suppose $\displaystyle \varphi$ is a linear functional on $\displaystyle B(H)$. Show that if there exists vectors $\displaystyle x_1,\dots,x_{n}$ and $\displaystyle y_1,\dots,y_2$ in $\displaystyle H$ such that $\displaystyle \varphi(T)=\sum^{n}_{k=1}\langle Tx_k,y_k\rangle$ with $\displaystyle T\in B(H)$ then $\displaystyle \varphi$ is WOT continuous.

(I'm missing something elementary...)