$\sum_{k=1}^{\infty}|a_kb_k| < \infty$ $\forall \sum_{k=1}^\infty|a_k|$ convergent. Prove $\{b_k\}_{k=1}^\infty$ is bounded. For continuous functions?
$\sum_{n\geq1} k_n x_n$ converges for every $x \in l_1$, then $\sup_{n\geq1} |k_n| < \infty$