Exercise 4, § 6

Let $(x_i)$ be a summable family in a Hausdorff topological ring A, and let s be its sum. Then for each $a \in A$ the family $(ax_i)$ [resp. $(x_i a)$] is summable and its sum is as (resp. sa). If $(x_\lambda)$ and $(y_\mu)$ are two summable families in A such that the family $(x_\lambda y_\mu)$ is summable, then $\sum_{\lambda, \mu} x_\lambda y_\mu = \left( \sum_\lambda x_\lambda \right) \left( \sum_\mu y_\mu \right)$. If moreover A is complete and if one of the $x_\lambda$ is a unit, then the summability of the family $(x_\lambda y_\mu)$ implies the summability of $(y_\mu)$.