Infinitesimal Hilbertianity of Locally CAT(k)-Spaces

We show that, given a metric space (Y, d) of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure mu on Y giving finite mass to bounded sets, the resulting metric measure space (Y, d, mu) is infinitesimally Hilbertian, i.e. the Sobolev space W-1,W-2(Y, d, mu) is a Hilbert space. The result is obtained by constructing an isometric embedding of the 'abstract and analytical' space of derivations into the 'concrete and geometrical' bundle whose fibre at x is an element of Y is the tangent cone at x of Y. The conclusion then follows from the fact that for every x is an element of Y such a cone is a CAT(0) space and, as such, has a Hilbert-like structure.