TITLE: Norm of the quotient map for a normed space
QUESTION [5 upvotes]: Let $X$ be a normed space and $F$ a closed subspace. On $X/F$ let us take the quotient norm $||[x]|| = \inf_{y \in F} ||x - y||$. Consider the quotient $q : X \rightarrow X/F$. I can see that, if $||x|| = 1$, then $||q(x)|| = \inf_{y \in F} ||x - y|| \leq ||x - 0|| = 1$ since $0 \in F$. This proves that $q$ is bounded and $||q|| \leq 1$. How may I show that $||q|| = 1$?

REPLY [4 votes]: Suppose $X \ne F$. Then $q(x) \ne 0$. For every $f \in F$,
$$
          \|x\|_{X/F} =\|q(x+f)\|_{X/F} \le \|q\|\|x+f\|.
$$
Hence,
$$
                          \|x\|_{X/F} \le \|q\|\inf_{f \in F}\|x+f\|=\|q\|\|x\|_{X/F}.
$$
So $1 \le \|q\|$.