Infinitesimal Hilbertianity of locally CAT($kappa$)-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 $W^{1,2}(Y,d,mu)$ 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 $xin Y$ is the tangent cone at $x$ of $Y$. The conclusion then follows from the fact that for every $xin Y$ such a cone is a CAT(0)-space and, as such, has a Hilbert-like structure.