On wikipedia it is mentioned that if we are on some (separable) Hilbert space $H$ and there is an ONB $(e_n)$ such that any compact operator $K$ can be written as $$ K = \sum_{n,m =0}^{\infty} K_{n,m} e_n \otimes e_m.$$
This made me wonder whether the result is still true if $K$ is a trace-class operator and the convergence is assumed to be in trace-norm as well? Does anybody know this?
By the way, the article was this one.