Added: are all types of mappings for vector spaces with "product" in their names always multilinear mappings between some vector spaces? Are there many counterexamples?
$F$ is a field. Any bilinear form on $F^n$ can be expressed as $$ B(\textbf{x},\textbf{y}) = \textbf{x}^\mathrm T A\textbf{y} = \sum_{i,j=1}^n a_{ij} x_i y_j $$ where $A$ is an n × n matrix.
I was wondering if $\textbf{x}^\mathrm T A\textbf{y}$ is a single product of some type between two vectors in $F^n$? What type is it?
More generally, is a multilinear form on $F^n$ a single product of some type on multiple vectors in $F^n$?
Is a multilinear form defined on the product space of a vector space a single product of some type on multiple vectors in the vector space?
Is a multilinear form defined from the product space of a vector space to another vector space a single product of some type on multiple vectors in the first vector space?
Thanks and regards!