In this part, $E$ denotes a $\mathbf{C}$-vector space of dimension $n$ and $u$ denotes an endomorphism of $E$. We assume that $u$ is diagonalizable. We denote by $\mathcal{B} = (v_{1}, v_{2}, \ldots, v_{n})$ a basis of $E$ formed of eigenvectors of $u$. Let $F$ be a vector subspace of $E$, different from $\{0_{E}\}$ and from $E$. We denote $$\mathcal{A} = \left\{H \text{ vector subspace of } E \text{ such that } u(H) \subset H \text{ and } F \cap H = \{0_{E}\}\right\}$$ and $$\mathcal{L} = \left\{p \in \mathbf{N}^{*} \mid \exists H \in \mathcal{A} : p = \operatorname{dim}(H)\right\}$$ Prove that $\mathcal{L}$ has a greatest element which we call $r$.
In this part, $E$ denotes a $\mathbf{C}$-vector space of dimension $n$ and $u$ denotes an endomorphism of $E$. We assume that $u$ is diagonalizable. We denote by $\mathcal{B} = (v_{1}, v_{2}, \ldots, v_{n})$ a basis of $E$ formed of eigenvectors of $u$. Let $F$ be a vector subspace of $E$, different from $\{0_{E}\}$ and from $E$.
We denote
$$\mathcal{A} = \left\{H \text{ vector subspace of } E \text{ such that } u(H) \subset H \text{ and } F \cap H = \{0_{E}\}\right\}$$
and
$$\mathcal{L} = \left\{p \in \mathbf{N}^{*} \mid \exists H \in \mathcal{A} : p = \operatorname{dim}(H)\right\}$$
Prove that $\mathcal{L}$ has a greatest element which we call $r$.