let $X= C^1([0,1], \mathbb{R})$ be the space of real value contnious function defined , differentiable and having a continious derivative on $[0,1]$. define a function fro $X \times X$ into $\mathbb{R}^{+}$ by
$d(f,g) = \sup_{x \in [0,1]} | f'(x) - g'(x)|$ where $f '$ stand for the derivative of $f$ .
Is $d$ a metric on $X$?
My attempt :yes, d is ametric on $X$. I take $f(x) = x^2$ and $g(x) =x$. Then all metric properties are satisfied
Is its true?
Any hints/ guidenace solution will be appreciated
thanks u