Exercise 15, § 6

Let $E, F$ be two Hausdorff locally convex space. We say that a continuous linear mapping $u : E \to F$ is linearly proper if, for every Hausdorff locally convex space $G$ and every closed vector subspace $V$ of $E \times G$ the image of $V$ by $u \times 1_G : E \times G \to F \times G$ is closed. Show that this condition is equivalent to the following : $u^{-1}(0)$ is a subspace of minimal type of $E$ and for every closed vector subspace $W$ of $E$, the set $u(W)$ is closed in $F$. (To show that the first condition implies the second, consider the mapping $v : E \to \{0\}$ and, giving $E$ the topology $\sigma(E, E')$, so that $E$ is immersed in ${E'}^*$ with $\sigma({E'}^*, E')$, take the image under the projection $v \times 1_{{E'}^*} : E \times {E'}^* \to {E'}^*$ of the closure in $E \times {E'}^*$ of the diagonal $\Delta$ of $E \times E$. To show that the second condition implies the first, show that it implies that, for the topologies $\sigma(E, E')$ and $\sigma(F, F')$, the mapping $u$ is a strict morphism and use exerc. 13, e). \*