Let E1, E2 be Hausdorff locally convex spaces with E2 quasi-complete, and T : E1 → E2 a continuous linear map. Then T maps bounded sets of E1 into relatively weakly compact subsets of E2 if and only if T is continuous with quasi-Mackey topology on E1. If E1 has quasi-Mackey topology and E2 is quasi-complete, then a sequentially continuous linear map T : E1 → E2 is an unconditionally converging operator.