Remark 4

Tag 03E9, permanent. Cite it as [Bourbaki, Tag 03E9].

Théories spectrales, chapter V, § 4. Représentations des groupes compacts

The Peter–Weyl theorem implies in particular that the natural continuous homomorphism of G into $\prod_{\pi\in\widehat{G}}\mathbf{U}(E_{\pi})$ is injective.

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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