Exercise 10, § 1

Let E be a complete Hausdorff topological vector space over a non-discrete valued division ring K. Denote by F a vector subspace of E, and by $\mathscr{T}$ the topology on F induced by the topology $\mathscr{T}'$ of E; let $\mathscr{B}$ be a fundamental system of closed, balanced neighbourhoods of 0 for the topology $\mathscr{T}$. Let $F_0$ be the vector subspace of E, generated by the closures $\overline{V}$ in E (relative to $\mathscr{T}'$) of the sets $V \in \mathscr{B}$; the sets $\overline{V}$ form a fundamental system of neighbourhoods of 0 for a topology $\mathscr{T}_0$ on $F_0$, compatible with the vector space structure of $F_0$; for this topology, $F_0$ is complete, and the topology induced by $\mathscr{T}_0$ on F is identical with $\mathscr{T}$.