I Tensor Products - Issue with Cooperstein, Theorem 10.3

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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
 

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