# Infinitesimal Hilbertianity of locally CAT($\kappa$)-spaces

## TL;DR

This paper proves that metric measure spaces with curvature bounded above in the Alexandrov sense are infinitesimally Hilbertian, meaning their Sobolev space W^{1,2} is a Hilbert space, by embedding derivations into tangent cones.

## Contribution

It establishes the infinitesimal Hilbertianity of Alexandrov curvature bounded spaces using an isometric embedding into tangent cones, linking geometric and analytical structures.

## Key findings

- Sobolev space W^{1,2} is Hilbert for these spaces
- Constructs an isometric embedding of derivations into tangent cones
- Tangent cones are CAT(0)-spaces with Hilbert-like structure

## Abstract

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 $x\in Y$ is the tangent cone at $x$ of $Y$. The conclusion then follows from the fact that for every $x\in Y$ such a cone is a CAT(0)-space and, as such, has a Hilbert-like structure.