As a closed nondegenerate symplectic 2-form ω. According to the Darboux's theorem, in a small neighbourhood around any point on M there exist suitable local coordinates p 1 , ⋯ ⋯ , p n , q 1 , ⋯ ⋯ , q n (canonical or symplectic coordinates) in which the symplectic form becomes: ω ω = ∑ ∑ i = 1 n d p i ∧ ∧ d q i . The form ω ω induces a natural isomorphism of the tangent space with the cotangent space : ⁠ T x M ≅ ≅ T x ∗ ∗ M ⁠. This is done by mapping a vector ξ ξ ∈ ∈ T x M to the 1-form ⁠ ω ω ξ ξ ∈ ∈ T x ∗ ∗ M ⁠, where ω ω ξ ξ ( η η ) = ω ω ( η η , ξ ξ ) for all ⁠ η η ∈ ∈ T x M ⁠. Due to the bilinearity and non-degeneracy of ⁠ ω ω ⁠, and the fact that ⁠ dim ⁡ ⁡ T x M = dim ⁡ ⁡ T x ∗ ∗ M ⁠, the mapping ξ ξ → → ω ω ξ ξ is indeed a linear isomorphism. This isomorphism is natural in that it does not change with change of coordinates on M . Repeating over all ⁠ x ∈ ∈ M ⁠, we end up with an isomorphism J − − 1 : Vect ( M ) → → Ω Ω 1 ( M ) between the infinite-dimensional space of smooth vector fields and that of smooth 1-forms. For every f , g ∈ ∈ C ∞ ∞ ( M , R ) and ⁠ ξ ξ , η η ∈ ∈ Vect ( M ) ⁠, J − − 1 ( f ξ ξ + g η η ) = f J − − 1 ( ξ ξ ) + g J − − 1 ( η η ) .