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Let M be a smooth manifold and $C^{\infty}(M)$ the smooth functions on it. Some authors are calling $C^{\infty}(M)$-multilinear mappings: $T:\mathcal{X}(M)^s\rightarrow\mathcal{X}$ Tensor fields of type (1,s).

(One can think for instance about the riemannian curvature tensor)

I think there is maybe an Isomorphism between both spaces. But I'm wondering, why some people are stressing this all the time? Do you have an idea??


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Yes, there is an isomorphism, generalizing the natural isomorphism \[ \operatorname{Hom}(E, F) \cong E^* \otimes F \] where $E$ and $F$ are finite dimensional vector spaces and $E^*$ denotes the dual space to $E$, and $\operatorname{Hom}(E,F)$ denotes the space of linear maps from $E$ to $F$.

I believe there are three main reasons to emphasize these isomorphisms:

  1. We think we understand linear maps from $E$ to $F$ very well, but tensor products can seem exotic. This helps make it clear that they are very closely related concepts, and thus tensor products should scare you only about as much as linear maps do.

  2. Seeing linear maps from $E$ to $F$ as elements of $E^* \otimes F$ sheds a new light on linear maps. For instance, it makes it obvious what is going on with matrix multiplication... this is something you already understand, obviously, but I think it is cool how it shows up here. It also makes it easier to understand the determinant. I think working out the details of these vague sentences is half the fun, so I leave it as an exercise. (OK, it's not that great, but it's pretty good, and it's worth doing yourself.)

  3. It is very often (notationally and/or conceptually) much easier to work with tensor products of bundles (tangent and cotangent bundles, for instance) than it is to work with spaces of linear maps.

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