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"?
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 $w^*\in W^*$.