What is the cotangent space $T_p$^* $R^n$ and why is it written as Hom($T_p R^n$, R)?

Conditions

• $p$ is a basepoint.

Reasoning, step by step

1. Identify the tangent space $T_p\mathbb{R}^n$.
2. Define the cotangent space $T_p^*\mathbb{R}^n$ as the dual space $(T_p\mathbb{R}^n)^*$.
3. Express the dual space as the set of linear maps $\operatorname{Hom}(T_p\mathbb{R}^n, \mathbb{R})$.

Example

The slide defines $T_p^*\mathbb{R}^n = (T_p\mathbb{R}^n)^*$, then unpacks this as $\operatorname{Hom}(T_p\mathbb{R}^n, \mathbb{R})$, the space of linear maps from the tangent space at $p$ to the field $\mathbb{R}$.

Common misconceptions

• Thinking the cotangent space is the same as the tangent space.
• Assuming the maps go to $\mathbb{R}^n$ instead of $\mathbb{R}$.

Watch the explanation

Connected concepts

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• How does the cotangent-space definition change for a general manifold M over a field F?
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