Is the mapping $\phi(f,x) = f(x)$, where $x \in E$, and $f \in L(E)$ and $E$ is a vector space, a bilinear function ?
In the definition of the book I use, Werner Greub Linear Algebra, the codomain of a bilinear function has to be a field, but in this example since $f$ is a linear mapping on $E$, the images of $f$ are in $E$, which is not a field, so from this argument, $\phi$ cannot be a bilinear function but in the book page 66, it says that if is a bilinear mapping, so where is the problem in my argument ?