Let $(E, \langle \cdot, \cdot \rangle)$ be a real pre-Hilbert space, with associated norm $\|\cdot\|$. Let $u$ be an endomorphism of $E$ satisfying,
$$\forall (x,y) \in E^2, \quad \langle u(x), y \rangle = \langle x, u(y) \rangle$$
Suppose that there exists a unit vector $x_0 \in F$ satisfying
$$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$
Show that $x_0$ is an eigenvector of $u$.