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I Tensor Products - Issue with Cooperstein, Theorem 10.3

  1. Apr 3, 2016 #1
    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:


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    ?temp_hash=97465cbe0a1cc9eba2d5bd63d741f3fa.png
    ?temp_hash=97465cbe0a1cc9eba2d5bd63d741f3fa.png



    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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    Last edited: Apr 3, 2016
  2. jcsd
  3. Apr 3, 2016 #2

    andrewkirk

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    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##.
     
  4. Apr 3, 2016 #3
    Thanks Andrew ... appreciate your help as usual ...

    Do you have an answer to my second question ..

    Peter
     
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