Theorem 2

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

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

Let E be a Banach space. The set $\mathscr{R}(E)$ of Riesz endomorphisms of E is open in $\mathscr{L}(E)$ . The mapping which associates with an element of $\mathscr{R}(E)$ the dimension of its nilspace is upper semi-continuous on $\mathscr{L}(E)$ .

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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