So we defined a (r,s) tensor field $w$ on a smooth manifold $M$ as a choice of (r,s) tensors for every point $p \in M$ living over the respected tangent spaces. So for every $p \in T_pM$, $w(p)$ is an Element of $\bigotimes^r T_pM \otimes \bigotimes^s T_pM^*$. We also mentioned in our lecture, that these tensor fields correspond uniquely to $C^\infty(M)$ mulitlinear maps: $TM^* \times \dots TM^* \times TM \times \dots \times TM \to C^\infty(M)$. We did not proove this theorem, but we kept defining tensor fields via their induced multilinear maps. Now I want to understand, how to define this correspondence. My idea was, that these multilinear maps themselves are elements of the tensor space $\bigotimes^r TM \otimes \bigotimes^s TM^*$ ($C^\infty(M)$ Tensor Module), by the usual identification of tensors and multilinear maps on the dual spaces. Now let $w$ be such a $C^\infty(M)$ multilinear maps as described above, then it can be written in the form:
$w=\sum h_1 \otimes \dots \otimes h_r \otimes g_1 \otimes \dots \otimes g_s$
where the tensor product is meant as a module tensor product.
The $h_i, g_j$ are vector and covector fields ($(1,0)$ and $(0,1)$ tensor fields), to get a tensor field in the definition I could define:
$w(p)= \sum h_1(p) \otimes \dots \otimes h_r(p) \otimes g_1(p) \otimes \dots \otimes g_s(p)$
where $h_1(p)$ is an abuse of notation, but uses implicitly the isomorphism from vector fields to $(1,0)$ tensor fields given by the canonical pairing.
I have meanwhile found a proof, which uses the fact that the multilinear map only depends on Values of the vector fields at $p$. I did not use that, so I am now wondering why my construction does not work, can you tell me in which step things go wrong?
