  Recently Gigli developed a Sobolev calculus on non-smooth spaces using module
theory. In this paper it is shown that his theory fits nicely into the theory
of differentiability spaces initiated by Cheeger, Keith and others. A
relaxation procedure for $L^p$-valued subadditive functionals is presented and
a relationship between the module generated by a functional and the one
generated by its relaxation is given. In the framework of differentiability
spaces, which includes so called PI- and $RCD(K,N)$-spaces, the Lipschitz
module is pointwise finite dimensional. A general renorming theorem together
with the characterization above shows that the Sobolev spaces of such spaces
are reflexive.
