Corollary 1

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

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

If $(a_\lambda)_{\lambda \in L}$ is a summable family of real numbers and if $(x_\mu)_{\mu \in M}$ is a summable family of points of $\mathbf{R}^n$ , then the family $(a_\lambda x_\mu)_{(\lambda, \mu) \in L \times M}$ is summable and we have

\sum_{(\lambda, \mu) \in L \times M} a_\lambda x_\mu = \left( \sum_{\lambda \in L} a_\lambda \right) \left( \sum_{\mu \in M} x_\mu \right).

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

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