Bourbaki

Proposition 7

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

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

If the series defined by the sequences (xn)(x_n) and (yn)(y_n) are convergent, then so are the series defined by the sequences (−xn)(-x_n) and (xn+yn)(x_n + y_n), and we have (7)

∑n=0∞(−xn)=−∑n=0∞xn, \sum_{n=0}^\infty (-x_n) = -\sum_{n=0}^\infty x_n,
(8)
∑n=0∞(xn+yn)=∑n=0∞xn+∑n=0∞yn. \sum_{n=0}^\infty (x_n + y_n) = \sum_{n=0}^\infty x_n + \sum_{n=0}^\infty y_n.
This is an obvious consequence of the continuity of −x-x on GG, and of x+yx + y on G×GG \times G.

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

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Cites

Nothing.

Transcribed from the printed volume. Not translated.