat¨®lica del Norte
%R 10.4067/S0716-09172012000200004
%X let e be a vector space, f aset, g be a locally convex space, b : e x f - g a map such that ¨°(-,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.
%K multiplier convergent series
%K uniform convergence
%K hahn-schur theorem.
%U http://www.scielo.cl/scielo.php?script=sci_abstract&pid=S0716-09172012000200004&lng=en&nrm=iso&tlng=en