Proposition 3

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

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

Let $(x_\lambda)_{\lambda \in L}$ be a summable family of points of $\mathbf{R}^m$ , $(y_\mu)_{\mu \in M}$ a summable family of points of $\mathbf{R}^n$ , and let $f$ be a bilinear mapping of $\mathbf{R}^m \times \mathbf{R}^n$ into $\mathbf{R}^p$ . Then the family $(f(x_\lambda, y_\mu))_{(\lambda, \mu) \in L \times M}$ is summable and we have

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

To show that the family $(f(x_\lambda, y_\mu))$ is summable, it is sufficient by Proposition 1 to establish that each of the $p$ families formed by the coordinates of the points $f(x_\lambda, y_\mu)$ in $\mathbf{R}^n$ is summable: in other words, we can restrict ourselves to the case where $f$ is a bilinear form; but for such a form $f$ we have

f(x, y) = \sum_{i, j} a_{ij} x_i y_j,

and therefore we are brought back to the case $f(x, y) = x_i y_j$ , and in this case the result has already been proved (Chapter IV, § 7, no. 3, Proposition 1).

By specializing the function $f$ we obtain in particular the following corollaries:

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

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Nothing.

Proposition 3

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