Weak Sequential Convergence and Weak Compactness in Spaces of Vector-Valued Continuous Functions

Let X be completely regular Hausdorff space, E a Hausdorff locally convex space, C(X, E) (Cb(X, E)) the space of all E-valued (all E-valued bounded) continuous functions on X, and βz a strict topology on Cb(X, E). It is proved that a sequence {ƒn} in (Cb(X, E), βz) converges weakly to 0 if and only if {ƒn} is uniformly bounded and ƒn,(x) → 0, pointwise on X with weak topology on E. Similar results are proved for (C(X, E), β∞c). Some characterizations of relatively weakly compact subsets of (Cb(X, E), βz) and (C(X, E), β∞c) are also given.