Infinitesimal Hilbertianity of Locally CAT (κ) -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 μ on Y giving finite mass to bounded sets, the resulting metric measure space (Y , d, μ) is infinitesimally Hilbertian, i.e. the Sobolev space W1 , 2(Y , d, μ) 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∈ Y is the tangent cone at x of Y. The conclusion then follows from the fact that for every x∈ Y such a cone is a CAT (0) space and, as such, has a Hilbert-like structure.