Proposition 3

Let X be a compact space, and let A be a bounded subset of $C(X)$. For A to be relatively compact for the topology of simple convergence, it is necessary and sufficient that it is relatively compact for $\sigma(C(X), C(X)')$.

The topology of simple convergence is Hausdorff and coarser than $\sigma(C(X), C(X)')$, hence the condition stated is sufficient (GT, I, § 9, No. 4, cor. 3).

Now suppose that A is relatively compact for the topology of simple convergence. Let $(f_n)_{n \in \mathbf{N}}$ be a sequence of elements of A. By prop. 2 (IV, p. 33), there exists a sequence $(f_{n_k})$ extracted from $(f_n)$ and converging simply to a continuous function f. By lemma 2, the bounded sequence $(f_{n_k})$ tends to f for $\sigma(C(X), C(X)')$. Then Šmulian’s theorem (IV, p. 36, th. 2) shows that A is relatively compact for $\sigma(C(X), C(X)')$.