Please, help. Why $K(H)$, the space of compact operators on a separable Hilbert space $H$, is separable as a topological space? Particular, $H = \ell_2$?
2026-03-26 03:10:58.1774494658
$K(\ell_2)$ is separable?
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The finite-rank operators are dense in $K(H)$. A finite-rank operator is a finite sum of operators of the form $x \mapsto \langle u, x \rangle v$ for $u, v \in H$. Approximate $u$ and $v$ by members of a dense sequence in $H$.