Let r be a linear form (not necessarily continuous) on the Operators on a separable Hilbert Space $H$ with finite Rank $F(H)$ such that $\operatorname{r}(AB)=\operatorname{r}(BA)$ whenever $B$ is any Operator and $A\in F(H)$. I want to show, that $r$ coincides with a scalar multiple of the usual Trace defined by $Trace(\sum<.,x>y)=\sum<y,x>$
If i had continuity, i think i could try something similar to what is done here: Characterization of the trace function.
Is this result still true, if $r$ is not assumed to be continuous, and how to prove that?