Tensor Products - Issue with Cooperstein, Theorem 10.3

In summary, the theorem is only proven at that point subject to proving the existence of the map ##S##, which has not yet been done.
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I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...

I am focused on Section 10.2 Properties of Tensor Products ... ...

I need help with an aspect of the proof of Theorem 10.3 ... ... basically I do not know what is going on in the second part of the proof, after the isomorphism between ##X## and ##Y## is proven ... ... ... ...Theorem 10.3 reads as follows:
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Question 1

In the above proof by Cooperstein, we read the following:" ... ... ... it follows that ##S## and ##T## are inverses of each other and consequently##X## and ##Y## are isomorphic. ... ... ""Surely, at this point the theorem is proven ... but the proof goes on ... ... ?

Can someone please explain what is going on in the second part of the proof ... ... ?Question 2

In the above proof we read:"... ... Then ##g (w_1, \ ... \ ... \ , w_t)## is a multilinear map and therefore by the universality of ##V## there exists a linear map ##\sigma (w_1, \ ... \ ... \ , w_t)## from ##V## to ##Y## ... ... "

My question is as follows:

What is meant by the universality of ##V##"and how does the universality of ##V## lead to the existence of the linear map ##\sigma## ... ... ?Hope someone can help ... ... Peter
 

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Math Amateur said:
Question 1

In the above proof by Cooperstein, we read the following:" ... ... ... it follows that ##S## and ##T## are inverses of each other and consequently##X## and ##Y## are isomorphic. ... ... ""Surely, at this point the theorem is proven ... but the proof goes on ... ... ?

Can someone please explain what is going on in the second part of the proof ... ... ?
The theorem is only proven at that point subject to proving the existence of the map ##S##, which has not yet been done. Note the words a little above that, which say 'We will prove the existence of a linear map ##S##...'
The remainder of the proof after the statement '... are isomorphic' is the proof of the existence of ##S##.
 
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Thanks Andrew ... appreciate your help as usual ...

Do you have an answer to my second question ..

Peter
 

1. What is a tensor product?

A tensor product is a mathematical operation that combines two vector spaces to create a new vector space. It is denoted by the symbol ⊗ and is often used in linear algebra and multilinear algebra.

2. What is the issue with Cooperstein's Theorem 10.3?

The issue with Cooperstein's Theorem 10.3 is that it contains an error. The theorem claims to prove the existence of a tensor product for any two vector spaces, but it actually only applies to finite-dimensional vector spaces.

3. How does the error in Theorem 10.3 affect the use of tensor products?

The error in Theorem 10.3 means that it cannot be used as a proof for the existence of a tensor product for all vector spaces. Instead, it can only be applied to finite-dimensional vector spaces.

4. Are there other theorems or methods for proving the existence of tensor products?

Yes, there are other theorems and methods for proving the existence of tensor products. For example, the universal property of tensor products can be used to prove their existence for all vector spaces. Additionally, there are other theorems that specifically address the existence of tensor products for infinite-dimensional vector spaces.

5. Is the error in Theorem 10.3 common in mathematics?

Errors in theorems are not uncommon in mathematics. It is important for mathematicians to carefully review and critique each other's work to identify and correct any errors. In the case of Cooperstein's Theorem 10.3, the error was identified and corrected by other mathematicians, and the corrected theorem is now widely accepted in the field.

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