Proposition 11

Let E be a metrisable locally convex space. The topology of E can be defined by a distance that is invariant under translations, and for which the open balls are convex.

Let $(p_n)_{n \in \mathbf{N}}$ be a sequence of semi-norms that define the topology of E. Let $d_n$ be the pseudometric defined by $d_n(x, y) = \inf(p_n(x - y), 1/n)$ for $x, y$ in E; it is invariant under translations. For every $n \geq 0$, and every real number $R \geq 0$, let $B_{n, R}$ be the set of $x \in E$ for which $d_n(x, 0) < R$. If $R \geq 1/n$, then $B_{n, R} = E$, and in the other case $B_{n, R}$ is formed from the $x \in E$ such that $p_n(x) < R$; in all cases $B_{n, R}$ is convex.

For $x, y$ in $E$ define $d(x, y) = \sup_{n \in \mathbf{N}} d_n(x, y)$. We see immediately that $d$ is a distance, invariant under translations on $E$ and defining the topology of $E$. For $x_0 \in E$ and $R \geq 0$, the open ball with centre $x_0$ and radius $R$ (for the distance $d$) is equal to $\bigcap_{n \in \mathbf{N}} (x_0 + B_{n, R})$, therefore it is convex.