%0 Journal Article
%T Uniform Convergence and the Hahn-Schur Theorem
%A Charles Swartz
%J Proyecciones (Antofagasta)
%D 2012
%I Universidad Cat¨®lica del Norte
%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_arttext&pid=S0716-09172012000200004