Exercise 18, § 2

In an A-module E, a submodule $M \neq E$ is called sheltered if there exists a least element in the set of submodules $\neq 0$ in $E/M$.

(a) Show that every submodule $N \neq E$ of E is an intersection of sheltered submodules (for all $x \notin N$, consider a maximal element in the set of submodules of E containing N and not containing $x$).

(b) Let $\mathscr{T}$ be a Hausdorff linear topology on E and let $\mathscr{T}^*$ be the linear topology with fundamental system of neighbourhoods of 0 the filter base generated by the submodules of E which are open under $\mathscr{T}$ and sheltered. Show that $\mathscr{T}^*$ is Hausdorff and that every submodule which is closed under $\mathscr{T}$ is also closed under $\mathscr{T}^*$ (note that every submodule which is closed under $\mathscr{T}$ is an intersection of submodules which are open under $\mathscr{T}$). Deduce that, if E is complete with respect to $\mathscr{T}^*$, it is complete with respect to $\mathscr{T}$ (General Topology, Chapter III, § 3, no. 5, Proposition 9).

(c) Suppose that $\mathscr{T}$ is linearly compact. Then show that $\mathscr{T}^*$ is linearly compact and is the coarsest Hausdorff linear topology coarser than $\mathscr{T}$; in particular $\mathscr{T}^*$ is a minimal linear topology (Exercise 17). (Let $\mathfrak{B}$ be a filter base consisting of submodules closed under $\mathscr{T}$ whose intersection is 0; show that, for every submodule U which is open under $\mathscr{T}$ and sheltered, there exists $M \in \mathfrak{B}$ such that $M \subset U$, using Exercise 16 (d)).

(d) Let F be a Hausdorff linearly topologized A-module and $\mathscr{T}_1$ its topology; show that, if $u : E \to F$ is a continuous linear mapping with respect to the topologies $\mathscr{T}$ and $\mathscr{T}_1$, $u$ is also continuous with respect to the topologies $\mathscr{T}^*$ and $\mathscr{T}_1^*$.