Example 1

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

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

Let E be a finite-dimensional vector space and $u$ an endomorphism of E. The spectrum of $u$ is finite and coincides with Sp $_s(u)$ . Its elements are the roots of the characteristic polynomial $\chi_u$ of $u$ ; they are the eigenvalues of $u$ . The spectral multiplicity $m_{\lambda}(u)$ of an element $\lambda \in$ Sp( $u$ ) is the multiplicity of $\lambda$ as a root of the polynomial $\chi_u$ (A, VII, p. 36, cor.). By Corollary 2 above, one has

\chi_{f(u)}(T) =\prod_{\lambda\in Sp(u)}(T-f(\lambda ))^{m_{\lambda}}

for every $f\in \mathscr{O}$ (Sp( $u$ )). This generalizes Prop. 10 of A, VII, p. 36.

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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