Properties of Tensor Products - Cooperstein, Theorem 10.3

[Math Amateur]

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:View attachment 5451In the above text from Cooperstein (Second Edition, page 355) we read the following:" ... ... The map \(\displaystyle f\) is multilinear and therefore by the universality of \(\displaystyle Y\) there is a linear map \(\displaystyle T \ : \ Y \longrightarrow X\) such that\(\displaystyle T(v_1 \otimes \ ... \ v_s \otimes w_1 \otimes \ ... \ w_t )\)

\(\displaystyle = (v_1 \otimes \ ... \ v_s ) \otimes (w_1 \otimes \ ... \ w_t ) \)

... ... ... "
My question is as follows:

What does Cooperstein mean by "the universality of \(\displaystyle Y\)" and how does the universality of \(\displaystyle Y\) justify the existence of the linear map \(\displaystyle T \ : \ Y \longrightarrow X\) ... and further, if \(\displaystyle T\) 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 is his Section 10.1 Introduction to Tensor Products as follows:https://www.physicsforums.com/attachments/5452
View attachment 5453

... ... BUT ... ... there is no equivalent of the logic surrounding the mapping \(\displaystyle j\) ... unless we are supposed to assume the existence of \(\displaystyle j\) and its relation to the existence of \(\displaystyle T\) ... ?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

 

[jvicens]

Dear Peter,

Thank you for bringing up this question regarding Theorem 10.3 in Cooperstein's book. The universality of Y in this case refers to the universal mapping property of the tensor product, which states that for any multilinear map f: V1 x V2 x ... x Vs x W1 x W2 x ... x Wt --> X, there exists a unique linear map T: Y --> X such that f(v1, v2, ..., vs, w1, w2, ..., wt) = T(v1 ⊗ v2 ⊗ ... ⊗ vs ⊗ w1 ⊗ w2 ⊗ ... ⊗ wt). In other words, the tensor product Y is the "best" object that can be used to represent the multilinear map f.

The existence of the linear map T can be justified by the definition of the tensor product, which states that Y is the quotient space of the free vector space generated by the elements of V1 x V2 x ... x Vs x W1 x W2 x ... x Wt by the subspace generated by the elements of the form (v1 ⊗ v2 ⊗ ... ⊗ vs ⊗ w1 ⊗ w2 ⊗ ... ⊗ wt) - f(v1, v2, ..., vs, w1, w2, ..., wt). This quotient space is constructed in such a way that the elements in Y correspond to the multilinear maps from V1 x V2 x ... x Vs x W1 x W2 x ... x Wt to X. Therefore, the linear map T: Y --> X with the specified form exists by construction.

I hope this helps clarify the concept of universality and the existence of the linear map T in Theorem 10.3. Please let me know if you have any further questions.