Proposition 7

— Let $u : E_1 \to E_2$ be a linear mapping. We assume that $u$ is continuous for the weakened topologies of $E_1$ and $E_2$.

(i) The mapping $u$ is continuous if $E_1$ and $E_2$ are assigned their Mackey topologies.

(ii) If $E_1$ is bornological or barrelled, then $u$ is continuous for the initial topologies of $E_1$ and $E_2$.

(iii) In order that $u$ be continuous for the initial topologies of $E_1$ and $E_2$, it is necessary and sufficient that the image under $'u$ of every equicontinuous subset of $E_2'$ be equicontinuous in $E_1'$.

The hypothesis implies that $'u$ is continuous for the weak topologies $\sigma(E_2', E_2)$ and $\sigma(E_1', E_1)$ (II, p. 46, corollary) hence the image under $'u$ of a convex, balanced and compact subset for $\sigma(E_2', E_2)$ is convex, balanced and compact for $\sigma(E_1', E_1)$, the topologies $\sigma(E_2', E_2)$ and $\sigma(E_1', E_1)$ being Hausdorff. Therefore, assertion (i) follows from GT, X, § 1, No. 4, prop. 3, b). Assertion (ii) is a consequence of (i) : for, if $E_1$ is bornological or barrelled, its initial topology is the Mackey topology, and the Mackey topology of $E_2$ is finer than the initial topology of $E_2$. Finally, the initial topology of $E_i$ is that of uniform convergence on equicontinuous subsets of $E_i'$ (III, p. 19, cor. 1 of prop. 7). This proves (iii).