Proposition 1

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

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

A family $(x_i)_{i \in I}$ of points $x_i = (x_{i,k})_{1 \leq k \leq n}$ of $\mathbf{R}^n$ is summable if and only if each of the $n$ families $(x_{i,k})_{i \in I}$ of real numbers is summable in $\mathbf{R}$ .

This follows from Chapter III, § 5, no. 4, Proposition 4.

The condition of Proposition 1 may be transformed as follows:

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

Cited by

Proposition 3, § 3
Theorem 1, § 3

Cites

Chapter III, § 5, no. 4, Proposition 4