I would like to show some sort of "Cauchy-Schwartz" inequality for symplectic maps.
i.e. given a symplectic map $\phi:\mathbb{R}^{2n}\rightarrow \mathbb{R}^{2n}$ and $u := \phi(e_1),v:=\phi(f_1)$, then $$1 = \omega_0(u,v) \leq\vert u\vert\cdot\vert v\vert$$
The equality is easy. Indeed, $$\omega_0(u,v) = \omega_0(\phi(e_1),\phi(f_1)) = \omega_0(e_1,f_1) = 1$$
But for the second inequality, I don't really know what to do