Can the infimum becomes minimum in the following definition?

Question

Let $\mathscr{X} $ be a n.v.s., $M$ be a subspace of $\mathscr{X} $, we know that the definition of norm in $\mathscr{X} /M$ inherited by $\mathscr{X} $ is $$\Vert x+M\Vert_0 =\inf_{y\in M}\Vert x-y\Vert .$$ Can the infimum becomes minimum in the definition? Or under a stronger assumption that $\mathscr{X} $ is a Banach space, or $M$ is closed, can the definition be strengthened to minimum? If not, what is the counterexample?