Let $\rho_{n}$ be some compact operator for all $n$ ( I know finite-dimensional operators are compact, forgive the redundancy) and let the rank of $\rho_{n}$ equal to $n$.
Now, let $F$ be some non-linear functional mapping from compact operators to the reals, think trace, quantum fidelity, trace norm etc. What are necessary and sufficient conditions for the following to hold?
$$\lim_{n}F(\rho_{n}) = F(\lim_{n\rightarrow \infty}\rho_{n})$$
I know that for infinite dimensional compact operators there are some issues owing to the fact that certain functionals are no longer continuous when acting on spaces of infinite dimensional operators.