In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with the projections $p_i$ and inclusions $q_i$ as defined, and $\|.\|_i$ the norm on $\mathcal{L}(E_i)$ and $\|.\|_c$ for $\mathcal{L}(\mathbf{C}^n)$. Show that, for all $i \in \llbracket 1 ; r \rrbracket$, there exists a constant $C _ { i } > 0$ such that: $$\forall u \in \mathcal { L } \left( E _ { i } \right) , \quad \left\| q _ { i } u p _ { i } \right\| _ { c } \leqslant C _ { i } \| u \| _ { i }$$
In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with the projections $p_i$ and inclusions $q_i$ as defined, and $\|.\|_i$ the norm on $\mathcal{L}(E_i)$ and $\|.\|_c$ for $\mathcal{L}(\mathbf{C}^n)$.
Show that, for all $i \in \llbracket 1 ; r \rrbracket$, there exists a constant $C _ { i } > 0$ such that:
$$\forall u \in \mathcal { L } \left( E _ { i } \right) , \quad \left\| q _ { i } u p _ { i } \right\| _ { c } \leqslant C _ { i } \| u \| _ { i }$$