Bourbaki

Remark 2

Tag 028F, permanent. Cite it as [Bourbaki, Tag 028F].

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

The image of the Gelfand transform separates the points of X(A)\mathsf{X}(A), since if χ1≠χ2\chi_1\not=\chi_2 are elements of X(A)\mathsf{X}(A), there exists x∈Ax\in A such that χ1(x)≠χ2(x)\chi_1(x)\not=\chi_2(x).

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.