TITLE: Density of finite rank operators on Hilbert space
QUESTION [5 upvotes]: Are finite rank operators on Hilbert space $H$ dense in $B(H)$ in the weak operator topology?

REPLY [5 votes]: Let $\mathcal{P}(H)$ be a directed system of all finite rank orthogonal projections in $H$. We take $p\leq q$ in $\mathcal{P}(H)$ if $\mathrm{Im}(p)\subseteq\mathrm{Im}(q)$. One may show that for all $a\in\mathcal{B}(H)$ we have
$$
a=\lim\limits_{p\in\mathcal{P}(H)}pap\tag{1}
$$
in the weak operator topology. Indeed,  take $x,y\in H$. Then for all orthogonal projections $p$ such that $\mathrm{Im}(p)\supseteq\mathrm{span}\{x,y,a(x)\}$ we will have $\langle (pap)(x),y\rangle=\langle a(x),y\rangle$. So we proved equality $(1)$.
Since $\{pap:p\in \mathcal{P}(H)\}\subseteq\mathcal{P}(H)$ we conslude that $\mathcal{P}(H)$ is dense in $\mathcal{B}(H)$ in the weak operator topology.