Terminologies used to describe tensor product of vector spaces

In summary, the tensor product of vector spaces is a mathematical operation that combines two vector spaces into a new one, capturing their relationships and interactions. Key terminologies include "bilinear map," which describes the linearity in both arguments; "tensor," referring to the elements of the resulting space; and "universal property," highlighting the unique mapping characteristics that define the tensor product. Other important concepts include "isomorphism," which indicates structural similarity between spaces, and "rank," which relates to the dimensions of the involved vector spaces. These terms collectively facilitate understanding and application of tensor products in various mathematical contexts.
  • #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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