Symplectic vector space
In mathematics, a symplectic vector space is a vector space V {\displaystyle V} over a field F {\displaystyle F} (for example the real numbers R {\displaystyle \mathbb {R} } ) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping ω : V × V → F {\displaystyle \omega :V\times V\to F} that is Bilinear Linear in each argument separately; Alternating ω ( v , v ) = 0 {\displaystyle \omega (v,v)=0} holds for all v ∈ V {\displaystyle v\in V} ; and Non-degenerate ω ( v , u ) = 0 {\displaystyle \omega (v,u)=0} for all v ∈ V {\displaystyle v\in V} implies that u = 0 {\displaystyle u=0} .