Let $H$ be a Hilbert space and for $\xi_1, \xi_2 \in H$ consider the bounded functional $$\omega_{\xi_1, \xi_2}: B_0(H) \to \mathbb{C}: x \mapsto \langle x \xi_1, \xi_2\rangle$$ Is it true that $$\overline{\operatorname{span}\{\omega_{\xi_1, \xi_2}: \xi_1, \xi_2 \in H\}}^{\|\cdot \|}= B_0(H)^*$$? I.e. can we approximate every functional on the compact operators by a linear combination of the above functionals? This is used in a proof I'm reading but I don't see why it should hold. It is unclear to me why one would even expect this to be true.
A reference or proof would be appreciated.
If your span were not dense, there would be a non-zero linear functional $\Lambda \in B_0(H)^{**}$ vanishing on every $\omega_{\xi_1,\xi_2}$. However, it is well known that $B_0(H)^{**}=B(H)$, so $\Lambda $ is in fact an operator such that $$ \langle \Lambda \xi _1, \xi _2\rangle =0, $$ for every $\xi _1$ and $\xi _2$. Therefore $\Lambda =0$, a contradiction.