A symplectic form on a smooth manifold is a smooth closed 2form 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 nondegenerate if for all implies that . Symplectic forms can exist on (or ) only if (or ) is evendimensional. An example of a symplectic form over a vector space is the complex Hilbert space with inner product given by
(4)
