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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 ##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.

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.

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