In this question, $E$ is a real vector space of dimension $n$ and $f$ is an endomorphism of $E$.
\textbf{V.D.1)} Show that if $f$ is diagonalisable then there exist $n$ hyperplanes of $E$, $(H_i)_{1 \leqslant i \leqslant n}$, all stable by $f$ such that $\bigcap_{i=1}^{n} H_i = \{0\}$.
\textbf{V.D.2)} Is an endomorphism $f$ of $E$ for which there exist $n$ hyperplanes of $E$ stable by $f$ and with intersection reduced to the zero vector necessarily diagonalisable?