Cotangent space

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

𝑇 * 𝑝 𝑀 = (𝑇 𝑝 𝑀) *

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 $𝑝 \in 𝑀$ be a point, and consider the ideal $𝐼 𝑝 ⊴ 𝐶 𝛼 (𝑀)$ of functions vanishing at $𝑝$ . Then

𝑇 * 𝑝 𝑀 : = 𝐼 𝑝 / 𝐼 2 𝑝

Proof of equivalence

#missing/proof

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