Corollary 2

Tag 07HN, permanent. Cite it as [Bourbaki, Tag 07HN].

Topological Vector Spaces, chapter II, § 6. Weak topologies

Let $u$ be a linear mapping of $F$ in $F_1$ that is continuous for $\sigma(F, G)$ and $\sigma(F_1, G_1)$ . We have then

(3)

\operatorname{Ker}({}^t u) = (\operatorname{Im}(u))^{\circ} ,

(4)

\overline{\operatorname{Im}({}^t u)} = (\operatorname{Ker}(u))^{\circ} .

Suppose that the dualities between $F$ and $G$ and between $F_1$ and $G_1$ are separating; then $u(F)$ is dense in $F_1$ (for $\sigma(F_1, G_1)$ ), if and only if ${}^t u$ is injective.

Apply prop. 6 with $A = F$ and $A_1 = \{0\}$ , using the fact that the weak topologies $\sigma(G, F)$ and $\sigma(F_1, G_1)$ are Hausdorff. The last assertion results from (4), interchanging $u$ and $^t u$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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