Di Marino, Simone Gigli, Nicola Pasqualetto, Enrico Soultanis, Elefterios 2020-11-06 We show that, given a metric space $$(\mathrm{Y},\textsf {d} )$$of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure $$\mu $$on $$\mathrm{Y}$$giving finite mass to bounded sets, the resulting metric measure space $$(\mathrm{Y},\textsf {d} ,\mu )$$is infinitesimally Hilbertian, i.e. the Sobolev space $$W^{1,2}(\mathrm{Y},\textsf {d} ,\mu )$$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 $$x\in \mathrm{Y}$$is the tangent cone at x of $$\mathrm{Y}$$. The conclusion then follows from the fact that for every $$x\in \mathrm{Y}$$such a cone is a $$\mathrm{CAT}(0)$$space and, as such, has a Hilbert-like structure. application/pdf https://folia.unifr.ch/global/documents/308959 https://folia.unifr.ch/documents/308959/files/sou_ihl.pdf eng info:eu-repo/semantics/altIdentifier/doi/10.1007/s12220-020-00543-7 info:eu-repo/semantics/openAccess License undefined The Journal of Geometric Analysis. - 2021, vol. 31, no. 8, p. 7621-7685 info:eu-repo/classification/udc/51 Infinitesimal Hilbertianity of Locally $$\mathrm{CAT}(\kappa )$$-Spaces http://purl.org/coar/resource_type/c_6501