Birkhoff-Kakutani Theorem/Topological Vector Space

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

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

Then $\struct {X, \tau}$ is metrizable $\struct {X, \tau}$ 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.

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

Let $\sequence {U_n}_{n \mathop \in \N}$ be a local basis for ${\mathbf 0}_X$ in $\struct {X, \tau}$.

Let $V_1 = U_1$.

From Open Neighborhood of Point in Topological Vector Space contains Sum of Open Neighborhoods: Corollary 2:
 * for $j \ge 2$, we can inductively pick an open neighborhood $V_j$ of ${\mathbf 0}_X$ such that:
 * $V_j + V_j \subseteq V_{j - 1} \cap U_{j - 1}$
 * so that $V_j + V_j \subseteq V_{j - 1}$.

Since $V_j \subseteq U_j$ for each $j \in \N$, and ${\mathbf 0}_X \in V_j$ for each $j \in \N$, $\sequence {V_n}_{n \mathop \in \N}$ is also a local basis for ${\mathbf 0}_X$ in $\struct {X, \tau}$.

Let $D$ be the set of real numbers with a terminating binary notation.

That is, the real numbers $r \in \R$ of the form:
 * $\ds r = \sum_{j \mathop = 1}^\infty \map {c_j} r 2^{-j}$

with $\map {c_j} r \in \set {0, 1}$ such that:
 * there exists $N \in \N$ such that $\map {c_j} r = 0$ for $j > N$.

From the Basis Representation Theorem, the coefficients $\map {c_j} r$ uniquely identify $r$.

Note that if $r, s \in D$ and $r + s < 1$, then $r + s \in D$.

Now, for $r \ge 1$, set $\map A r = X$.

For $r \in D$, set:
 * $\ds \map A r = \sum_{j \mathop = 1}^\infty \map {c_j} r V_j$

where $\ds \sum_{j \mathop = 1}^\infty$ denotes linear combination.

Since $\map A 1 = X$, we have that:
 * $\set {r \ge 0 : x \in \map A r} \ne \O$

for each $x \in X$.

Hence:
 * $\inf \set {r \ge 0 : x \in \map A r}$ is finite.

So, we can define $f : X \to \hointr 0 \infty$ by:
 * $\map f x = \inf \set {r \ge 0 : x \in \map A r}$

for each $x \in X$.

Define:
 * $\map d {x, y} = \map f {x - y}$

We aim to show that $d$ is a metric.

We require the following lemma.