Let $(V,\omega)$ be a finite-dimensional real symplectic [vector space](/page/Vector%20Space), and let $W\subset V$ be a linear subspace. Define the symplectic complement of $W$ by
paragraph
admin
\begin{align*}
W^\omega := \{v\in V : \omega(v,w)=0\text{ for every }w\in W\}.
\end{align*}
latex_env
admin
Then
paragraph
admin
\begin{align*}
\dim W+\dim W^\omega=\dim V.
\end{align*}