Corollary 2

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

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

The function defined by $u\mapsto$ dim Ker( $u$ ) is locally constant on $\mathscr{E}(E; F)$ .

Indeed, one has dim Ker( $u$ ) $=-$ ind( $u$ ) for every $u\in \mathscr{E}(E; F)$ .

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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