I've read the definitions in the book of Kostrikin and in the book of Sterling Berberian. The book of Berberian gives a definition for the product of two spacees, Kostrikin gives it more generally. The Berberian definition is:
$V\otimes W=\mathcal{L}(V^*,W)$, that is, the linear mappings from the dual of $V$ to $W$. Then he defines $v\otimes w\in V\otimes W$ such that for $f\in V^*$, $(v\otimes w)(f)=f(v)w$.
Let's denote this definition with $\otimes_1$.
The Kostrikin construction is given here, let's denote it with $\otimes_2$
The question is: Is there any way to extend the Berberian definition to $n$ spaces in a way that it coincides with Kostrikin's?