I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...(adsbygoogle = window.adsbygoogle || []).push({});

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

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# I Tensor Products - Issue with Cooperstein, Theorem 10.3

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