Let $F$ be a vector subspace of a symplectic space $(E , \omega)$. The $\omega$-orthogonal of $F$ is defined as
$$F ^ { \omega } = \{ x \in E \mid \forall y \in F , \omega ( x , y ) = 0 \}.$$
Justify that $F ^ { \omega }$ is a vector subspace of $E$.