Exercise 16, § 2

(a) Let $E$ be a Hausdorff linearly topologized A-module. Show that the following properties are equivalent:

$(\alpha)$ $E$ is linearly compact (Exercise 15).

$(\beta)$ For every continuous linear mapping $u$ from $E$ to a Hausdorff linearly topologized A-module $F$, $u(E)$ is a closed submodule of $F$.

$(\gamma)$ With every linear topology (Hausdorff or not) on $E$ coarser than the given topology, $E$ is complete.

$(\delta)$ $E$ is complete and there is a fundamental system $(U_\lambda)$ of open neighbourhoods of 0 in $E$, consisting of submodules and such that the $E/U_\lambda$ are linearly compact discrete A-modules (cf. § 3, Exercise 5).

(To see that $(\gamma)$ implies $(\delta)$ consider an open submodule $F$ of $E$ and a filter base $\mathfrak{B}$ on $E/F$ consisting of affine linear varieties. For all $V \in \mathfrak{B}$, let $M_V$ be the inverse image in E of the direction submodule of V in E/F; consider on E the linear topology in which the $M_v$ form a fundamental system of neighbourhoods of 0.)

(b) Let E be a Hausdorff linearly topologized A-module and M a linearly compact submodule of E. Show that, for every closed submodule F of E, $M + F$ is closed in E (consider the image of M in E/F).

(c) Let E be a linearly compact A-module and F a Hausdorff linearly topologized A-module. Show that for ever closed submodule M of $E \times F$, the projection of M onto F is closed. Obtain the converse (cf. General Topology, Chapter I, § 10, no. 2).

(d) Let E be a linearly compact A-module and $u$ a continuous linear mapping of E to a Hausdorff linearly topologized A-module F. Show that, for every filter base $\mathscr{B}$ on E consisting of affine linear varieties, the image under $u$ of the set of cluster points of $\mathscr{B}$ is the set of cluster points of $u(\mathscr{B})$. In particular, for every closed submodule M of E, $\bigcap_{N \in \mathscr{B}} (M + N) = M + \bigcap_{N \in \mathscr{B}} N$.