  The tensorization problem for Sobolev spaces asks for a characterization of
how the Sobolev space on a product metric measure space $X\times Y$ can be
determined from its factors. We show that two natural descriptions of the
Sobolev space from the literature coincide, $W^{1,2}(X\times Y)=J^{1,2}(X,Y)$,
thus settling the tensorization problem for Sobolev spaces in the case $p=2$,
when $X$ and $Y$ are infinitesimally quasi-Hilbertian, i.e. the Sobolev space
$W^{1,2}$ admits an equivalent renorming by a Dirichlet form. This class
includes in particular metric measure spaces $X,Y$ of finite Hausdorff
dimension as well as infinitesimally Hilbertian spaces.
  More generally for $p\in (1,\infty)$ we obtain the norm-one inclusion
$\|f\|_{J^{1,p}(X,Y)}\le \|f\|_{W^{1,p}(X\times Y)}$ and show that the norms
agree on the algebraic tensor product $W^{1,p}(X)\otimes W^{1,p}(Y)\subset
W^{1,p}(X\times Y)$. When $p=2$ and $X$ and $Y$ are infinitesimally
quasi-Hilbertian, standard Dirichlet form theory yields the density of
$W^{1,2}(X)\otimes W^{1,2}(Y)$ in $J^{1,2}(X,Y)$ thus implying the equality of
the spaces. Our approach raises the question of the density of
$W^{1,p}(X)\otimes W^{1,p}(Y)$ in $J^{1,p}(X,Y)$ in the general case.
