Bourbaki

Remark

Tag 027Q, permanent. Cite it as [Bourbaki, Tag 027Q].

Théories spectrales, chapter I, § 3. Algèbres de Banach commutatives

Let A be a commutative Banach algebra. It follows from this theorem that the topology of simple convergence on X′(A)\mathsf{X}'(A) coincides with the restriction to X′(A)\mathsf{X}'(A) of the weak topology σ(A′,A)\sigma (A',A) of the dual A′A' of A.

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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Machine translation of the English, by gpt-5.4, gpt-5-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.