Let $H$ be separable Hilbert space. We say the sequence $\{A_n\}$ of subsets in $B(H)$ has DS property if the following two items hold:
1) $A_n$'s are all finite.
2) Let $x$ be in $B(H)$. There exists a sequence $\{a_n\}$ converging strongly to $x$ where $a_n\in A_n$ for all $n$
Q. Does there exist any sequence $\{A_n\}$ in $B(H)$ satisfying in DS property?
This amounts to ask whether $B(H)$ is sot-separable.
Fix an orthonormal basis $\{e_n\}_{n\in\mathbb N}$. Let $P_n$ be the orthogonal projection onto the span of $\{e_1,\ldots,e_n\}$. Then $P_n\to I$ sot, and since $\|P_n\|\leq1$ for all $n$, it follows that $P_nTP_n\to T$ for all $T\in B(H)$. It is also clear that $P_nTP_n$ has rank at most $n$.
If we write $\{E_{kj}\}$ for the matrix units associated with the basis $\{e_n\}$ (i.e., $E_{kj}$ is the operator $E_{kj}x=\langle x,e_j\rangle\,e_k$), then $$ P_nTP_n\in\text{span}\,\{E_{kj}:\ k,j\in\{1,\ldots,n\}. $$ Now the set $\text{span}_{\mathbb Q}\,\{E_{kj}:\ k,j\in\{1,\ldots,n\}$ is countable and, for any $T\in B(H)$, $$ P_nTP_n\in\overline{\text{span}_{\mathbb Q}\,\{E_{kj}:\ k,j\in\{1,\ldots,n\}}. $$ It follows that $$ \text{span}_{\mathbb Q}\,\{E_{kj}:\ k,j\in\mathbb N\} $$ is countable and sot-dense in $B(H)$.
