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:
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 $$S$$ 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

 

[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.