Exercise 11, § 1

In a topological vector space E over a non-discrete topological division ring K there exists a fundamental system $\mathfrak{B}$ of closed neighbourhoods of 0, satisfying the conditions (EV$_{\text{II}}$) and (EV$_{\text{III}}$) as well as the two following: (EV$_{\text{Ia}}$) For each $V \in \mathfrak{B}$, there exists $W \in \mathfrak{B}$ and a neighbourhood U of 0 in K such that UW $\subset V$. (EV$_{\text{Ib}}$) For every $x \in E$ and every $V \in \mathfrak{B}$, there exists $\lambda \neq 0$ in K such that $\lambda x \in V$. Conversely, let E be a vector space over K and let $\mathfrak{B}$ be a filter base on E satisfying the conditions (EV$_{\text{Ia}}$), (EV$_{\text{Ib}}$), (EV$_{\text{II}}$) and (EV$_{\text{III}}$). Show that there is a topology on E (and one such only), that is compatible with the vector space structure of E, and for which $\mathfrak{B}$ is a fundamental system of neighbourhoods of 0.