Let µ be a normal scalar sequence space which is a K-space under the family of semi-norms M and let X be a locally convex space whose topology is generated by the family of semi-norms X. The space µ{X} is the space of all X valued sequences x = {xk} such that {q(xₖ)} ∈ µ{X} for all q ∈ X. The space µ{X} is given the locally convex topology generated by the semi-norms πpq(x) = p({q(xₖ)}), p ∈ X, q ∈ M.We show that if µ satisfies a certain multiplier type of gliding hump property, then pointwise bounded subsets of the β-dual of µ{X} with respect to a locally convex space are uniformly bounded on bounded subsets of µ{X}.