Let H be a separable, infinite dimensional Hilbert space. Let X and Y be (not necessarily closed) subspaces such that X is a cofinite dimensional subspace of Y. Let X′ be the closure of X and Y′ the closure of Y. Does X′ have cofinite dimension in Y′?
(I sleepily posted this question with "coinfinite" instead of "cofinite", which was accidental.)
Yes.
The sum of a closed subspace $U$ and a finite-dimensional subspace $V$ of a normed space $H$ is closed: Since $U$ is closed, $H/U$ is a normed space. It follows that $\pi(V)$ is closed in $H/U$ because it is finite-dimensional. Therefore $U+V = \pi^{-1}\pi[V]$ is closed, where $\pi \colon H \to H/U$ is the projection.
Since $Y = X + Z$ with $Z$ finite-dimensional, we have $Y \subseteq \overline{X} + Z$ and hence $\overline{X} \subseteq \overline{Y} \subseteq \overline{X} +Z$.