# “Cauchy-Schwartz” for symplectic forms

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

• Typo: It is Schwarz. – user371663 May 21 '18 at 13:54
• Hint: the standard complex structure $J_0$ and the standard scalar product $g_0$ are such that $g_0(-, -) = \omega_0(-, J_0 -)$, and $J_0$ is both $\omega_0$-symplectic and $g_0$-orthogonal. – Jordan Payette May 21 '18 at 20:14