Let E be a vector space, F aset, G be a locally convex space, b : E X F → G a map such that b(·,y): E → G is linear for every y ∈ F; we write b(x, y) = x · y for brevity. Let λ 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 ∈ F .A series ΣjXj in X is λ multiplier convergent with respect to w(E, F) if for each t = {tj} ∈ λ ,the series Σj=1∞ tjxj  is w(E,F) convergent in E. For multiplier spaces λ satisfying certain gliding hump properties we establish the following uniform convergence result: Suppose Σjxij   is λ multiplier convergent with respect to w(E, F) for each i ∈ N and for each t ∈ λ the set {Σj=1∞ tjxij  : i} is uniformly bounded on any subset B C F such that {x · y : y ∈ B} is bounded for x ∈ E.Then for each t ∈ λ the series Σj=1∞ tjxij · y converge uniformly for y ∈ B,i ∈ N. This result is used to prove a Hahn-Schur Theorem for series such that limi Σj=1∞ tjxij · y exists for t ∈ λ,y ∈ 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.