Definition 1

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

Théories spectrales, chapter III, § 6. Propriétés spectrales des endomorphismes des espaces de Banach

One says that $\lambda$ is of finite spectral multiplicity for $u$ if the spectral projector $e_{\lambda}(u)$ is of finite rank; in this case, the integer $m_{\lambda}(u) =$ dim(Im( $e_{\lambda}(u)$ )) is called the spectral multiplicity of $\lambda$ for $u$ .

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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