Proposition 11

In a complete normed space E, every absolutely summable family is summable.

For if $(x_i)$ is an absolutely summable family in E, then for each $\varepsilon > 0$ there is a finite subset J of the index set I such that, for each finite subset H of I which does not meet J, we have $\sum_{i \in H} ||x_i|| \leq \varepsilon$; hence a fortiori $\left\| \sum_{i \in H} x_i \right\| \leq \varepsilon$, and this proves the proposition, since E is complete (Cauchy’s criterion, Chapter III, § 5, no. 2, Theorem 1).

A series whose general term is $x_n$ is said to be absolutely convergent in E if the series whose general term is $||x_n||$ is convergent in $\mathbf{R}$, or (equivalently) if the family $(x_n)$ is absolutely summable; consequently (Chapter III, § 5, no. 7, Proposition 9):