I would like to revisit this question, which can be equivalently stated as:
Proposition. Let $(a_n)$ be a sequence of real (or complex) numbers such that $\sum a_n b_n$ converges for every $(b_n) \in \ell^2$. Then $(a_n) \in \ell^2$.
The proof given here by Bruno Stonek is the only one I know, which applies the uniform boundedness principle to the linear functionals $(b_n) \mapsto \sum_{n=1}^m a_n b_n$ in $(\ell^2)^*$. But I am inclined to agree with GEdgar's comment on Davide Giraudo's answer that this approach, though slick, is "far too advanced".
One could write the contrapositive as:
Let $(a_n)$ be a sequence of real numbers which is not $\ell^2$. Then there exists $(b_n) \in \ell^2$ such that $\sum a_n b_n$ diverges.
A natural way to try to prove that statement would be to explicitly construct such a $(b_n)$, by somehow manipulating $(a_n)$. Our current proof is far from constructive in that sense, since the uniform boundedness principle essentially just says that the set of such $(b_n)$ is comeager in the Hilbert space $\ell^2$, and then uses Baire to assert it is nonempty.
So my question:
Can anyone think of a way to explicitly construct $(b_n)$? Or alternatively, is there some reason to think that such a construction might be impossible?
For instance, as a wild guess, perhaps one could show that any map sending each $(a_n) \notin \ell^2$ to an appropriate $(b_n)$ would have to be nonmeasurable in some sense.