Proposition 10

Let E, F be two locally convex spaces. Suppose that :

1) There exists a family of Fréchet spaces $(E_\alpha)$, and for each $\alpha$ a linear mapping $g_\alpha : E_\alpha \to E$, such that the topology of E is the final locally convex topology for the family $(g_\alpha)$.

2) There exists a sequence of Fréchet spaces $(F_n)$ and for each n a continuous linear injection $j_n : F_n \to F$ such that $F = \bigcup_n j_n(F_n)$.

Then every linear mapping u of E in F, whose graph is closed in $E \times F$, is necessarily continuous.

To prove that $u$ is continuous, it is sufficient to show that for every $\alpha$, the mapping $u \circ g_\alpha : E_\alpha \to F$ is continuous (II, p. 27, prop. 5). Now the graph of $u \circ g_\alpha$ is the inverse image of the graph of $u$ under the continuous mapping $g_\alpha \times 1_F : E_\alpha \times F \to E \times F$, and therefore is, by hypothesis, closed in $E_\alpha \times F$. We can, therefore, restrict ourselves to the case when E itself is a Fréchet space. But then the proposition is a particular case of I, p. 20, prop. 1.