tensor definition confusion as multilinear map

Question

I'm currently learning tensor as multilinear maps. $$T: \underset{p}{\underbrace{V^*\times \cdots \times V^*}}\times \underset{q}{\underbrace{V\times V \times \cdots V\times V}} \rightarrow K\tag 1$$ I'm a little confused by $\times$ here. Does it mean cartesian product of vector spaces?

Answer 1

The cross designates a Cartesian product, which is just a set-theoretic construction, but the result is just viewed as a Cartesian product, not as a vector space. There is in this context a lot of reason to talk about the individual components of the product, for instance it will be assumed that $T$ becomes a linear map when all components except one are fixed; this talks about $V^*$ or $V$ as a vector space, but not of the Cartesian product as a vector space. Indeed, one never needs to use scalar multiplication (which would multiply all components by that factor simultaneously) nor even of addition of elements of the product.