# Does convergence in Hilbert-Schmidt norm imply convergence of singular values?

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?