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 regarding a property of tensor products ... ...

The relevant part of Theorem 10.3 reads as follows:

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 byand 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 ..."the universality of [itex]Y[/itex]"

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:

... ... 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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# I Properties of Tensor Products - Cooperstein, Theorem 10.3

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