TITLE: Do non-$\ell^2$ sequences have an $\ell^2$ functional that takes them to infinity?
QUESTION [6 upvotes]: Suppose $\{a_n\}_{n=1}^{\infty}$ is a sequence of real numbers (suppose also positive for simplicity) so that $$\sum_{n=0}^{\infty} a_n^2 = \infty$$
i.e. the sum diverges.
Can you necessarily find a square summable sequence $\{b_n\}_{n=1}^{\infty}\in\ell^2$ so that $\sum_{n=0}^{\infty} a_n b_n = \infty$ ?

REPLY [2 votes]: If $a_n$ is unbounded, then it is easy.  Otherwise wlog $|a_n| \le 1$.  Choose integers $0=j_0 < j_1 < j_2 < \cdots$ such that
$$ \sum_{n=j_{k-1}+1}^{j_k} a_n^2\in [2^k, 2^k+1) .$$
Now let
$$ b_n = 2^{-2k/3} a_k \quad\text{if } j_{k-1} < n \le j_k .$$