Setup and Thoughts to Date
Let $B$ be the closed unit ball in ${L}^2_m[0,1]$, where $m$ is the Lebesgue measure, equipped with the weak topology. I know that $B$ is separable, so there exists a separating sequence $(f_n \in L^2_m[0,1])_{n \in \mathbb{N}}$ such that (here's I've used Riez to identify $L^2_m[0,1]$ with its topological dual) $$ d^{\star}(x,y)\triangleq \sum_{n \in \mathbb{N}}\frac{f_n(x-y)}{2^n}, $$ defines a metrizes $B$ under the weak topology on $L^2_m[0,1]$.
The Banach-Alaoglou theorem, implies that $B$ is a compact metric space under this metric. Therefore the open cover $$ \mathcal{U}\triangleq \left\{ \overline{Ball\left(x;\frac1{2}\right)} \right\}_{x \in B}, $$ admits a finite subcover.
Question My question is, what is the cardinality of this finite sub-cover?