Dual norm and distance Let $Z$ be a subspace of a normed linear space $X$ and $x\in X$ has distance $d=\inf\{||z-y||:z\in Z\}$ to $Z$.
I would like to find a function $f\in X^*$ that satifies
$||f||\le1$, $f(x)=d$ and $f(z)=0$
Is it correct that $||f||:=\sup\{|f(x)| :x\in X, ||x||\le 1\}$ because I cannot conclude from this definition that $||f||\le1$
May you could help me with that, thank you very much.

I'll list the ingredients and leave the cooking to you: 


*

*The function $d:X\to [0,\infty)$ is sublinear in the sense used in the Hahn-Banach theorem

*There is a linear functional $\phi$ on the one-dimensional space $V=\{t x:t\in\mathbb R\}$ such that $\phi(x)=d(x)$ and $|\phi(y)|\le d(y)$ for all $y\in V$. If you get stuck here, mouse over the hidden text. 

*From 1, 2, and Hahn-Banach you will get $f$. 



   $\phi(tx)=td(x)$

