- #1
Math Amateur
Gold Member
MHB
- 3,998
- 48
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with the proof of Lemma 10.1 on the uniqueness of a tensor product ... ... Before proving the uniqueness (up to an isomorphism) of a tensor product, Cooperstein defines a tensor product ... as follows:View attachment 5380Cooperstein then states that he is going to show that the tensor product is essentially unique (up to an isomorphism) ... and then presents Lemma 10.1 ... ... as follows ... ...
https://www.physicsforums.com/attachments/5381My questions are as follows:Question 1
I am assuming that the point of the Theorem is to show that \(\displaystyle V\) is isomorphic to \(\displaystyle Z\) and, I think, also that \(\displaystyle \gamma\) is isomorphic to \(\displaystyle \delta\) ... ... is that correct?
Now I am assuming that once we have shown that \(\displaystyle ST = I_V\) ... ... and \(\displaystyle TS = I_Z\) ... ...
then we have shown that \(\displaystyle S\) (and \(\displaystyle T\) for that matter) is a bijection ... and hence \(\displaystyle V\) and \(\displaystyle Z\) are isomorphic ... is that correct?
... ... BUT ... ...... what exactly are we showing if we demonstrate that \(\displaystyle T \gamma = \delta\) ... ... and also that \(\displaystyle S \delta = \gamma\) ... ... unless it is that, since from (i), \(\displaystyle S\) and \(\displaystyle T\) are bijections then \(\displaystyle \delta\) and \(\displaystyle \gamma\) are isomorphic
... ... is that correct?
Question 2
In Cooperstein's proof of Lemma 10.1 we read:
" ... ... Since \(\displaystyle (V, \gamma)\) is a tensor product of \(\displaystyle V_1, \ ... \ ... \ , \ V_m\) over \(\displaystyle \mathbb{F}\)
and \(\displaystyle \delta\) is a multilinear map from \(\displaystyle V_1, \ ... \ ... \ , \ V_m \) to \(\displaystyle Z\), then there exists a
unique linear map \(\displaystyle T \ : \ V \longrightarrow Z\) such that \(\displaystyle T \gamma = \delta\) ... ... "Can someone please explain to me why the above statement is true ... ...Peter==========================================================**** EDIT ****
I have been reflecting on Question 2 ...
... Now ... in Cooperstein's definition of the tensor product ... we read ...
" ... a pair \(\displaystyle (V , \gamma)\) consisting of a vector space \(\displaystyle V\) over \(\displaystyle \mathbb{F}\)
and a multilinear map \(\displaystyle \gamma \ : \ V_1 \times \ ... \ \times
V_m \longrightarrow V\) is a tensor product of
\(\displaystyle V_1, \ ... \ ... \ , \ V_m\) over \(\displaystyle \mathbb{F}\) if, whenever \(\displaystyle W\) is a vector space over \(\displaystyle \mathbb{F}\)
and \(\displaystyle f \ : \ V_1 \times \ ... \ \times V_m \longrightarrow W\) is a multilinear map,
then there exists a unique bilinear map \(\displaystyle T \ : \ V \longrightarrow W\) such that \(\displaystyle T \circ \gamma = f\) ... ... "
Well ... presumably \(\displaystyle Z\) can stand in for \(\displaystyle W\) and \(\displaystyle \delta\) can stand in for \(\displaystyle f\) ... ... which gives the situation shown in Figure 1 below ... ...
View attachment 5382
The conditions of the definition of the tensor product imply there exists a unique linear map \(\displaystyle T \ : \ V \longrightarrow Z\) such that \(\displaystyle T \circ \gamma = \delta\) ... ...
Is that correct? Can someone please confirm ... or point out errors and shortcomings ... ?
Peter
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with the proof of Lemma 10.1 on the uniqueness of a tensor product ... ... Before proving the uniqueness (up to an isomorphism) of a tensor product, Cooperstein defines a tensor product ... as follows:View attachment 5380Cooperstein then states that he is going to show that the tensor product is essentially unique (up to an isomorphism) ... and then presents Lemma 10.1 ... ... as follows ... ...
https://www.physicsforums.com/attachments/5381My questions are as follows:Question 1
I am assuming that the point of the Theorem is to show that \(\displaystyle V\) is isomorphic to \(\displaystyle Z\) and, I think, also that \(\displaystyle \gamma\) is isomorphic to \(\displaystyle \delta\) ... ... is that correct?
Now I am assuming that once we have shown that \(\displaystyle ST = I_V\) ... ... and \(\displaystyle TS = I_Z\) ... ...
then we have shown that \(\displaystyle S\) (and \(\displaystyle T\) for that matter) is a bijection ... and hence \(\displaystyle V\) and \(\displaystyle Z\) are isomorphic ... is that correct?
... ... BUT ... ...... what exactly are we showing if we demonstrate that \(\displaystyle T \gamma = \delta\) ... ... and also that \(\displaystyle S \delta = \gamma\) ... ... unless it is that, since from (i), \(\displaystyle S\) and \(\displaystyle T\) are bijections then \(\displaystyle \delta\) and \(\displaystyle \gamma\) are isomorphic
... ... is that correct?
Question 2
In Cooperstein's proof of Lemma 10.1 we read:
" ... ... Since \(\displaystyle (V, \gamma)\) is a tensor product of \(\displaystyle V_1, \ ... \ ... \ , \ V_m\) over \(\displaystyle \mathbb{F}\)
and \(\displaystyle \delta\) is a multilinear map from \(\displaystyle V_1, \ ... \ ... \ , \ V_m \) to \(\displaystyle Z\), then there exists a
unique linear map \(\displaystyle T \ : \ V \longrightarrow Z\) such that \(\displaystyle T \gamma = \delta\) ... ... "Can someone please explain to me why the above statement is true ... ...Peter==========================================================**** EDIT ****
I have been reflecting on Question 2 ...
... Now ... in Cooperstein's definition of the tensor product ... we read ...
" ... a pair \(\displaystyle (V , \gamma)\) consisting of a vector space \(\displaystyle V\) over \(\displaystyle \mathbb{F}\)
and a multilinear map \(\displaystyle \gamma \ : \ V_1 \times \ ... \ \times
V_m \longrightarrow V\) is a tensor product of
\(\displaystyle V_1, \ ... \ ... \ , \ V_m\) over \(\displaystyle \mathbb{F}\) if, whenever \(\displaystyle W\) is a vector space over \(\displaystyle \mathbb{F}\)
and \(\displaystyle f \ : \ V_1 \times \ ... \ \times V_m \longrightarrow W\) is a multilinear map,
then there exists a unique bilinear map \(\displaystyle T \ : \ V \longrightarrow W\) such that \(\displaystyle T \circ \gamma = f\) ... ... "
Well ... presumably \(\displaystyle Z\) can stand in for \(\displaystyle W\) and \(\displaystyle \delta\) can stand in for \(\displaystyle f\) ... ... which gives the situation shown in Figure 1 below ... ...
View attachment 5382
The conditions of the definition of the tensor product imply there exists a unique linear map \(\displaystyle T \ : \ V \longrightarrow Z\) such that \(\displaystyle T \circ \gamma = \delta\) ... ...
Is that correct? Can someone please confirm ... or point out errors and shortcomings ... ?
Peter
Last edited: