I'm struggling with this question:
Show that $\mathcal{T}_{(r,s)} \cong {\frak{M}}_{(r,s)}$. That is, show that the set of $C^{\infty}(M)$-module of all smooth $(r,s)$ tensor fields are isomorphic to the set of $C^{\infty}(M)$-module of multilinear maps from $E^1(M) \times...\times E^1(M)\times{\frak{X}}(M)\times...\times {\frak{X}}(M)$ to $C^{\infty}(M)$.
The text i'm using (Warner's Foundations of Differentiable Manifolds and Lie Groups) shows, for simplicity, the case where $(r,s)=(0,1)$ by showing that for any $\eta \in {\frak{M}}_{(0,1)}$, $\eta(X)(m)$ depends only on $X|m$. I'm trying to see what happens in the case where $(r,s)=(1,1)$ in detail, or $(r,s)=(1,2)$.
I'm guessing that I should claim that
$\eta(\omega ,X)(m)$ depends only on $\omega|m$ and $X|m$
and I should prove this claim using the fact that it's suffice to show that $\omega|m=0$ and $X|m=0$ implies $\eta(\omega ,X)(m)=0$.
I'm stuck here, and I would appreciate it if someone could give me some detailed solution, as I am trying to understand the concept of tensor fields in detail by examining examples in detail.
Oh, and also, I saw somewhere that the dual of ${\frak{X}}(M)$ is isomorphic to $E^1(M)$. I would love some brief explanation for this too, thanks!