How can I prove that the space $(C [a, b], \|\cdot\|_1)$ is not a Banach space? If we are working with $\|f\|_1=\int_0^1|f(t)|$ on the space of all real continuous functions in the $[a,b]$ interval. Is there a way to prove that $(C [a, b], \|\cdot\|_1)$ is not complete?
I think it is easier if I took a sequence of functions in $[0,1]$ and prove that it is a Cauchy sequence that converges to a discontinuous function hence get a contradiction. But I don´t know how to make it in general ( for the $[a,b]$ interval)
This is my first idea.
$f_n: [0,1] \rightarrow \mathbb{R},\text{ given by }f_n(x)= \ \begin{cases} 1 & \text{ $ x ≤ \frac{1}{2}$}\\[2ex] \left(\cfrac{1}{2}n+1-nx\right) &\text{$\frac{1}{2} < x ≤ \frac{1}{2} + \frac{1}{n}$} \\[2ex] 0 & \text{$\frac{1}{2} + \frac{1}{n} < x $} \end{cases}$
I think you need to correct the norm.
Once that's done, how about $f_n(x)=x^n$. Then each $f_n$ is continuous, but the sequence converges to a discontinuous $f$.