# Definition (Vector Space) Given $(Z, \Omega)$, a ([[Non-Degeneracy (Quadratic Forms)#Weak Non-Degeneracy|weak]]) [[Symplectic Vector Space]] and $E$ and $F$ its [[vector subspace|subspaces]], then we define: $ E^\Omega = \{z \in Z | \Omega(z, e) = 0 \, \forall\, e \in E\} $ which is called the *symplectic orthogonal complement of $E$*. # Properties 1. $E^\Omega$ is closed 2. $E \subset F \implies F^\Omega \subset E^\Omega$ 3. $E^\Omega \cap F^\Omega = (E + F)^\Omega$ 4. If $Z$ is finite dimensional, then $\dim E + \dim E^\Omega = \dim Z$ 5. If $Z$ is finite dimensional $(E^\Omega)^\Omega = E$. This is also true in infinite dimensions if $E$ is closed. 6. If $E$ and $F$ are closed, then $(E \cap F)^\Omega = E^\Omega + F^\Omega$