Proposition 1

— Let $\mathfrak{S}$ be a filter base on a vector space $E$ formed from sets that are absorbent, symmetric and convex. Then the set $\mathfrak{B}$ of transforms of the sets of $\mathfrak{S}$ by homotheties of ratio $> 0$ is a fundamental system of neighbourhoods of 0 for a locally convex topology on $E$.

Clearly $\mathfrak{B}$ is a filter base satisfying (EV_I) and (EV_{II}) of I, p. 7, prop. 4; it also satisfies (EV_{III}) since if $V \in \mathfrak{S}$ then $\frac{1}{2}V + \frac{1}{2}V = V$.

Note that if $\mathscr{T}$ is the locally convex topology on E having $\mathfrak{B}$ for a fundamental system of neighbourhoods of 0, then the sets $(1/n) \ V$, where $n$ varies in the integers $> 0$ and V varies in $\mathfrak{S}$, form a fundamental system of neighbourhoods of 0 for the topology $\mathscr{T}$. Then $\mathscr{T}$ is Hausdorff, if and only if, for every $x \neq 0$ in E there exists an integer $n$ and a set $V \in \mathfrak{S}$, such that $nx \notin V$; if, further, $\mathfrak{S}$ is enumerable, then the topology $\mathscr{T}$ is a metrisable locally convex topology. Conversely, it is clear that if $\mathscr{T}$ is a metrisable locally convex topology, then there exists an enumerable fundamental system of closed symmetric convex neighbourhoods of 0 for $\mathscr{T}$.