Proyecciones Journal of Mathematics Vol. 31, N^o 2, pp. 149-164, June 2012. Universidad Católica del Norte Antofagasta - Chile

 

Uniform Convergence and the Hahn-Schur Theorem

 

Charles Swartz

New Mexico State University, U.S.A.

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ABSTRACT

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 ë 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 ë multiplier convergent with respect to w(E, F) if for each t = {tj} G ë ,the series Xj=! tj Xj is w(E,F) convergent in E. For multiplier spaces ë satisfying certain gliding hump properties we establish the following uniform convergence result: Suppose _j XX ij is ë multiplier convergent with respect to w(E, F) for each i G N and for each t G ë 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 ë 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 ë,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.

Key Words : Multiplier convergent series, uniform convergence, Hahn-Schur Theorem.

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Received : January 2012. Accepted : February 2012

Charles Swartz

Department of Mathematical Sciences

New Mexico State University

Las Cruces, NM 88003, U. S. A.

e-mail : cswartz@nmsu.edu