# understanding of the tensor product $V^*\otimes W^*$

In S.S.Chern's Lectures on Differential Geometry, I don't understand the following text in Chapter 2, which introduces the tensor product:

The tensor product $V^*\otimes W^*$ of the vector spaces $V^*$ and $W^*$ refers to the vector space generated by all elements of the form $v^*\otimes w^*$, $v^*\in V^*$, $w^*\in W^*$. It is a subspace of ${\mathcal L}(V,W;{\mathbb F})$. We need to point out that any element in $V^*\otimes W^*$ is a finite linear combination of elements of the form $v^*\otimes w^*$, but generally cannot be written as a single term $v^*\otimes w^*$ (the reader should construct examples).

Here are my questions:

• What does the first sentence mean? Does it mean $$V^*\otimes W^*:=\operatorname{span}\{v^*\otimes w^*|v^*\in V^*, w^*\in W^*\}$$ or $$V^*\otimes W^*:=\{v^*\otimes w^*|v^*\in V^*, w^*\in W^*\} ?$$
• In the context, it is only defined that $$v^*\otimes w^*(v,w)=v^*(v)\cdot w^*(w).$$ What's the "finite linear combination of elements of the form $v^* \otimes w^*$" supposed to be defined? And what's the example "the reader needs to construct"?
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By the way: when $V$ and $W$ are finite-dimensional, the span of these elements is all of $L(V, W; F)$. – Dylan Moreland Jul 12 '11 at 2:36

The exposition you're quoting may be somewhat confusing if you're used to how tensor products are usually defined in a general setting (see e.g. the Wikipedia article). Usually, the tensor product is defined as a new vector space, either through a universal property or by explicit construction using equivalence classes. In your case however, it's being regarded as a subspace of an existing vector space, the space of all bilinear functions from $V\times W$ to $\mathbb F$. That allows the book to speak of linear combinations without introducing these as formal expressions, since it's already known how to form linear combinations of bilinear functions from $V\times W$ to $\mathbb F$.
To answer your questions specifically: Yes, the vector space generated by a set of elements is the span of those elements. There's a subtle difference in that the formulation "the vector space generated by $S$" can also be used to refer to the free vector space over $S$, whereas the formulation using the span can't be thus used and only serves to identify a subspace of a vector space already otherwise defined.
Concerning the examples to be constructed, consider linearly independent functions $v^*_1,v^*_2\in V^*$ and $w^*_1,w^*_2\in W^*$, and form $v^*_1\otimes w^*_1+v^*_2\otimes w^*_2$; I think you'll find that you can't express this in the form $v^*\otimes w^*$ for any $v^*\in V^*$ and $w^*\in W^*$.