Differential forms on vector spaces For a differential form on Euclidean space
$$dx_i:\mathbb{R}^n\rightarrow \mathbb{R}$$
$$dx_i:(\textbf{v})\rightarrow v_i$$
Looking more genereally at vector spaces, for a differential form on a vector space is it also true that
$$dx_i:V\rightarrow K$$
$$dx_i:(\textbf{v})\rightarrow v_i$$
Where $\textbf{v}\in V$, $V$ is a vector space over field $K$?
Sure, let $K$ be a field, $V$ a $K$-vector field and $(e_1,\ldots,e_n)$ be a basis for $V$, then $(\mathrm{d}x_1,\ldots,\mathrm{d}x_n)$ is defined to be the dual basis of $(e_1,\ldots,e_n)$, every linear form on $V$ is a linear combination of the $\mathrm{d}x_i$.
Sure, let $K$ be a field, $V$ a $K$-vector field and $(e_1,\ldots,e_n)$ be a basis for $V$, then $(\mathrm{d}x_1,\ldots,\mathrm{d}x_n)$ is defined to be the dual basis of $(e_1,\ldots,e_n)$, every linear form on $V$ is a linear combination of the $\mathrm{d}x_i$.