Show that $\infty$-norm and $C^1$-norm are not equivalent.

Question

Show that $\infty$-norm and $C^1$-norm are not equivalent.

For the $C^1([a,b],\mathbb{R})$ space, show that

$\displaystyle ||g||_\infty=sup_{a\leq t\leq b}|g(t)|$

and

$\displaystyle ||g||_{C^1}=sup_{a\leq t\leq b}|g(t)|+sup_{a\leq t\leq b}|g'(t)|$

are not equivalent.

My attempt:

$\displaystyle |g'(t)|>0 \implies sup_{a\leq t\leq b}|g'(t)|>0$

So then

$\displaystyle ||g||_\infty=sup_{a\leq t\leq b}|g(t)| \leq sup_{a\leq t\leq b}|g(t)|+sup_{a\leq t\leq b}|g'(t)| \leq p ||g||_{C^1}$ for constant $p \geq 1$.

If the two norms are not equivalent then I'm assuming that

$||g||_\infty \ngeq q ||g||_{C^1}$ for any constant $0<q<1$. How do I show this is true?

Is this a good approach or does anyone have a better idea?

Answer 1

A plan for show this kind of non-inequality is to show a sequence of functions where the left side is bounded, but the right diverges. Therefore there can be no such constant.

Can you think of a simple sequence of functions that are bounded in their extreme values, but have unbounded slopes?