Corollary 1

— The topology of E is identical with the topology of uniform convergence on equicontinuous subsets of $E'$.

More generally, let F be a locally convex space; every $u \in \mathscr{L}(E; F)$ defines a map $j(u):(x, f) \mapsto f(u(x))$ from $E \times F'$ into K (i.e. into $\mathbf{R}$ or $\mathbf{C}$). This enables us to define, on the space $\mathscr{L}(E; F)$, the topology of uniform convergence on a set of subsets of $E \times F'$. In particular :