Remark 1

Tag 026E, permanent. Cite it as [Bourbaki, Tag 026E].

Théories spectrales, chapter I, § 2. Algèbres normées

The spectrum of an element in a unital Banach algebra may be any nonempty compact subset F of $\mathbf{C}($ cf. Example 3 of I, p. 17; for $A =\mathscr{C}(F;\mathbf{C})$ and $f\in A$ the canonical injection of F into $\mathbf{C}$ , we have Sp $_A(f) = F$ ).

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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