Remark

Tag 03B2, permanent. Cite it as [Bourbaki, Tag 03B2].

Théories spectrales, chapter V, § 2. Représentations des groupes localement compacts

Let $\chi$ be a character of G. If G is not compact, the $\chi$ -isotypical component of the regular representation of G in $L^2(G)$ is trivial. If G is compact, the $\chi$ -isotypical component of the regular representation of G is of dimension 1 and the function $\chi \in L^2(G)$ is a basis of it.

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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Cites

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