Let $A$ be a Hilbert-Schmidt operator between two hilbert spaces $H_1$ and $H_2$, with singular value decomposition: $A = \sum_n \lambda_n u_n \otimes v_n$.
Now let $A_i$ be a sequence of operators converging to $A$ in Hilbert-Schmidt norm. Do the singular values of the $A_i$ converge to the singular values of $A$, and if so in what precise sense? What can we say about the singular vectors?
Finally, does this require that $H_1$ and $H_2$ be separable?