Proposition 11

Every Hausdorff locally convex space F with the property (GDF) is barreled.

Let V be a barrel in F, q is gauge, which is a semi-norm on F; let H be the Hausdorff space associated with the space F equipped with the topology defined by this single semi-norm. The completion $\widehat{H}$ of H is a Banach space; let $\pi$ be the canonical mapping of F into $\widehat{H}$; we are going to show that $\pi$ is continuous (for the original topology on F); this will establish the proposition, because V, the inverse image under $\pi$ of the unit ball of $\widehat{H}$, will then be a neighborhood of 0 in F. To establish the continuity of $\pi$ it will suffice, by virtue of (GDF), to show that the graph of $\pi$ is closed in $F \times \widehat{H}$; in other words, we must see that if $\mathscr{F}$ is a filter on F, convergent to $x \in F$, and its image $\pi(\mathscr{F})$ converges to $y \in \widehat{H}$, then $y = \pi(x)$. Now, every element $x'$ of the polar $V^\circ$ of V in $F'$ may be extended in only one way to a continuous linear form on $\widehat{H}$ (again denoted $x'$), and the set of these forms is the unit ball of the dual of $\widehat{H}$; it therefore suffices to show that $\langle y, x' \rangle = \langle \pi(x), x' \rangle$ for every $x' \in V^\circ$. But this follows from the relations