Corollary

Let $\lambda \in K$. The following subspaces of $E$ are equal : a) the eigen subspace of $u$ relative to $\lambda$; b) the eigen subspace of $u^*$ relative to $\overline{\lambda}$;

c) the primary subspace of u relative to $\lambda$ (in other words, by LIE, VII, § 1, No. 1, the set of all vectors $x$ of E for which there exists an integer $n \geqslant 0$ such that $(u - \lambda.1_E)^n(x) = 0$); d) the primary subspace of $u^*$ relative to $\overline{\lambda}$.

It is clear that $w = u - \lambda.1_E$ is a normal endomorphism of E, hence the endomorphisms $w, w^* = u^* - \overline{\lambda}.1_E, w^n$ and $(w^*)^n$ of E have the same kernel by prop. 8.