Is the space C([0,1]) with the norm integral from 0 to 1 of |f(t)|dt a Banach space?
2026-04-11 16:50:15.1775926215
Show space C([0,1]) with norm integral is a Banach space
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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?