Bourbaki

Theorem 4

Tag 02J5, permanent. Cite it as [Bourbaki, Tag 02J5].

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

A sequence

H⟶fG⟶gK H\overset{f}{\longrightarrow}G\overset{g}{\longrightarrow}K

of commutative locally compact topological groups is exact if, and only if, the dual sequence

K^⟶^gG^⟶^fH^ \widehat{K}\widehat{\longrightarrow}^g\widehat{G}\widehat{\longrightarrow}^f\widehat{H}

is exact.

We shall begin by proving a few lemmas. It will be noted that each of them is moreover an easy consequence of the assertion of theorem 4.

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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Machine translation of the English, by gpt-5.4, gpt-5-6-mini, against glossary version 34. Not checked by a person. That model is a cut down version of the one the rest of this language was written by.