What does a "pairing" in the context of vector spaces?

Question

The statement of an exercise begins by saying:

Suppose that $\beta:V\otimes V\rightarrow \mathbb{R}$ is a non-degenerated symmetric pairing.

where $V$ is vector space finite dimensional.

My interest is knowing that it is a "pairing". My first thought is that this was a bilinear form, but I think it is not correct because the bilinear forms are defined in $V\times V$ not on $V\otimes V$.

I ask for your help with the definition of a 'pairing' as far as possible. Please, provide the names of some books where they use this term.

Answer 1

A pairing is the same thing as a bilinear map (usually with codomain the base field).

As for the notation, there's a natural equivalence between bilinear maps $A \times B \to C$ and linear maps $A \otimes B \to C$ (this is the point of the tensor product), so they often get conflated.