Relationship between a tensor and the tensor product I am trying to wrap my head around what exactly a tensor is. I am familiar with the notion of a tensor product, both the basis-dependent and basis-independent definitions, and there seems to be a link between them and tensors. I have seen some sources define tensors as elements of a tensor product, but there seems to be more to it than this. Can a tensor be defined in terms of tensor products, and if so, what else can be said?
I have already gone through the following questions:

*

*What is the difference between tensors and tensor products?


*What, Exactly, Is a Tensor?
but my confusion still remains.
 A: Short answer: To a mathematician, a tensor is just an element of a tensor product. As you ask in the comments, it is correct to say that a tensor is to a tensor product what a vector is to a vector space.
Medium answer: A tremendous amount of confusion here is caused by the fact that the word "tensor" has (at least) five different meanings which are rarely distinguished, and different communities of practitioners (e.g. mathematicians vs. physicists) default to different ones. Here they are:

*

*A tensor is an element of a tensor product.

*A tensor of type $(m, n)$ relative to a (usually finite-dimensional) vector space $V$ is an element of the tensor product $V^{\otimes n} \otimes (V^{\ast})^{\otimes m}$.

*A tensor is a multidimensional array of numbers (a hypermatrix).

*A tensor is a tensor field (e.g. the Riemann curvature tensor or the stress-energy tensor).

*A tensor is an object which transforms in a certain way under change of coordinates.

These concepts are of course related (2 is a special case of 1, 3 is how you represent a tensor in sense either 1 or 2 using a choice of basis, 5 is equivalent to either 2 or 4 depending on context, 4 is a generalization of 2) but they are not the same and much confusion results from not carefully distinguishing them.
Additionally sense 2 is more sophisticated than it appears, because we have the freedom to interpret a tensor of type $(m, n)$ as a function between tensor products in $2^{m+n}$ different ways. For example, a tensor of type $(1, 1)$ can be thought of as any of

*

*an element of $V \otimes V^{\ast}$,

*a linear map $V \to V$,

*a linear map $V^{\ast} \to V^{\ast}$, or

*a linear functional $V^{\ast} \otimes V \to K$ where $K$ is the underlying field

and any of these representations may be appropriate depending on context.
