Definition 8

In a normed space $E$, a family $(x_i)$ of points of $E$ is said to be absolutely summable if the family $(||x_i||)$ of norms of the $x_i$ is summable in $\mathbf{R}$.

This concept appears to depend on the norm chosen on $E$; but by Proposition 7 of no. 3 and the comparison principle for summable families of real numbers, a family which is absolutely summable with respect to a norm $p$ on $E$ is absolutely summable with respect to any norm on $E$ which is equivalent to $p$.

If $(x_i)_{i \in I}$ is a family of points of $E$ which is summable and absolutely summable, we have

Indeed, for each finite subset $J$ of $I$ we have $\left\| \sum_{i \in J} x_i \right\| \leq \sum_{i \in J} \|x_i\|$, and the inequality (4) follows by passing to the limit with respect to the directed set of finite subsets of $I$.