A symplectic form on a smooth manifold 
is a smooth closed 2-form 
on 
which is nondegenerate such that at every point 
, the alternating bilinear form 
on the tangent space 
is nondegenerate.
A symplectic form on a vector space 
over 
is a function 
(defined for all 
and taking values in 
) which satisfies
 |
(1)
|
 |
(2)
|
and
 |
(3)
|
is called non-degenerate if 
for all 
implies that 
. Symplectic forms can exist on 
(or 
) only if 
(or 
) is even-dimensional. An example
of a symplectic form over a vector space is the complex Hilbert
space with inner product 
given by
 |
(4)
|
