Weak topologies on subspaces of $C(S)$

Let $S$ be a locally compact Hausdorff space, $E$ a linear subspace of $C(S)$. It is shown that the unit ball of $E$ is compact in the strict topology if and only if both of the following two conditions are satisfied: (1) $E$ is the Banach space dual of $M(S)/{E^0}$ in the integration pairing, and (2) the bounded weak star topology on $E$ coincides with the strict topology. This result is applied to several examples, among which are ${l^\infty }$ and the space of bounded analytic functions on a plane region.