symplectic vector space

A symplectic vector space $(V,\omega )$is a finite dimensional real vector space $V$ equipped withan alternating non-degenerate 2-tensor, i.e.,a bilinear map $\omega :V\times V\to ℝ$that satisfies the following properties:

1. Alternating: For all $v,w\in V$, $\omega (v,w)=-\omega (w,v)$.

2. Non-degenerate: If $\omega (v,w)=0$ for all $w\in V$, then $v=0$.

The tensor $\omega$ iscalled a for $V$.

A linear automorphism $T\in \mathrm{Aut}(V)$ is called linear symplectomorphism when $T^{*}\omega =\omega$, i.e.

$$\omega (Tv,Tw)=\omega (v,w)\mathit{\hspace{1em}}\forall v,w\in W.$$

Linear symplectomorphisms of $(V,\omega )$ form a group (under composition of linear map) that is denoted by $\mathrm{Sp}(\mathrm{V},\omega )$.

One can show that a symplectic vector space is always even dimensional [1].