Why does $A_1(T_p \mathbb{R}^n)={T_p}^{*}(\mathbb{R}^n)$

Question

Hi i am reading about manifolds and there it is mentioned that $A_1(T_p \mathbb{R}^n)={T_p}^{*}(\mathbb{R}^n)$ where $A_k(V)$ is the space of alternating k-linear functions. Why does the above equality holds? Does the equality hold in general that is $A_k(T_p \mathbb{R}^n)={T_p}^{*}(\mathbb{R}^n)$. I know that on the rhs we have the cotangent space at $p$. Is this by definition or is there a proof for this? Is this because the basis for both lhs and rhs is same?

Answer 1

$A_1(\mathbb{R}^n)$ is the space of all alternating 1-linear functions. That is functions from $V\to \mathbb{R}$ and that we know equals to the dual space $V^*$. Hence proved