# Tensor Products: Universal Property or Maps

I've seen two introductions to the tensor product - in one, tensor products are maps to a field:

$V^\ast\otimes V^\ast:V\times V\rightarrow F$ because this is $({v^\ast}' w')(v^\ast w)$ which is the product of two real numbers/elements of the field

In the universal property, tensor products are being mapped (by a linear function $\hat{f}$, and $f$is multilinear):

$f:A\times B\to C$

$\otimes:A\times B\to A\otimes B$

$\hat{f}:A\otimes B\to C$

An image is given in the article "How to lose your fear of tensor products" for the second, and the first approach from XylyXylyX on youtube "What is a tensor? Lesson 5". How are these related, and why? Is the first the only way a tensor product can act, and if so how is that derived from the universal property?

Any insight is welcome!

• The first way is just a tensor product, in the second way of the dual of $V$ with itself. This is only a special case. The general situation is the solution of a universal problem for modules over rings, and is better understood from the point of view of categories theory. May 2 '18 at 23:57
• I edited the question slightly. Does this mean the first definition is only one of many ways to expand this product, and that it could map to anything not just a field? Can I find a way to think about ALL tps's or do you just have to see if it satisfies a universal property? May 3 '18 at 1:14
• That's right, it could map to a vcetor space over the same field (or more generally to a module over the same base ring). The canonical way to think about, in my opinion, is the universal property, as it can have many different realisations: This Wikipedia article:Tensor product of modules](en.wikipedia.org/wiki/Tensor_product_of_modules) might shed a light on this other:Tensor product. May 3 '18 at 7:29
• So the first definition as a map to a field is just a random construct that satisfies the universal property? Also, if f hat is invertible does that mean the tensor product is the set of all linear transformations on C? May 10 '18 at 3:41