I know the usual definition of Tensor product (cartesian product of basis of $U$ and $V$, with bilinear property) but I started confusing myself as I was reading this new definition.
The tensor product $u\otimes v$ of two finite-dimensional vector spaces $u$ and $v$(over the same field) is the dual of the vector space of all bilinear forms on $u\oplus v$. For each pair of $x$ and $y$, with $x$ in $u$ and $y$ in $v$, the tensor product $z=x\otimes y$ of $x$ and $y$ is the element of $u\otimes v$ defined by $z(w)=w(x.y)$ for every bilinear form $w$.
What does this function $z$ exactly look like? Does this suggest that the tensor product $z=x\otimes y$ is a function that sends all bilinear forms (which form a vector space) to something that looks like $w(x,y)$, which is a scalar value? How can I understand this definition?
I'd really appreciate any help.