context
A tensor product of $V$ and $W$ is a couple $(T,h)$ where $T$ is a vector space and $h:V\oplus W\to T$ is a linear surjective map satisfying the following universal property :
Let $f:V\oplus W\to U$ be a bilinear map. There exists a linear map $g:T\to U$ such that $f=g\circ h$.
In the following, $L_k(V_1,\ldots V_k;R)$ is the set of $k$-multilinear maps $V_1\times \ldots V_k$ to R.
Finite dimensional
I assume every $V_i$ is finite dimensional vector space (over the real numbers if that matters).
My guess
I guess that $L_k(V_1,\ldots V_k;R)$ is a tensor product for $V=L_{k-1}(V_1,\ldots V_{k-1};R)$ and $W=V_k^*$.
This is what I want to prove.
I guess that the maps $h$ is the map $h:L_{k-1}(V_1,\ldots,V_{k-1};R)\oplus V_k^*\to L_k(V_1,\ldots V_k;R)$ is given by
$$ h(\xi,\xi_k)(v_1,\ldots v_k)=\xi(v_1,\ldots v_{k-1})\xi_k(v_k). $$
Question
I can prove that $h$ is bilinear.
QUESTION: how to prove the universal property of $h$ ?
EDIT: answer in the case $V^*\times W^*$ here: tensor product and space of multilinear forms
EDIT: $h$ is probably not surjective, but does not need to.