I would like to find
- a vector space $E$
- a norm on $E$
- a sequence $(u_n)$ which converges for this norm but such that its components doesn't converge to the component of the limit.
First $E$ should be infinite dimensional.
Second I am tying to look at $E=\mathbb{R}[X]$.
Take $u_n=(\underbrace{0,\ldots,0}_{\text{n zeros}},1/n,1/(n+1),\ldots,1/(n+p),\ldots)$.
or
$u_n=(1/1,1/2,\ldots,1/n,0,\ldots,0,\ldots)$.
We have $||u_n||_1=\sum_{k=1}^n1/k$
