Exercise 18, § 2

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

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

(b) Let $\mathscr{C}$ be a separated linear topology on $E$, and let $\mathscr{C}^*$ be the linear topology having as fundamental system of neighbourhoods of $0$ the filter basis generated by the submodules of $E$ open for $\mathscr{C}$, and sheltered. Now prove that $\mathscr{C}^*$ is separated and that every submodule closed for $\mathscr{C}$ is also closed for $\mathscr{C}^*$ (remark that every submodule closed for $\mathscr{C}$ is an intersection of submodules open for $\mathscr{C}$). Deduce that if $E$ is complete for $\mathscr{C}^*$, it is complete for $\mathscr{C}$ (Top. gén., Chap. III, 3rd ed., § 3, No. 5, Prop. 9).

(c) Suppose that $\mathscr{C}$ is linearly compact. Now prove that $\mathscr{C}^*$ is linearly compact and is the least fine of the separated linear topologies less fine than $\mathscr{C}$; in particular $\mathscr{C}^*$ is a minimal linear topology (Exercise 17). (Let $\mathscr{B}$ be a filter basis formed of submodules closed for $\mathscr{C}$, having $0$ for intersection; prove that for every submodule $U$ open for $\mathscr{C}$ and sheltered, there exists $M \in \mathscr{B}$ such that $M \subset U$, using Exercise 16 d)).

(d) Let $F$ be a separated linearly topologized $A$-module, and let $\mathscr{C}_1$ be its topology; prove that if $u : E \to F$ is a linear mapping continuous for the topologies $\mathscr{C}$ and $\mathscr{C}_1$, $u$ is also continuous for the topologies $\mathscr{C}^*$ and $\mathscr{C}_1^*$.