I am given a metric $d$ on the closed unit ball $S$ of $B(H)$ on a separable Hilbert space $H$, and I am to prove that it induces the weak operator topology (WOT) on $B(H)$. To this end, I am allegedly to prove that a net $\{f_\lambda\}_{\lambda ∈ \Lambda}$ in $S$ converges in the WOT to some $f ∈ S$ iff $d(f_\lambda, f) → 0$. Can anyone give a thorough detailed explanation of why this is indeed sufficient? (I am a little rusty on my general metric topology.)
2026-03-27 18:09:28.1774634968
Metrisability of WOT on unit ball of bounded operators on a Hilbert space
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