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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:
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
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
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