Theorem 1

Tag 02SM, permanent. Cite it as [Bourbaki, Tag 02SM].

Théories spectrales, chapter III, § 4. Perturbations dans les espaces de Banach

Let E and F be Banach spaces. The set $\mathscr{F}(E; F)$ of Fredholm mappings of E into F is open in the Banach space $\mathscr{L}(E; F)$ , and the mapping $u\mapsto$ ind( $u$ ) of $\mathscr{F}(E; F)$ into $\mathbf{Z}$ is locally constant.

The theorem follows from the following more precise statement:

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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