If $(a_n)$ is such that $\sum_{n=1}^\infty a_nb_n$ converges for every $b\in\ell_2$, then $a\in\ell_2$ Please help me with this question. I've been thinking about it for almost two days.
Let $a_n$ a real series that have the following property:
for every series $b_n$ in $l_2$: $\sum_{n=1}^\infty a_nb_n$ converges.
prove that $a_n$ in $l_2$.

Using Holder inequality we say that
$$\sum_{n} a_n b_n\leq \left(\sum_n |a_n|^p\right)^{1/p}\left(\sum_n |b_n|^2\right)^{1/2} $$
with $1/p+1/2= 1$ and from that we get $p=2.$ So the result.
