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I Properties of Tensor Products - Cooperstein, Theorem 10.3

  1. Mar 29, 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 regarding a property of tensor products ... ...


    The relevant part of Theorem 10.3 reads as follows:


    ?temp_hash=33b5ae1cfdc2f7ac325b528ac71daf3b.png



    In the above text from Cooperstein (Second Edition, page 355) we read the following:


    " ... ... The map [itex]f[/itex] is multilinear and therefore by the universality of [itex]Y[/itex] there is a linear map [itex]T \ : \ Y \longrightarrow X[/itex] such that


    [itex]T(v_1 \otimes \ ... \ v_s \otimes w_1 \otimes \ ... \ w_t )[/itex]

    [itex]= (v_1 \otimes \ ... \ v_s ) \otimes (w_1 \otimes \ ... \ w_t ) [/itex]

    ... ... ... "



    My question is as follows:

    What does Cooperstein mean by "the universality of [itex]Y[/itex]" and how does the universality of [itex]Y[/itex] justify the existence of the linear map [itex]T \ : \ Y \longrightarrow X[/itex] ... and further, if [itex]T[/itex] does exist, then how do we know it has the form shown ...

    Hope someone can help ...

    Peter



    *** Note ***

    Presumably, Cooperstein is referring to some "universal mapping property" or "universal mapping problem" such as he describes in his Section 10.1 Introduction to Tensor Products as follows:


    ?temp_hash=33b5ae1cfdc2f7ac325b528ac71daf3b.png
    ?temp_hash=33b5ae1cfdc2f7ac325b528ac71daf3b.png


    ... ... BUT ... ... there is no equivalent of the logic surrounding the mapping [itex]j[/itex] ... unless we are supposed to assume the existence of [itex]j[/itex] and its relation to the existence of [itex]T[/itex] ... ?


    Indeed reading Cooperstein's definition of a tensor product ... it reads like the tensor product is the solution to the UMP ... but I am having some trouble fitting the definition and the UMP to the situation in Theorem 10.3 ...

    Again, hope someone can help ...

    Peter
     

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  3. Mar 29, 2016 #2

    Samy_A

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    I will state it with 2 vector spaces for simplicity.

    The tensor product ##V \otimes W##, where ##V,W## are vector spaces, has the following universal property:
    If ##f: V \times W \to U## is a bilinear map to a vector space ##U##, then there is a unique linear map ##T : V \otimes W \to U## such that ##f(v,w)=T(v \otimes w)##.

    (Prop 5.1 in https://people.maths.ox.ac.uk/hitchin/hitchinnotes/Differentiable_manifolds/Chapter_2.pdf)
    (See also https://en.wikipedia.org/wiki/Tensor_product#Universal_property )
     
  4. Mar 29, 2016 #3
    Thanks Samy ... that looks pretty directly applicable to Theorem 10.3 ...

    Appreciate your help ...

    Peter
     
  5. Apr 3, 2016 #4



    Hi Samy,

    I was too quick to indicate that I understood what you said and the implications of what you said ... ...

    ... indeed ... I need some further help ...

    I cannot see how what you have said leads to T having the property that

    [itex]T(v_1 \otimes \ ... \ v_s \otimes w_1 \otimes \ ... \ w_t )[/itex]

    [itex]= (v_1 \otimes \ ... \ v_s ) \otimes (w_1 \otimes \ ... \ w_t ) [/itex]


    Can you explain why T has the above property?


    Hope you can help ...

    Peter
     
  6. Apr 4, 2016 #5

    Samy_A

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    It is a direct application of the universality of the tensor product. In sloppy language, that principle means that a multilinear map on a cartesian product of vector spaces defines ("can be extended to") a linear map on the tensor product of these vector spaces.

    In the proof, they define ##f: V_1 \times \dots \times V_s \times W_1 \times \dots \times W_t \to X=V \otimes W## by ##f(v_1,\dots ,v_s,w_1,\dots ,w_t)=(v_1 \otimes \dots \otimes v_s ) \otimes (w_1 \otimes \dots \otimes w_t)##.

    This ##f## is clearly multilinear. Therefore, there exists a linear ##T: V_1 \otimes \dots \otimes V_s \otimes W_1 \otimes \dots \otimes W_t \to X## such that ##T(v_1 \otimes \dots \otimes v_s \otimes w_1 \otimes \dots \otimes w_t) =f(v_1,\dots ,v_s,w_1,\dots ,w_t)=(v_1 \otimes \dots \otimes v_s ) \otimes (w_1 \otimes \dots \otimes w_t) ##
     
    Last edited: Apr 4, 2016
  7. Apr 4, 2016 #6
    Oh! , OK Samy, Thanks ...

    Most important principle ... seems to be used again and again in proofs of properties of tensor products ...

    Peter
     
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