*Submitted Paper*

**Inserted:** 5 dec 2018

**Last Updated:** 6 dec 2018

**Year:** 2018

**Abstract:**

We show that, given a metric space $({\rm Y},{\sf d})$ of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure $\mu$ on ${\rm Y}$ giving finite mass to bounded sets, the resulting metric measure space $({\rm Y},{\sf d},\mu)$ is infinitesimally Hilbertian, i.e. the Sobolev space $W^{1,2}({\rm Y},{\sf d},\mu)$ 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 $x\in{\rm Y}$ is the tangent cone at $x$ of ${\rm Y}$. The conclusion then follows from the fact that for every $x\in{\rm Y}$ such a cone is a ${\rm Cat}(0)$ space and, as such, has a Hilbert-like structure.

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