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 chi = {<FONT FACE=Symbol>c k</FONT>} such that {q(<FONT FACE=Symbol>c k</FONT>)} <FONT FACE=Symbol>Î</FONT>µ{X} for all q <FONT FACE=Symbol>Î</FONT> X. The space µ{X} is given the locally convex topology generated by the semi-norms <FONT FACE=Symbol>ðp</FONT>pq(chi) = p({q(<FONT FACE=Symbol>c k</FONT>)}), p <FONT FACE=Symbol>Î</FONT> X, q <FONT FACE=Symbol>Î</FONT> M. We show that if µ satisfies a certain multiplier type of gliding hump property, then pointwise bounded subsets of the â-dual of µ{X} with respect to a locally convex space are uniformly bounded on bounded subsets of µ{X}