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 $$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$ For every unit vector $y \in F$ orthogonal to $x_0$, we set, for every real $t$, $$\begin{aligned} & \gamma(t) = x_0 \cos t + y \sin t \\ & \varphi(t) = \langle u \circ \gamma(t), \gamma(t) \rangle \end{aligned}$$ Deduce that $u(x_0)$ is orthogonal to $y$.
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
$$\langle u(x_0), x_0 \rangle = \sup_{x \in F, \|x\|=1} \langle u(x), x \rangle$$
For every unit vector $y \in F$ orthogonal to $x_0$, we set, for every real $t$,
$$\begin{aligned} & \gamma(t) = x_0 \cos t + y \sin t \\ & \varphi(t) = \langle u \circ \gamma(t), \gamma(t) \rangle \end{aligned}$$
Deduce that $u(x_0)$ is orthogonal to $y$.