I know that there are quite a few threads dealing with this question already. I have pored through them for quite some time and they have been informative. However there are still some clarifications that I seek. If this is too much of a duplicate, then I wouldnt mind if it is closed. I will sum up what I have understood.These are the definitions that I have seen:
Tensor Product of two vector spaces $V,W$ over a field $\mathbb F$ is another vector space $T$ over $\mathbb F$ equipped with a bilinear mapping $\otimes:V \times W \to T$ ; $\otimes (v,w) \mapsto v\otimes w$ such that for any vector space $U$ over the same field and any bilinear map $f:V \times W \to U$, $ \exists !$ linear map $f^* :T \to U$ where $f^*\circ \otimes =f $.
An (r,s) tensor is a multilinear function $B_{i_1 \ldots i_r}^{j_1 \ldots j_s}$ on $(V^*)^r \times (V)^s$ onto the field $\mathbb F$, denoted as $V \otimes \ldots \otimes V \otimes V^* \otimes \ldots \otimes V^*$ .
Now my question is the first is a definition of Tensor Product of Vector spaces, whereas the 2nd defines just a "tensor"??Also is the 2nd definition something that restricts itself to merely copies of a single vector space and its dual only??
Further a Torsion Tensor on a Riemannian Manifold M is defined as a vector valued two-form on $\chi(M) \times \chi (M) \to \chi(M) $ where $\chi (M)$ is the set of all $C^{\infty}$ vector fields on M. So in this case what kind of tensor is this??Its said to be (1,2) tensor , but I am not able to see that. The closest I would say is a (0,2) tensor. And does vector-valued mean this is the 1st definition in work??
Also am I right in saying the two definitions are similar only in the case of Finite Dimensional spaces whereas the 1st definition is the more general one??
Please help me differentiate between the two definitions and put the Torsion in proper context.