$\{y(n)\} \in l_1$ if $\sum_{n = 1}^{\infty} x(n) y(n)$ converges for all $x(n) \in c_0$
Convergence of $\sum_{n=1}^{\infty}x_n\overline{y_n}$ for every $(x_n)\in c_0$ implies $\sum_{n=1}^{\infty}|y_n|<\infty$