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I am reading Chapter 13: Differential Forms ... ... and am currently focused on Section 13.1 Tensor Fields ...

I need some help in order to fully understand some statements by Browder in Section 13.1 ... ...

Section 13.1 reads as follows:

In the above text we read the following:

" ... ... We observe also that \(\displaystyle \text{dx}_P^j (h^1, \cdot \cdot \cdot , h^n) = h^j\) ... ... "

My question is what is the nature of the \(\displaystyle h^i\) ... given that \(\displaystyle \text{dx}_P^j\) is a constant mapping from an open subset \(\displaystyle U\) of \(\displaystyle \mathbb{R}^n\) it seems that the \(\displaystyle h^i\) are real numbers ...

... BUT ... then it seems strange that \(\displaystyle \text{dx}_P^j (h^1, \cdot \cdot \cdot , h^n) = h^j\) ... that is ... how can the function evaluate to a real number when it is a constant mapping into \(\displaystyle \mathbb{R}^{ n \ast }\) ... ... it should evaluate to a linear functional, surely ....

... indeed ... should Browder have written ...

\(\displaystyle \text{dx}_P^j (h^1, \cdot \cdot \cdot , h^n) = \tilde{e}^j \)

Can someone please clarify the above ...

Peter