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.