WOT continuity

Aug 2009
122
19
Pretoria
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...)