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)/EO 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 and the space of bounded analytic functions on a plane region. The space C(S) of bounded, continuous, complex valued functions on the locally compact Hausdorff space S is paired by integration with M(S), the space of regular Borel measures on S having finite variation. We call a vector subspace of C(S) weakly normal if its unit ball is compact in the weak topology of this pairing, and normal if its unit ball is compact in the strict topology (see ?1 for definitions). In ?1 of this paper we show that every normal subspace of C(S) is weakly normal, every weakly normal subspace is norm closed; and that a subspace is normal if and only if its unit ball is compact in the compact-open topology. We discuss several examples of normal subspaces, in particular the spaces Hfl(G) of bounded analytic functions on the plane region G, and lw(I') of bounded complex valued functions on the index set r. If E is a linear subspace of C(S), and E is its annihilator in M(S), then it is easy to see that the mapping which takes f in E to the linear functional m+E?0 jf dm (m in M(S)) is a linear isometry of E into the dual of M(S)/EO. We show that E is weakly normal if and only if this map is onto. In the second section we prove our main result (Theorem 2): A subspace E of C(S) is normal if and only if it is weakly normal and the bounded weak star topology induced on it by M(S)/EO coincides with the strict topology. In the last section we show that if S is separable and E is a normal subspace of C(S), then a subset of E is strictly closed if and only if it is strictly sequentially closed. This result was first proved by Paul Hessler for H'(G). Our work continues that of Rubel and Shields [10] and Rubel and Ryff [9], who studied the particular class of normal subspaces HO(G). The coincidence of the Received by the editors January 8, 1970 and, in revised form, June 29, 1970. AMS 1969 subject classifications. Primary 4625; Secondary 4601, 4630.