Proposition 6

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

General Topology, chapter III, § 5. Infinite sums in commutative groups

Let $(x_i), (y_i)$ be two summable families, with the same index set, in a group $G$ . Then the families $(-x_i), (nx_i)$ ( $n \in \mathbf{Z}$ ), $(x_i + y_i)$ are summable, and we have

\begin{align*} \Sigma(-x_i) &= -\Sigma x_i, \\ \Sigma(nx_i) &= n\Sigma x_i, \\ \Sigma(x_i + y_i) &= \Sigma x_i + \Sigma y_i. \end{align*}

For $x \to -x$ and $x \to nx$ are continuous homomorphisms of $G$ into $G$ ; on the other hand, if $(x_i)$ and $(y_i)$ are summable, then the family $(x_i, y_i)$ is summable in $G \times G$ , and since $(x, y) \to x + y$ is a continuous homomorphism of $G \times G$ into $G$ , we deduce (6).

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

Cited by

Theorem 3, § 7

Cites

Nothing.