Proposition 1

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

General Topology, chapter IX, Appendix 2. Produits infinis dans les algèbres normées

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

It is known (VII, p. 16, prop. 2) that there exists a number $c > 0$ such that, for every finite family $(\mathbf{x}_i)_{i \in I}$ of points of $A$ , one has

(1)

\sum_{i \in I} \|\mathbf{x}_i\| \leq c \cdot \sup_{J \subset I} \left\| \sum_{i \in J} \mathbf{x}_i \right\|.

Let $(a_n)_{n \in \mathbf{N}}$ be any sequence of elements of $A$ . For every finite subset $I$ of $\mathbf{N}$ , put

p_I = \prod_{i \in I} (\mathbf{e} + a_i) \qquad s_I = \sum_{i \in I} a_i, \qquad \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.

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