Density of $V \cap W$ in $W$ for a subspace $W$ of a completion of a pre-Hilbert space $V$

Question

I am interested in density of $V \cap W$ in $W$ for a subspace $W$ of a completion of a pre-Hilbert space $V$. Does there exist $W$ for which $V \cap W$ is not dense in $W$? If so, what are $W$ for which $V\cap W$ is dense in $W$?

Answer 1

Let $H$ be a Hilbert space and let $W, V$ be two subspaces, which are dense, and such that $U \cap V =\{ 0\}$.

You can find such examples Here.

Then $H$ is the completion of $V$, and $ \{ 0\}= V \cap W$ is not dense in $W$.

The question you are asking can be rephrased as: Given a Hilbert space $H$. and two subspaces $V, W$, such that $V$ is dense in $H$, when is $V$ dense in $W$, which seems to be too general to answer. Note that in any of the above examples, you can replace $W$ by any of its subspaces.