Corollary 2 (Gelfand-Mazur Theorem)

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

Théories spectrales, chapter I, § 2. Algèbres normées

Let A be a normed algebra over $\mathbf{C}$ . If A is a field, then $A =\mathbf{C}\cdot 1$ .

If $x\in A$ , there exists $\lambda \in \mathbf{C}$ such that $x-\lambda$ is not invertible (cor. 1), whence $x-\lambda = 0$ and $x\in \mathbf{C}\cdot 1$ .

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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