The space $C[a,b]$ with $\|f\|:=\int_a^b |f(x)|d(x)$ is a normed space but not a Banach space 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

The first question is straight forward if you apply the definition of the norm, and I encourage you to do it on your own in order to assimilate the concept of norm.
For the second question, you need to produce a sequence of functions $(f_n\in C[a,b]^{\mathbb N})$ such that $f_n$ converges (with respect with the norm defined in the exercise) towards a function that is not in $C[a,b]$.
Hint : (Assuming, without loss of generality, that a=-1, b=1) You may want to try with $f_n:x\longrightarrow 1 \mathrm {\ if\ }x<0,(1 - nx)$ if $0\leqslant x<\frac 1 n$, $0$ otherwise.
This sequence is Cauchy with respect to the $L_1$ norm in $C[a,b]$ but you can prove that its limit is not cotinuous (it is $\mathbb {1}_{[-1,0]}$).
