Bourbaki

Remark 1

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

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

In general, the image of the Gelfand transform is neither closed in C0(X(A))\mathscr{C}_0(\mathsf{X}(A)), nor dense in C0(X(A))\mathscr{C}_0(\mathsf{X}(A)) (Exercise 7 of I, p. 193).

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.