Let X, Y be locally convex spaces and L(X, Y ) the space of continuous linear operators from X into Y . We consider 2 types of multiplier convergent theorems for a series ? T? in L(X, Y ). First, if ? is a scalar sequence space, we say that the series ? T? is ? multiplier P convergent for a locally convex topology ? on L(X, Y ) if the series ? t?T? is ? convergent for every t = {t?} ? ?. We establish conditions on ? which guarantee that a ? multiplier convergent series in the weak or strong operator topology is ? multiplier convergent in the topology of uniform convergence on the bounded subsets of X. Second, we consider vector valued multipliers. If E is a sequence space of X valued sequences, the series ? T? is E multiplier convergent in a locally convex topology ? on Y if the series ? T?x? is ? convergent for every x = {x?} ? E. We consider a gliding hump property on E which guarantees that a series ? T? which is E multiplier convergent for the weak topology of Y is E multiplier convergent for the strong topology of Y .