Remark

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

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

Recall that, for every function $f$ on G, one defines the function $\check{f}$ on G by $\check{f}(g) =f(g^{-1})$ for every $g\in G$ (INT, VII, p. 12, § 1, No. $^o1$ , formula (12)).

One verifies that if $f\in \mathscr{L}^1$ (G), the linear mapping $u=\boldsymbol{\delta }^{(p)}_G(f)$ satisfies the relation

u(\widecheck{\varphi}) =f*\check{\varphi}

for every $\varphi \in L^p(G, \mu')$ .

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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