Birkhoff-Kakutani Theorem/Topological Vector Space/Corollary

Corollary

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

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

Then $\struct {X, \tau}$ is metrizable if and only if $\struct {X, \tau}$ is first-countable and Hausdorff.

Further, if $\struct {X, \tau}$ is metrizable then there exists an invariant metric $d$ on $X$ such that:

$(1): \quad$ $d$ induces $\tau$

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

Proof

Necessary Condition

Suppose that $\struct {X, \tau}$ is metrizable.

From Metric Space is First-Countable, $\struct {X, \tau}$ is first-countable.

From Metric Space is Hausdorff, $\struct {X, \tau}$ is Hausdorff.

$\Box$

Sufficient Condition

Suppose that $\struct {X, \tau}$ is first-countable and Hausdorff.

By Birkhoff-Kakutani Theorem: Topological Vector Space, there exists an invariant pseduometric $d$ on $X$ such that:

$(1) \quad$ $d$ induces $\tau$

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

From Topological Vector Space is Hausdorff iff T1 and T1 Space is T0 Space, we have that $\struct {X, \tau}$ is a Kolmogorov space.

Hence from Pseudometric Space is Metric Space iff Kolmogorov, $\struct {X, d}$ is a metric space.

That is, $d$ is in fact an invariant metric.

So $\struct {X, \tau}$ is metrizable and there exists an invariant metric $d$ on $X$ such that:

$(1): \quad$ $d$ induces $\tau$

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