Let $A_n$ be a series of matrices, and let $A$ be another matrix. Let $S(B)$ be an SVD operator that takes a matrix and returns the left singular vectors matrix ordered by largest singular value to smallest singular value. Also, assume all singular values for $A$ are unique.
Is there some matrix norm $\| \cdot \|$ under which if $\|A_n - A\| \to 0$ as $n \to \infty$ then $\|S(A_n) - S(A)\| \to 0$?
Does it happen for the Frobenius norm?