Natural Isomorphism b/w Dual Spaces Tensor Prod & Multilinear Form Space

  • #1
caffeinemachine
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I am trying to prove the following.

Let be finite dimensional vector spaces over a field .
There is a natural isomorphism between and .

Define a map as

for all .
It can be seen that is a multilinear map.
By the universal property of tensor product, there exists a unique linear map such that .

We also know that

Thus we just need to show that to show that and are isomorphic.

My Problem: I want to show the triviality of the kernel in a basis free manner. But here I am stuck.

Can anybody help?
 
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  • #2
caffeinemachine said:
I am trying to prove the following.

Let be finite dimensional vector spaces over a field .
There is a natural isomorphism between and .

Define a map as

for all .
It can be seen that is a multilinear map.
By the universal property of tensor product, there exists a unique linear map such that .

We also know that

Thus we just need to show that to show that and are isomorphic.

My Problem: I want to show the triviality of the kernel in a basis free manner. But here I am stuck.

Can anybody help?
Instead of trying to show that is injective, I think it would be easier to show that it is surjective. This should somehow be equivalent to the fact that a matrix is a sum of rank matrices.
 
  • #3
Opalg said:
Instead of trying to show that is injective, I think it would be easier to show that it is surjective. This should somehow be equivalent to the fact that a matrix is a sum of rank matrices.
Hello Opalg,

Sorry for the late reply. I somehow forgot about this post.

I can show that is surjective by choosing a basis. I am getting more and more convinced that this cannot be done without choosing a basis.
 
  • #4
caffeinemachine said:
I am getting more and more convinced that this cannot be done without choosing a basis.
I tend to agree. For one thing, I believe that the result is false if the spaces are infinite-dimensional. So you somehow need to make use of the fact that the spaces are finite-dimensional, and the obvious way is to choose bases for them.
 

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