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 $$W^\omega = \{x\in V\, | \, \omega(x,y)=0\,\, \mbox{for all}\,\, y\in W\}.$$

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.

For the symplectic complement, we have the following dimension theorem.

Theorem [1,2] Let $(V,\omega)$ be a symplectic vector space, and let $W$ be a vector subspace of $V$ . Then $$\dim V = \dim W^\omega + \dim W.$$

Bibliography

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.