We let $L=B(\mathcal{H})$ for some Hilbert space $\mathcal{H}$. Dixmier lets $L_*\subseteq L^*$ be the norm closure of the set of weakly continuous functionals on $L$ and wants to prove that every functional in $L_*$ is ultraweakly continuous. His proof (or most of it) is here. ($\omega_{\xi,\eta}$ denotes the linear functional $T\mapsto\langle T\xi,\eta\rangle$.) I'm mainly concerned about the part about weakly continuous linear functionals, so I will focus here on what I don't understand. Hope you can help me out!
That $\phi$ is of the form described follows from a preceding theorem. I don't see why he would want to consider $x_k'$ and $y_k'$ with the finite rank operator properties as given, and even then, there is a part in it that doesn't seem entirely valid to me. On the first page, he concludes an equality on the entirety of $L$ by linearity and continuity, but he only proved an equality for rank one operators, so wouldn't the equality hold only for compact operators? Is there something I'm missing out on here?
Nonetheless, I only do not get how he uses spectral theory; is it on the absolute value of the finite rank operator, and how does that allow the $x_k'$ and $y_k'$ to be orthogonal?
If someone could please explain to me everything from the beginning of the proof to the part about the orthogonal systems, I would be most MOST grateful!
