The distance of a vector to a subspace

Question

Suppose $Y$ is a Banach space, $M$ is a subspace of $Y$ , is it true that $\sup\{d(x,M):||x||=1,x \in Y\} = 1$?

Given that $d(x,M):=\inf\{||x-y||:y \in M\}$

Answer 1

In general no: If $M \subset Y$ is dense, then the supremum is simply $0$. However, in the case that $M$ is a proper closed subspace of $Y$, the supremum is indeed $1$ by Riesz's lemma.