If a subspace M is closed and dense in an inner product space V, does V = M?

Question

The question said, if M is closed and dense in V, what conclusions can be drawn about M and V? I am assuming that these sets must be equal just by intuition and trying to visualize it. This is because every sequence is convergent to some point still in M, yet every $m\in M$ is within a neighborhood of any $v \in V$. So, to me, it seems clear that they must be equal. Are there any counterexamples to this, though?

Answer 1

If $M$ is dense, $\overline{M}=V$, if $M$ is close then $M=\overline{M}$, for transitivity $M=V$