How can I use the Banach-Steinhaus' Uniform boundedness principle in order to prove the following claim:
If $x_n$ is a sequence of complex numbers such that the series $\sum_1^\infty x_n \chi_n$ converges for every sequence $ \chi_n \in l_p $ ($1 \leq p < \infty $ ) , then $x_n \in l_q $ where $ \frac{1}{p} + \frac{1}{q} = 1 $ .
Thanks in advance!