symplectic vector space
A symplectic vector space $(V,\omega )$ is a finite dimensional real vector space $V$ equipped with an alternating nondegenerate 2tensor, i.e., a bilinear map $\omega :V\times V\to \mathbb{R}$ that satisfies the following properties:

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

2.
Nondegenerate: If $\omega (v,w)=0$ for all $w\in V$, then $v=0$.
The tensor $\omega $ is called 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].
References
 1 D. McDuff, D. Salamon, Introduction to Symplectic Topology, Clarendon Press, 1997.
Title  symplectic vector space 

Canonical name  SymplecticVectorSpace 
Date of creation  20130322 13:32:22 
Last modified on  20130322 13:32:22 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  11 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 53D05 
Defines  symplectic vector space 
Defines  linear symplectomorphism 