Remark

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

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

By definition, in order that one have $((u))>0$ , it is necessary and sufficient that the bijective linear mapping $\widetilde{u}$ be bicontinuous, that is to say, that $u$ be a strict morphism (TG, III, p. 16). One then has

((u)) =\|\widetilde{u}^{-1}\|^{-1} \tag{10}

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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