Exercise 26, § 2

A locally convex space $E$ is said to have the property (GDF) if every linear mapping $u$ from $E$ into a Banach space $F$ which satisfies the following property, is continuous : in the product space $E \times F$, every limit of a convergent sequence of points in the graph $\Gamma$ of $u$ again belongs to $\Gamma$. Every Fréchet space has the property (GDF) (I, p. 19, cor. 5); this is also true of every inductive limit of a family of Fréchet spaces (II, p. 34, prop. 10). Show that every Hausdorff locally convex space with the (GDF) property is barrelled. (Let $V$ be a barrel in $E$, $q$ its gauge and $H$ the Hausdorff space associated with $E$ endowed with this semi-norm; show that the canonical mapping $\pi$ from $E$ into the completion $\hat{H}$ is continuous by using the property (GDF) and the fact that every linear form $x' \in V^0$ can be extended uniquely to a continuous linear form on $\hat{H}$, the set of these forms being the unit ball in the dual of $\hat{H}$.)