User:Caliburn/s/fa/First-Countable Topological Vector Space is Metrizable with Translation-Invariant Metric

Theorem

Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \tau}$ be a first-countable topological vector space over $\GF$.

Then there exists a metric $d$ on $X$ inducing $\tau$ such that:

$(1) \quad$ $d$ is translation-invariant

$(2) \quad$ the open balls in $\struct {X, d}$ are balanced.