Let $V$ be a finite-dimensional real [vector space](/page/Vector%20Space) and let $\omega:V\times V\to\mathbb R$ be a bilinear, skew-symmetric, nondegenerate form. Then there is an integer $n\ge 0$ and there exist vectors $e_1,\dots,e_n,f_1,\dots,f_n\in V$ such that $(e_1,\dots,e_n,f_1,\dots,f_n)$ is a basis of $V$ and
for all $1\le i,j\le n$. In particular, $\dim V=2n$ is even.
paragraph
admin
Consequently, if $\omega_0:\mathbb R^{2n}\times\mathbb R^{2n}\to\mathbb R$ denotes the standard symplectic form on $\mathbb R^{2n}$ with coordinates $(q_1,\dots,q_n,p_1,\dots,p_n)$, defined by