Symplectic vector space

A symplectic vector space is a pair $(V,\omega )$ where $V$ is a vector space and $\omega :V\times V\to \mathbb{R}$ is a non-degenerate skew-symmetric bilinear form. In more detail, this means that Skew-symmetric $\omega (x,y)=-\omega (y,x)$ Non-degenerate For every $v\in V$, $\omega (v,w)=0\forall w\in V⟹v=0$

Morphisms

A morphism in the category of symplectic vector spaces is a linear symplectomorphism.

Subspaces

For a subspace $W\subseteq V$, we call the subspace

• isotropic if $W\subset W^{\omega }$
• coisotropic if $W^{\omega }\subset W$
• symplectic if $W\cap W^{\omega }=\{0\}$
• Lagrangian if $W=W^{\omega }$ Where $W^{\omega }$ denotes the symplectic complement.