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 $||q|| = 1$?
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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 $1 \le \|q\|$.
Disintegrating By Parts
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@JonasMeyer : By a strange thought where I was thinking about annihilators instead. :) I'll remove that. – Disintegrating By Parts Jun 03 '15 at 18:14
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"Then $q(x)\neq 0$." You mean to take $x\in X\setminus F$? – Jonas Meyer Jun 03 '15 at 18:15
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@JonasMeyer : That's it. There is that strange technicality that I want to avoid where $q \equiv 0$ because $X=F$. Otherwise $|q|=0$. – Disintegrating By Parts Jun 03 '15 at 18:17