I have a sequence of finite-dimensional, compact sets in $L^2(\Omega)$, where $\Omega\subset \mathbf{R}^2$ is closed and bounded. The dimension grows monotonically with the sequence, and there is no assumption that the sets are nested or anything. I would like to say something about the "limiting set", but sequence doesn't converge under standard "set metrics" like the Hausdorff metric (for example, in the Hausdorff psuedometric, a limit of compact sets, if it exists, is compact, so long as the underlying metric space is complete).

Does anyone know a way to talk about convergence of finite-dimensional compact sets to an infinite-dimensional set?

P.S. This is my first post. I read the rules, but my apologies in advance for any mistakes.