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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 (!) ?

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up vote 1 down vote accepted

As $\mathbb{R}$-modules (real vector spaces) both $\mathcal{V}^1(X)$ and $\Omega^1(X)$ are usually infinite dimensional, but as modules over the ring $C^1(X,\mathbb{R})$ they are finitely generated and free (with basis as you've listed).

It is easy to see why: take any $\omega \in \mathcal{V}^1(X)$, and note that there are $\{ \omega_1, \omega_2, \ldots, \omega_n \} \subseteq C^1(X,\mathbb{R})$ such that $\omega(x) = (\omega_1(x), \omega_2(x), \ldots, \omega_n(x))$ for all $x \in X$, i.e. $\omega = \sum_{i =1}^n \omega_i e_i$. The proof for $\Omega^1(X)$ is similar.

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Thanks for the explanation; I was not properly discerning the distinction between these spaces when being considered as $C^1(X, \mathbb{R})$-modules versus $\mathbb{R}$-modules. – ItsNotObvious Apr 4 '12 at 18:15

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