Terminologies used to describe tensor product of vector spaces

  • #1
cianfa72
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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 is written as . It is basically a bi-linear map defined on the cartesian product .

From a technical point of view, I believe it is actually the tensor product of the bidual , then using the canonical isomorphism we are allowed to understand it as the product tensor of the two vector .

In general when we talk of the product tensor we have in mind the dyadic tensor given by the product tensor of the canonically associated bi-duals and .

Other thing is the product tensor of vector spaces such as . This is again the full set of bi-linear application from the cartesian product (which is its own a vector space).

Does it make sense ? Thanks.
 
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  • #2
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 for and consider the vector space with basis modulo the subspace generated by .
 
  • #3
martinbn said:
Or you can start with a basis for and consider the vector space with basis .
Yes, but how do you define the product tensors ? They are defined by how they act on the associated dual-vectors , i.e. an element is a bi-linear map from the cartesian product of dual-spaces.
 
  • #4
cianfa72 said:
Yes, but how do you define the product tensors ? They are defined by how they act on the associated dual-vectors , i.e. an element is a bi-linear map from the cartesian product of dual-spaces.
Call them if you prefer. It is just a set of one element for each pair of basis vectors.
 
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  • #5
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.
 
  • #6
Was that your question?
 
  • #7
My question was to understand how the tensor product is defined for vectors . We can define it in terms of bi-duals and even though it seems to me like a tautology/circular (how is defined the tensor product ) ?
 
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  • #8
is the linear map on , for ; a ring; possibly a field, in the right category ( V Space, module, etc.), as image of a bilinear map B , that assumes the value
It's a matter of diagram-chasing. Maybe @fresh_42 can elaborate after he's done with his European vacation.
 
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