Proposition 1

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

General Topology, chapter IX, Appendix 0. Infinite products in normed algebras

Let $A$ be a normed algebra of finite rank over $\mathbf{R}$ . If $(e + u_n)$ is a multipliable sequence in $A$ , whose product is a unit of $A$ , then the series whose general term is $u_n$ is absolutely convergent.

From Chapter VII, § 3, no. 1, Proposition 2, there exists a number $c > 0$ such that, for every finite family $(x_i)_{i \in I}$ of points of $A$ , we have

\sum_{i \in I} \|x_i\| \leq c \sup_{J \subset I} \left| \sum_{i \in J} x_i \right|.

Let $(a_i)_{i \in \mathbf{N}}$ be an arbitrary sequence of elements of $A$ . For each finite subset $I$ of $\mathbf{N}$ , put

p_I = \prod_{i \in I} (e + a_i), \quad s_I = \sum_{i \in I} a_i, \quad \sigma_I = \sum_{i \in I} \|a_i\|.

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

Cited by

Proposition 2, § 3

Cites

Chapter VII, § 3, no. 1, Proposition 2