Let $\{a_n\}$ be a sequence of complex numbers such that $\sum \limits _{n=1}^{\infty} a_nb_n$ converges for every complex sequence $b_n \in \ell^p$.
Show that $\{a_n\} \in \ell^q$ where $1/p+1/q=1$ and $p>1$.
How can we use the closed graph theorem to solve this problem?