# using uniform boundedness principle

I have a sequence of numbers $x_n$ that satisfy that for every $y_n \in c_0$ (when $c_0$ is a Banach space of all the complex sequences that satisfy $\lim_{n\rightarrow \infty }{a_n} =0$ ) the series $\sum_1^\infty{x_ny_n}$ convergence .How do I show that $x_n \in l^1$ ?

Maybe using uniform boundedness principle will help me ?

• $\sum_{1}^{\infty}x_ny_n$ converges, here $y_n\in c_0$ and $x_n$ is just a number? Mar 4 '13 at 18:33
For each integer $N$, consider $T_N\colon c_0\to \Bbb R$ given by $$T_N(y):=\sum_{j=1}^Nx_jy_j.$$