Theorem 1 (Grothendieck)

— Let E be a locally convex space, and let \mathscr{S} be an adapted and covering bornology on E. Let F \subset E^* be the space of those linear forms on E whose restriction to each set belonging to \mathscr{S} is continuous. If F is assigned the \mathscr{S}-topology, then the canonical injection from E'_\mathscr{S} into F extends to an isomorphism from the completion \hat{E}'_\mathscr{S} of E'_\mathscr{S} onto F.

Since every simple limit of linear forms on E is a linear form (III, p. 16, prop. 4) and since the bornology \mathscr{S} on E is covering, it follows from GT, X, § 1, No. 6, cor. 2 that the space F with the \mathscr{S}-topology is Hausdorff and complete. It is clear that E'_\mathscr{S} is a topological vector subspace of F; hence it is enough to prove that E'_\mathscr{S} is everywhere dense in F. This follows from the following lemma: