Corollary

Let S be a topological space and A a bounded subset of $C^b(S)$. The following conditions are equivalent : (i) A is relatively compact for $\sigma(C^b(S), C^b(S'))$; (ii) if $(f_n)_{n \in \mathbf{N}}$ is a sequence of elements of A and $(x_m)_{m \in \mathbf{N}}$ is a sequence of points of S such that the iterated limits

exist, then $\gamma = \delta$.

Let X be the Stone-Čech compactification of S (GT, IX, § 1, No. 6) and $\alpha$ the canonical mapping from S into X. Put D = $\alpha(S)$. The mapping $\phi : f \mapsto f \circ \alpha$ is an isomorphism from the normed space $C(X)$ onto the normed space $C^b(S)$; put $\tilde{A} = \phi^{-1}(A)$. Since X is compact and D is dense in X, the prop. 2 (IV, p. 33) shows that condition (ii) is equivalent to the compactness of $\tilde{A}$ for the topology of simple convergence. The equivalence of (i) and (ii) then follows from prop. 3. \*