A Riesz representation theory for completely regular Hausdorff spaces and its applications

Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let C-b(X, E) be the space of all E-valued bounded, continuous functions on X, equipped with the strict topology beta. We develop the Riemman-Stieltjes-type integral representation theory of (beta, parallel to . parallel to(F))-continuous operators T : C-b (X, E) -> F with respect to the representing Borel operator measures. For X being a k-space, we characterize strongly bounded (beta, parallel to . parallel to(F))-continuous operators T : C-b (X, E) -> F. As an application, we study. (beta, parallel to . parallel to(F))-continuous weakly compact and unconditionally converging operators T : C-b (X, E) -> F. In particular, we establish the relationship between these operators and the corresponding Borel operator measures given by the Riesz representation theorem. We obtain that if X is a k-space and E is reflexive, then (C-b (X, E), beta) has the V property of Pelczynski.