Lemma 7

Tag 02J1, permanent. Cite it as [Bourbaki, Tag 02J1].

Théories spectrales, chapter II, § 1. Transformation de Fourier

Let $(G_i)_{i\in I}$ and $(H_i)_{i\in I}$ be finite families of locally compact topological groups. For $i\in I$ , let $\theta_i: G_i\times H_i\rightarrow \mathbf{U}$ be a mapping which puts the groups $G_i$ and $H_i$ in duality. The mapping $\theta$ defined by

\theta ((g_i),(h_i)) =\prod_{i\in I}\theta_i(g_i, h_i)

puts the groups $\prod G_i$ and $\prod H_i$ in duality.

This results from Proposition 2 of II, p. 206 and from the preceding definition.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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