understanding of the "tensor product of vector spaces"

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:

1. $V\otimes W:=\operatorname{span}\{[v,w]\mid v\in V,w\in W\}$ where $[v,w]:{\mathcal L}(V\times W;{\mathbb R})\to {\mathbb R}$ such that $$[v,w](f)\mapsto f(v,w)$$
2. $V\otimes W:=Z/E$ 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?

Solution 1:

The first definition comes from the philosophy that students are bad at understanding abstract definitions and would prefer to see the tensor product defined as a space of functions of some kind. This is the reason that some books define the tensor product of $V$ and $W$ to simply be the space of bilinear functions $V \times W \to k$ ($k$ the underlying field), but this defines what in standard terminology is called the dual $(V \otimes W)^{\ast}$ of the tensor product.

For finite-dimensional vector spaces, $(V^{\ast})^{\ast}$ is canonically isomorphic to $V$, and that is the property that Gowers is taking advantage of in the first definition, which is basically a definition of $((V \otimes W)^{\ast})^{\ast} \cong V \otimes W$. The second definition is essentially the standard definition.

To answer your last question, no, we do not. $Z$ is infinite-dimensional whenever the underlying field is infinite. It is really, really huge, in fact pointlessly huge; the relations are there for a reason.

Solution 2:

I'm going to go way out on a limb and instead of answering the questions actually posed, I'll propose a way to think about..... um OK, here it is: what's the difference between an ordered pair of vectors and a tensor product of two vectors? It is this: If you multiply one of the two vectors by $c$ and the other by $1/c$, then you've got a different ordered pair of vectors, but you've got the same tensor product.