TITLE: Separability of the space of bounded operators on a Hilbert space
QUESTION [8 upvotes]: 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.

REPLY [2 votes]: For the first question you may consider a Hilbert basis for $H$ and consider the map $\Phi:l^{\infty}\to B(H)$ such that $\Phi((a_i)_i)=T; \  Te_i=a_ie_i$. It is an isometry.