Proposition 9

— Let F, G be two vector spaces in separating duality. If $\hat{F}$ is the completion of the space F for the topology $\sigma(F, G)$ and if we consider the canonical injection $j : F \to G^*$, where $G^*$ has the topology $\sigma(G^*, G)$, then the continuous extension $\hat{j} : \hat{F} \to G^*$ of j is an isomorphism of topological vector spaces.

For, we see that $G^*$, endowed with $\sigma(G^*, G)$, is Hausdorff and complete (II, p. 51, cor. 2); if we identify F by j with a vector subspace of $G^*$ then the topology induced on F by $\sigma(G^*, G)$ is $\sigma(F, G)$, and F is dense in $G^*$ in the topology $\sigma(G^*, G)$ (II, p. 43, cor. 4); from which the proposition follows.

Vector spaces that are complete for a weak topology are therefore the duals $G^*$ of arbitrary vector spaces G endowed with $\sigma(G^*, G)$; after II, p. 51, cor. 2, they are (topologically) isomorphic to products $\mathbf{R}^I$ of real lines. To simplify the language, we shall call them products of lines (for an intrinsic characterisation of these spaces see II, p. 85, exerc. 13 and II, p. 81, exerc. 1).

We note that on $G^*$, the $\sigma(G^*, G)$ topology is minimal among the weak topologies that are Hausdorff; for, a weak topology that is coarser than $\sigma(G^*, G)$ is necessarily of the form $\sigma(G^*, H)$ where $H \subset G$ (II, p. 43, cor. 3); but if $H \neq G$, then there exists a linear form $x^* \in G^*$ that is non null and is orthogonal to $H$ (A, II, § 7.3, prop. 8), therefore $\sigma(G^*, H)$ is not Hausdorff.

We deduce from this remark that, if $F, G$ are two vector spaces, then a linear bijection $u : G^* \to F^*$, that is continuous for the topologies $\sigma(G^*, G)$ and $\sigma(F^*, F)$, is necessarily bicontinuous.