Theorem 1

Tag 0BZL, permanent. Cite it as [Bourbaki, Tag 0BZL].

General Topology, chapter VII, § 3. Infinite sums in the groups $\mathbf{R}^n$

A family $(x_i)_{i \in I}$ of points of $\mathbf{R}^n$ is summable if and only if the family $(||x_i||)$ of Euclidean norms of the $x_i$ is summable in $\mathbf{R}$ .

This follows without trouble from Proposition 1, the condition for summability of a family of real numbers (Chapter IV, § 7, no. 2, Theorem 3), the inequalities

\sup_{1 \leq k \leq n} |x_{i,k}| \leq ||x_i|| \leq \sum_{i=1}^n |x_{i,i}|,

and the comparison principle (Chapter IV, § 7, no. 1, Theorem 2).

One can also proceed somewhat differently, by first establishing the following proposition:

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Corollary, § 3
Proposition 2, § 3

Cites

Chapter IV, § 7, no. 1, Theorem 2
Chapter IV, § 7, no. 2, Theorem 3
Proposition 1