Let X be a completely regular Hausdorff space, and E and F be Banach spaces. Let $C_{rc}(X,E)$ be the Banach space of all continuous functions $f:X{ ightarrow}E$ such that f(X) is a relatively compact set in E. We establish an integral representation theorem for bounded linear operators $T:C_{rc}(X,E){ ightarrow}F$. We characterize continuous operators from $C_{rc}(X,E)$, provided with the strict topologies ${eta}_z(X,E)$ ($z={sigma},{ au}$) to F, in terms of their representing operator-valued measures.