Let $(V,\omega)$ be a finite-dimensional symplectic [vector space](/page/Vector%20Space) over $\mathbb R$ with $\dim V=2n$. For a subspace $W\subset V$, let
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\begin{align*}
W^\omega=\{v\in V:\omega(v,w)=0\text{ for every }w\in W\}
\end{align*}
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denote its symplectic orthogonal complement. Then the following hold:
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If $W$ is isotropic, then $\dim W\le n$. If $W$ is coisotropic, then $\dim W\ge n$. If $W$ is Lagrangian, then $\dim W=n$. Conversely, if $W$ is isotropic and $\dim W=n$, then $W$ is Lagrangian.