In Gowers's article "How to lose your fear of tensor products", he uses two ways to construct the tensor product of two vector spaces $V$ and $W$. The following are the two ways I understand:
- $V\otimes W:=\operatorname{span}\{[v,w]\mid v\in V,w\in W\}$ where $[v,w]:{\mathcal L}(V\times W;{\mathbb R})\to {\mathbb R}$ such that $[v,w](f)\mapsto f(v,w)$
- $V\otimes W:=Z/E$ where $Z:=\operatorname{span}\{[[v,w]]\mid v\in V,w\in W\}$ and $E$ is the subspace of $Z$ generated by all vectors of one of the following four forms: $\begin{align} & [[v,w+w']]-[[v,w]]-[[v,w']]\\ & [[v+v',w]]-[[v,w]]+[[v',w]] \\ & [[av,w]]-a[[v,w]] \\ & [[v,aw]]-a[[v,w]] \end{align}$
Here are my questions:
- Are the definitions I wrote above correct?
- They look so different. How are they essentially the same?
- The set $\operatorname{span}\{[v,w]\mid v\in V,w\in W\}$ in (1) and $Z$ in (2) seem to be the "same". Do we have $Z\cong Z/E$ here?