Bounded sequence perpendicular to dense subset of $\ell^2$ Consider the real Banach space $\ell^2$ of square summable sequences and let $\mathcal{A}\subset \ell^2$ be a dense subspace. Suppose I have a bounded sequence $\psi=(\psi_n)_{n\geq 1}\in \ell^\infty$ which has the following property:
$$\sum\limits_{n=1}^\infty \psi_na_n=0$$
for any $(a_n)_{n\geq 1}\in \mathcal{A}$. I would like to conclude that $\psi=0$. If I would know that $\psi\in \ell^2$, then this would be easy enough, but unfortunately I only know that $\psi\in\ell^\infty$. Does anyone have a clue how to approach this?

In my specific case I know that each sequence in $\mathcal{A}$ decays exponentially, but I do not know if this useful in this case.

This question is sort of a follow-up question of the following question
Scaled series of shifted continuous function
As I stated there, I encountered this problem in my research, where I consider linear differential equations with infinitely many delayed and/or forward terms.
It's not true.  Consider $\psi = (1,1,1,\ldots)$ with $\mathcal A$ the set of
sequences $a$ of finite support such that $\sum_i a_i = 0$.  To see this is dense in
$l^2$, take any $x \in l^2$ and $\epsilon &gt; 0$.  Take $N \in \mathbb N$ such that $\sum_{n &gt; N} |x_n|^2 &lt; \epsilon^2/2$.  Let $s = \sum_{n \le N} x_n$, and take $M \in \mathbb N$ such that $M \epsilon^2 &gt; 4 |s|^2$.  Let
$$ a_n = \cases{x_n &amp; for $n \le N$\cr
        -s/M &amp; for $N+1 \le n \le N+M$\cr
          0 &amp; otherwise} $$
Then $a \in \mathcal A$ and $\|x - a\| &lt; \epsilon$.
Every element of $l^{\infty}$ defines a bounded linear map on $l^2$. If it vanishes on a dense subset it is zero.