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$ℓ^2$ 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.

Gabriel Romon
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  • Take a sequence that is everywhere zero except for $n=2^m$. For such $n$s, take $a_n=\frac{1}{\sqrt{n}}$. Here we have a counter-example. – Jack D'Aurizio Apr 02 '15 at 10:38
  • What about : $$u_n=\cases{ n^{-1} & \text{if } n \text{ is not a power of $p$ } \cr n^{-1/p} & \text{if } n \text{ is a power of $p$}}$$

    The sum $\sum |u_n|^2$ converges. But, $u_n=\mathcal{O}(n^{-1/p})$ This is true for any $p\in\mathbb N$. So $\mathcal{O}(n^{-\varepsilon})$ sequences can be in $ℓ^2$ for any $\varepsilon>0$.

     – Kitegi Apr 02 '15 at 10:43
  • There is the well-known $\limsup_n|u_n|^{2/n}\leq1$. On the other hand, there is not going to be a "biggest" due to the lack of slowest converging series. – OR. Apr 02 '15 at 11:38

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