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This was originally a problem in Stratila and Zsido's "Lectures on von Neumann algebras" (E.1.2). I've spent so much time working on it, and right now I cannot see how the result can be so simple.

The problem goes like this: Let $\omega$ be a weakly continuous linear functional on $B(\mathscr{H})$. Then there exist two families of mutually orthogonal vectors $\{\xi_1,\ldots,\xi_n\},\ \{\eta_1,\ldots,\eta_n\}$ in $\mathscr{H}$ such that $$\omega(T)=\sum_{i=1}^n\langle T\xi_i,\eta_i\rangle,\quad T\in B(\mathscr{H}),$$$$\|\omega\|=\sum_{i=1}^n\|\xi_i\|\|\eta_i\|.$$

I've tried altering the proof that any weakly continuous linear functional can be written in the above form with no extra assumptions on the vectors, and gotten as far as proving that the $\xi_i$'s can be chosen to be mutually orthogonal (orthonormal, in fact), but that's about it. Does anybody have any suggestions of what to do? I thought about using some facts about compact operators, but seeing as it is not a prerequisite of understanding the section containing the problem, I'm assuming the proof can be elementary (even though it's marked as one of the harder exercises).

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