Can you show this space isn't complete? Let $X=C^0([0,1])$ and $||\cdot||:X\to\Bbb R$ be defined as $$||f||=\max_{x\in[0,1]}x^2|f(x)|.$$
Show that $||\cdot||$ isn't a Banach space.
(I can't find any Cauchy sequence that does not converge.  Can you find one?)

Let $f_n(x)=\max(n,x^{-1})$. If $N\le m<n$
$$
x^2(f_n(x)-f_m(x))=\begin{cases}
0 & 1/m\le x\le 1,\\
x-m\,x^2 & 1/n\le x<1/m,\\
(n-m)x^2 & 0\le x<1/n,
\end{cases}
$$
and
$$
\|f_n-f_m\|\le\max\Bigl(\frac{1}{4\,m},\frac{n-m}{n^2}\Bigr)\le\frac{1}{N}.
$$
This proves that $f_n$ is a Cauchy sequence, but it does not converge to a function in $C^0([0,1])$.
