For precise definitions of the spaces referenced below, please refer to this question.
For $X \subset \mathbb{R}^n$, I understand that the $n$-dimensional tangent space $T_pX$ has a natural/canonical basis $((e_1)_p, \dots, (e_n)_p)$ where each $(e_i)_p$ is the standard basis vector $e_i$ translated to the point $p$. I also understand from linear algebra that the $n$-dimensional cotangent space $T^*_pX$ has a basis which is dual to the canonical basis given above and that the elements of this basis are typically denoted by $(dx^1)_p, \dots, (dx^n)_p$. In the cotangent space, these basis elements can be interpreted as the differentials of the canonical projection functions. Now, in this context, I am trying to parse the following claim from a text where, for notional ease, the point $p$ is suppressed:
The list $(e_1, \dots, e_n)$ is a module basis for $\mathcal{V}^1(X)$ and the list $(dx^1, \dots dx^n)$ is a module basis for $\Omega^1(X)$
I'm not sure how to make sense of this statement because, by definition, $ \mathcal{V}^1(X) = C^1(X,\mathbb{R}^n) \;\;\; \text{and} \;\;\; \Omega^1(X)= C^1(X, (\mathbb{R}^n)^{\prime}) $ where $(\mathbb{R}^n)^{\prime}$ denotes the continuous dual space of $\mathbb{R}^n$. These seem to me (and indeed, later in the text it is demonstrated) that both of these are infinite dimensional vector spaces.
So my question is, Is there a way to interpret the above statement so it makes sense, particularly in light of the fact that these spaces are infinite dimensional (!) ?