Let $H$ be a (separable) infinite dimensional Hilbert space, and $B(H)$ the space of bounded operators on $H$. Is $B(H)$ separable in the operator norm topology? What about in the strong and weak operator topologies? In the latter cases (which are not metrizable, correct?), what about second countability?
I recall hearing that the answer to the first question was no, but I cannot see why.