Can you show this space isn't complete?

Question

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?)

Answer 1

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 $C^0([0,1])$.