Definition 2

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

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

The isometric isomorphism $\mathscr{F}_G$ of $F^2(\widehat{G})$ onto $L^2(G)$ inverse to the isomorphism induced by the Fourier cotransform is called the Fourier transform of G. The image of an element of $F^2(\widehat{G})$ is called its Fourier transform.

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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