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Let $H$ be a separable Hilbert space, $B$ the closed unit ball in $H$, and $(x_n)$ be a dense sequence in $B$, $(e_n)$ be an orthonormal basis, and $\varepsilon>0$. Then $B$ is covered by the (weakly)-open sets $O_{n,m} = \{ y: |\langle e_m, x_n-y\rangle| < \varepsilon \}$. So there must exist a finite subcover since the weak topology is metrizable in this case.

Is there a rough idea how such a finite sub-cover would look like?

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    $\begingroup$ you should specify your question a bit, it seems too general. $\endgroup$ Commented Jun 13, 2019 at 8:25

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The sets $O_{m,n}$ with $m$ fixed already cover the ball. Then find finitely many values of $n$ such that the numbers $\langle e_m, x_n\rangle$ are $\epsilon$-dense in $[-1,1]$ (or in $\mathbb{D}$ in the case of complex scalars).

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