Proposition 2

— Let E be a vector space, $(F_\alpha)_{\alpha \in A}$ a family of locally compact spaces having the property (GDF), and for each $\alpha \in A$ let $h_\alpha$ be a linear mapping of $F_\alpha$ into E. If E is equipped with the finest locally convex topology that makes the $h_\alpha$ continuous, then E has the property (GDF).

Let $u$ be a linear mapping of E into a Banach space B, such that every limit in $E \times B$ of every convergent sequence of points of the graph $\Gamma$ of $u$ also belongs to $\Gamma$. It suffices to show that, for every $\alpha \in A$, $u \circ h_\alpha$ is continuous on $F_\alpha$ (TVS, II, §4, No. 4, Prop. 5). Now, let $(x_n)$ be a sequence of elements of $F_\alpha$ having a limit $a$ and such that the sequence $\left(u(h_\alpha(x_n))\right)$ has a limit $b \in B$. Since $h_\alpha$ is continuous, $h_\alpha(a)$ is a limit of the sequence $(h_\alpha(x_n))$ in E; therefore by hypothesis $b = u(h_\alpha(a))$ and, since $F_\alpha$ has the property (GDF), $u \circ h_\alpha$ is continuous.