Let $H$ be an inifinite dimensional separable K-Hilbert space with $K$ could be $\mathbb{R}$ or $\mathbb{C}$. Are the compact operators $K(H)$ separable?
It's well known that $\overline{F(H)}=K(H)$, where $F(H)$ is the Banach space of the finite rank operators. But $F(H)$ is an uncountable set in general, right? If it's correct, $K(H)$ isn't separable in general.