Let $F$ be a vector subspace of a symplectic space $(E , \omega)$, with $\omega$-orthogonal $F^{\omega} = \{ x \in E \mid \forall y \in F , \omega ( x , y ) = 0 \}$. Is the subspace $F ^ { \omega }$ necessarily in direct sum with $F$?
Let $F$ be a vector subspace of a symplectic space $(E , \omega)$, with $\omega$-orthogonal $F^{\omega} = \{ x \in E \mid \forall y \in F , \omega ( x , y ) = 0 \}$. Is the subspace $F ^ { \omega }$ necessarily in direct sum with $F$?