Terminologies used to describe tensor product of vector spaces

[cianfa72]

TL;DR Summary

About the terminologies used to describe tensor product of vector spaces

Hi,
I'm in trouble with the different terminologies used for tensor product of two vectors.

Namely a dyadic tensor product of vectors ##u, v \in V## is written as ##u \otimes v##. It is basically a bi-linear map defined on the cartesian product ##V^* \times V^* \rightarrow \mathbb R##.

From a technical point of view, I believe it is actually the tensor product of the bidual ##u^{**} \otimes v^{**}##, then using the canonical isomorphism ##V \cong V^{**}## we are allowed to understand it as the product tensor of the two vector ##u, v \in V##.

In general when we talk of the product tensor ##u \otimes v## we have in mind the dyadic tensor given by the product tensor of the canonically associated bi-duals ##u^{**}## and ##v^{**} \in V^{**}##.

Other thing is the product tensor of vector spaces such as ##V \otimes V##. This is again the full set of bi-linear application from the cartesian product ##V^* \times V^* \rightarrow \mathbb R## (which is its own a vector space).

Does it make sense ? Thanks.

 

[martinbn]

The tensor product exists and is unique (up to isomorphism), so it doesn't matter how you realize it. For example the way you describe it. Or you can start with a basis ##\{e_i\}## for ##V## and consider the vector space with basis ##\{e_i\otimes e_j\}## modulo the subspace generated by ##\{(a+b)\otimes c - a\otimes c - b\otimes c, etc\}##.

 

[cianfa72]

Or you can start with a basis ##\{e_i\}## for ##V## and consider the vector space with basis ##\{e_i\otimes e_j\}##.

Yes, but how do you define the product tensors ##e_i\otimes e_j## ? They are defined by how they act on the associated dual-vectors ##\{e^i\}##, i.e. an element ##e_i\otimes e_j## is a bi-linear map from the cartesian product of dual-spaces.

 

[martinbn]

Yes, but how do you define the product tensors ##e_i\otimes e_j## ? They are defined by how they act on the associated dual-vectors ##\{e^i\}##, i.e. an element ##e_i\otimes e_j## is a bi-linear map from the cartesian product of dual-spaces.

Call them ##e_{ij}## if you prefer. It is just a set of one element for each pair of basis vectors.

 

[WWGD]

The tensor product is often described through its Universal Property of linearizing multilinear maps. Given an inner-product, the isomorphism between V, V* becomes a natural one.

 

[WWGD]

Was that your question?

 

[cianfa72]

My question was to understand how the tensor product ##u \otimes v## is defined for vectors ##u,v \in V##. We can define it in terms of bi-duals ##u^{**}## and ##v^{**}## even though it seems to me like a tautology/circular (how is defined the tensor product ##u^{**} \otimes v^{**}##) ?

 

[WWGD]

## m \otimes_{R} n## is the linear map on ##M \otimes_{R} N \rightarrow P##, for ##P##; ##R## a ring; possibly a field, in the right category ( V Space, module, etc.), as image of a bilinear map B , that assumes the value## B(m,n)##
It's a matter of diagram-chasing. Maybe @fresh_42 can elaborate after he's done with his European vacation.