Corollary 1

With the notations of prop. 5, let H be a set of linear mappings of E in the locally convex space G. Suppose that E is the sum of its subspaces $g_\alpha(F_\alpha)$; then H is equicontinuous for $\mathscr{T}$, if, and only if, for every $\alpha$, the set $f \circ g_\alpha$ where $f$ varies in H, is equicontinuous in $F_\alpha$.

Remembering I, p. 9, prop. 6 the argument is similar to that of (ii) prop. 5. Let W be a symmetric convex neighbourhood of 0 in G and note that if the set $f \circ g_\alpha$, where $f \in H$ is equicontinuous, then the intersection $\bigcap_{f \in H} g_\alpha^{-1}(f^{-1}(W))$ is a symmetric convex neighbourhood of 0 in $F_\alpha$. As this intersection is the same as $g_\alpha^{-1}(\bigcap_{f \in H} f^{-1}(W))$ and the set $\bigcap_{f \in H} f^{-1}(W)$ is symmetric and convex, everything depends on showing that it is also absorbent. Now, by hypothesis, every $x \in E$ can be written as $\sum_{i=1}^n g_{\alpha_i}(z_{\alpha_i})$, where $z_{\alpha_i} \in F_{\alpha_i}$. To show that there exists $\lambda > 0$ such that $f(\lambda x) \in W$ for all $f \in H$, it is sufficient to consider the case $x = g_\alpha(z_\alpha)$ with $z_\alpha \in F_\alpha$ (since we can pass to the general case by replacing W by $W/n$). But this case follows from the fact that $g_\alpha^{-1}(\bigcap_{f \in H} f^{-1}(W))$ is a neighbourhood of 0 in $F_\alpha$.