Show space C([0,1]) with norm integral is a Banach space

Question

Is the space C([0,1]) with the norm integral from 0 to 1 of |f(t)|dt a Banach space?

Answer 1

What do you think of the sequence of functions $(f_n)$ defined by

$f_n(x)= 1$ on $[0,\frac{1}{2}-\frac{1}{n}]$

$f_n(x)= n(\frac{1}{2}-x)$ on $[\frac{1}{2}-\frac{1}{n},\frac{1}{2}]$

$f_n(x) = 0$ on $[ \frac{1}{2} , 1]$

Is it a Cauchy sequence for your norm? Does it converge to a continuous function?