Cotangent space

The cotangent space f a differentiable manifold at a point is the dual of the tangent space, i.e. it consists of linear functionals on the tangent space. #m/def/geo/diff

However, for various reasons one might want to define the cotangent space independently (which enables defining the tangent space as the dual of the cotangent space), as given below. See also Cotangent bundle.

Alternative definition

Let be a point, and consider the ideal of functions vanishing at . Then

Proof of equivalence

#missing/proof

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