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 \(\displaystyle X\) and \(\displaystyle Y\) is proven ... ... ... ... Theorem 10.3 reads as follows:
https://www.physicsforums.com/attachments/5477
View attachment 5478
View attachment 5479Question 1


In the above proof by Cooperstein, we read the following:" ... ... ... it follows that \(\displaystyle S\) and \(\displaystyle T\) are inverses of each other and consequently \(\displaystyle X\) and \(\displaystyle Y\) 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 \(\displaystyle g (w_1, \ ... \ ... \ , w_t)\) is a multilinear map and therefore by the universality of \(\displaystyle V\) there exists a linear map \(\displaystyle \sigma (w_1, \ ... \ ... \ , w_t)\) from \(\displaystyle V\) to \(\displaystyle Y\) ... ... "

My question is as follows:

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

 

[Deveno]

He is referring to the universal mapping property of the tensor product, which guarantees the existence of such a map, since the tensor product (by its defining UMP) turns a multilinear map:

$g:V_1 \times \cdots \times V_s \to Y$

into a LINEAR map:

$L: V_1 \otimes \cdots \otimes V_s \to Y$

(equivalently, we have $L$ is the unique linear map such that $L \circ \otimes = g$)

Note that we get a *different* $g$ for each element of $W_1 \times \cdots \times W_t$. Thus for each:

$\omega \in W_1 \times \cdot \times W_n$, we have the multilinear function $\omega \mapsto L_{\omega}$, which when we tensor the $W$'s, induces the linear function $\sigma$.

The proof *would* be done when we show $S,T$ inverses, but the existence of $S$ needs to be established, which is what the second half of the proof is doing. This existence is shown by invoking the UMP three separate times, for different vector spaces.