Let $-\infty\lt a\lt b \lt \infty$. Let $C[a,b]$ is space of continuous functions and a function $\|.\|:C[a,b]\to R$ is given by $\|f\|:=\int_a^b |f(x)|d(x)$
a). Show that $\|\cdot\|$ is a norm in $C[a,b]$
b). Show that $C[a,b]$ with this norm is not a Banach space.
Well the 1st question is Kind of easy. Non-negativity, Homogenity, and Triangle inequality is easily proven. At the 2nd question I have a Problem.
I know the $$C[0,1]$$ space with Norm $$\|f\| = \int_0^1|f(x)|d(x)$$ is not complete that means Not Banach-Space. But how do I prove the generality?
Does anyone has a suggestion?
Thanks