understanding of the "tensor product of vector spaces"

Question

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?

Answer 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.

Answer 2

As for your first question: yes.

As for your second question: for instance, elements of the basis of the "first" $V\otimes W$, make the expression

$$ [v, w+w'] - [v,w]- [v,w'] $$

to be equal to zero. Indeed, by definition, this guy evaluated on any bilinear map $f: V \times W \longrightarrow \mathbb{R}$ is

\begin{align} ([v, w+w'] - [v,w]- [v,w']) (f) &= [v, w+w'] (f) - [v,w] (f)- [v,w'] (f) \\ &= f(v,w+w') - f(v,w) - f(v,w') \\ &= 0 \end{align}

by the bilinearity of $f$.

In the same way, you can verify that elements of the basis of the "first" $V\otimes W$ make all the expression you're quotienting out in the second $V\otimes W$ to be zero. Hence, it is true that

$$ [v, w+w'] - [v,w]- [v,w'] = 0 $$

as well as

$$ [[v, w+w']] - [[v,w]]- [[v,w']] = 0 $$

in $Z/E$.

As for your third question: no.

Answer 3

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.

Answer 4

Edit. This answer had been moved here from this question, question which had been closed and then reopen. After the reopening I rolled back the original answer with the help of Will Orrick.

Answer 5

Gowers's first defintion.

$\def\bbf{\mathbf{R}}$Given any vector spaces $V,W,X$ over the field $\bbf$, the notation $L(V;X)$ is commonly used as the set of all linear functions from $V$ to $X$ and $L(V,W;X)$ denotes the set of all bilinear functions from the product space $V\times W$ to $X$.

In Gowers's article, $B$ denotes "the set of all bilinear maps from $V\times W$ to $\bbf$". By the above convention, $B=L(V, W;\bbf)$. His first definition can be interpreted as follows:

Given any $(v,w)\in V\times W$, define $[v,w]:B\to\bbf$ as a function with $ [v,w](f):=f(v,w)\;. $ $\def\span{\operatorname{span}}$Define $$V\otimes W=\span\{[v,w]\mid (v,w)\in V\times W\}$$ where the "span" takes place in the vector space of all real-valued functions on $B$. (See the last sentence of the first paragraph in the section "Back to the main discussion" of his arcticle.)

According to the definition above, we have $V\otimes W=B^*=(L(V,W;\bbf))^*$ where $B^*$ is the dual of $B$.

Gowers's second definition.

In order to talk about the notion of "span", one needs a vector space in the first place. Your interpretation of the set $Z$ is not quite right. He writes in the article that

^{We can define a rather large vector space $Z$ by taking formal linear combinations of these symbols. By that I mean that Z consists of all expressions of the form $$a_1[[v_1,w_1]]+ a_2[[v_2,w_2]]+...+ a_n[[v_n,w_n]]$$ with obvious definitions for addition and scalar multiplication.}

Note that the space $Z$ is not isomorphic to $V\times W$. For instance, $(v,w)$ and $(2v,2w)$ are linearly dependent in $(v,w)$ but $[[v,w]]$ and $[[2v,2w]]$ are linearly independent in $Z$. One can think $Z$ as a vector space that has the Cartesian product $V\times W$ as a basis. So every element in $V\times W$, including $(0,0)$, identified as $[[0,0]]$, is a basis element of $Z$. This is why Gowers says it "rather large". It is in this huge space that the quotient is taken. This also gives a NO to your third question.

The two definitions are equivalent in the sense that one can establish an isomorphism between the spaces in the two definitions. As Gowers writes in his article:

^{The usual notation for the tensor product of two vector spaces V and W is V followed by a multiplication symbol with a circle round it followed by W. Since this is html, I shall write V@W instead, and a typical element of V@W will be a linear combination of elements written v@w. You can regard v@w as an alternative notation for [v,w], or for [[v,w]]+E - it doesn't matter which as the above discussion shows that the space spanned by [v,w] is isomorphic to the space spanned by [[v,w]]+E, via the (well-defined) linear map that takes [v,w] to [[v,w]]+E and extends linearly.}