Equivalent definition of continuity/differentiability of a differential form?

Question

I just started learning about differential forms, and i found this definition in my textbook: Let $A \subseteq \mathbb{R}^n$. We say that a differential form $\omega : A \rightarrow (\mathbb{R}^n)^{*}$ is $C^k$ if all the components of $\omega$ with respect to the canonical basis of $(\mathbb{R}^n)^{*}$ are $C^k$. Do we get the same thing if we make $(\mathbb{R}^n)^{*}$ a normed space by putting $|a_1 {e^*}_1 + \cdot \cdot \cdot + a_n {e^*}_n| = \sqrt{a_1^2 + \cdot \cdot \cdot + a_n^2}$, and then defining continuity and differentiability with respect to this norm?

Answer 1

There is an isomorphism $(\Bbb R^n)^* \simeq \Bbb R^n$ sending the standard dual basis to the standard basis, so the Euclidean norm on the first space is just the Euclidean norm on the second. Thus, it makes no difference to define $C^k$ regularity via components or directly in the dual. (Also, all norms in finite-dimensional spaces are equivalent to one another, so the Euclidean norm does not play a special role in these definitions.)