symplectic complement
Definition [1, 2] Let $(V,\omega )$ be a symplectic vector space and let $W$ be a vector subspace of $V$. Then the symplectic complement of $W$ is
$${W}^{\omega}=\{x\in V\omega (x,y)=0\text{for all}y\in W\}.$$ 
It is easy to see that ${W}^{\omega}$ is also a vector subspace of $V$. Depending on the relation^{} between $W$ and ${W}^{\omega}$, $W$ is given different names.

1.
If $W\subset {W}^{\omega}$, then $W$ is an isotropic subspace (of $V$).

2.
If ${W}^{\omega}\subset W$, then $W$ is an coisotropic subspace.

3.
If $W\cap {W}^{\omega}=\{0\}$, then $W$ is an symplectic subspace.

4.
If $W={W}^{\omega}$, then $W$ is an Lagrangian subspace.
Theorem [1, 2] Let $(V,\omega )$ be a symplectic vector space, and let $W$ be a vector subspace of $V$. Then
$$dimV=dim{W}^{\omega}+dimW.$$ 
References
 1 D. McDuff, D. Salamon, Introduction to Symplectic Topology, Clarendon Press, 1997.
 2 R. Abraham, J.E. Marsden, Foundations of Mechanics, 2nd ed., Perseus Books, 1978.
Title  symplectic complement 

Canonical name  SymplecticComplement 
Date of creation  20130322 13:32:25 
Last modified on  20130322 13:32:25 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  8 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 15A04 
Defines  symplectic complement 
Defines  isotropic subspace 
Defines  coisotropic subspace 
Defines  symplectic subspace 
Defines  Lagrangian subspace 