Proposition 2

Let E be a complete normable space and $u$ an endomorphism of E. Let H be a finite subset of Sp$_s(u)$. The set H is open and closed in Sp($u$). The spectral projector $e_H(u)$ is of finite rank, has image $\sum_{\lambda\in H}N_{\lambda}(u)$ and kernel $\bigcap_{\lambda\in H}I_{\lambda}(u)$.

This follows from No.$^o3$ of I, p. 131 since, for every $\lambda \in$ Sp$_s(u)$, the subspaces $N_{\lambda}(u)$ and $I_{\lambda}(u)$ are respectively the image and the kernel of the spectral projector $e_{\lambda}(u)$.