The max norm $|. |_{\infty}$ is generating a metrics 
I want to prove that the max norm $\|\cdot \|_{\infty}$ generates in the
  space of continuos functions $C([a,b])$ a metrics.

Well, I am not sure how to properly make it. I just know that if I have two functions $f$ and $g$ then for all $x \in [a,b]$ a I can find the distance between those two functions such that $\|f - g\|$. For the $x \in [a,b]$ I can find out the biggest possible distance between $f(x)$ and $g(x)$ by $max_{x \in [a,b]}|f(x) - g(x)|$. Is this the metrics or do I have to do more or something else?

You need to check that it satisfies the four conditions of a metric:


*

*$d(f,g)\ge 0$,

*$d(f,g)=0 \Leftrightarrow f=g $,

*$d(f,g)=d(g,f)$,

*$d(f,g) \le d(f,h) + d(h,g) $,


where, as you mentioned above, $d(f,g)=\max_{[a,b]} |f(x)-g(x)|$.
