$ℓ^2$ is the space of complex sequences $u_n$ such that $\sum |u_n|^2$ converges.
I'm wondering if there are asymptotic results known about such sequences. We have trivially $u_n=o(1)$.
Are better bounds known ? What if other constraints are added (monotonocity e.g) ?
I would say $u_n=O(\frac{1}{n})$ holds, but no counter-example comes to mind.