I'm asking for your help in the next problem, I can't think how to do it!
Prove that if $$\sum_{n=1}^{\infty}|(A\phi_n,\phi_n)|<\infty$$ for all orthonormal bases, then $A$ is in the trace class.
I think the fact that the hypothesis is true for all orthonormal bases is the hint, but I don't know! Thank you.