# functional analysis – Showing that \$sum_{n=1}^{infty} a_n c_n\$ converges for all \$a=(a_n)in l^1(mathbb{N})\$ then \$c\$ is bounded.

Let $$c=(c_n)subset mathbb{C}$$ and suposse that $$sum_{n=1}^{infty} a_n c_n$$ converges for all $$a=(a_n)in l^1(mathbb{N})$$. Show that $$c$$ is bounded.

Hi. I cannot conclude on this problem. I attach the possible ideas I have.

Idea: For all $$k$$, $$|c_k|leq sum_{n=1}^{k} |a_n||c_n|leq sum_{n=1}^{infty} b_nc_n$$ with $$a=(frac{overline{c_1}}{|c_1|},ldots, frac{overline{c_k}}{|c_k|},0,ldots, )in l^1(mathbb{N})$$
and I would like to consider $$b$$ as $$b=(frac{overline{c_1}}{|c_1|},ldots, frac{overline{c_k}}{|c_k|},ldots)$$ but this $$bnotin l^1(mathbb{N})$$… and that does not allow me to obtain a bound for $$| c_k|$$