Tensor Products - Issue with Cooperstein, Theorem 10.3

[Math Amateur]

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:

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

 

[andrewkirk]

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 $S$.

 

[Math Amateur]

Thanks Andrew ... appreciate your help as usual ...

Do you have an answer to my second question ..

Peter