My aim is to prove that the space $(C_\mathbb{R}[0,1],||•||)$, where $||•||=\int_{0}^{1} t(1-t)|f(t)|dt$, is not complete.
I have already proved that the sequence $(f_n)$ defined by
$f_n(t)=\begin{cases} 0 & t\in [0,1/2] \\ n(t-1/2) & t\in [1/2,1/2+1/n]\\ 1 & t\in [1/2+1/n,1] \end{cases} $
is Cauchy. I have also proved that $||f_n-f||\rightarrow 0$ as $n\rightarrow \infty $, where $f$ is the function which is 0 on $ [0,1/2]$ and 1 on $(1/2,1]$. However $f_n(t)-f(t)$ is not in $C_\mathbb{R}[0,1]$, so the norm is not really defined here...does this make the statement '$f_n $ converges to $f$ with respect to $||•||$' invalid (and therefore mean this does not prove the incompleteness of $(C_\mathbb{R}[0,1],||•||)$)?
Many thanks
If you want to make your argument correct, consider your space as a subspace of the space of all integrable functions on $[0,1]$ with the same norm. Then the limit would not be in this subspace, so that the subspace is not closed and hence not complete.