I have the following two norms on $C[a,b]$ :
$$||x||_1= \int_a^b |x(t)|dt$$
$$||x||_\infty = sup_{t \in [a,b]}|x(t)|$$
$\forall x \in C[a,b]$. I need to prove that these are not equivalent. They are both bounded as far as I can tell, so can't prove that only one is bounded. Drawing them I can see they are not equivalent, as two functions may have the same integral but different suprema and this can vary at different rates, but am having difficulty writing a proper counterexample. Can anyone suggest anything?
[Math] Show these two norms are not equivalent
functional-analysisnormed-spaces
Best Answer
Hint. Equivalence of norms $||\cdot||_1$ and $||\cdot||_\infty$ would mean that there exist constants $c_1,c_2$ such that $$c_1||x||_1\leq||x||_\infty\leq c_2||x||_1,\ \forall x.$$
$c_1$ exists and is equal to $1$. To show that $c_2$ does not exists show that there exist $x$ with $||x||_1=1$ and $||x||_\infty$ arbitrarily large.