Points of differentiability of the norm in Lipschitz-free spaces

[EN] We consider convex series of molecules in Lipschitz-free spaces, i.e. elements of the form such that . We characterise these elements in terms of geometric conditions on the points , of the underlying metric space, and determine when they are points of Gâteaux differentiability of the norm. In particular, we show that Gâteaux and Fréchet differentiability are equivalent for finitely supported elements of Lipschitz-free spaces over uniformly discrete and bounded metric spaces, and that their tensor products with Gâteaux (resp. Fréchet) differentiable elements of a Banach space are Gâteaux (resp. Fréchet) differentiable in the corresponding projective tensor product.R. J. Aliaga was partially supported by the Spanish Ministry of Economy, Industry and Competitiveness under Grant MTM2017-83262-C2-2-P, and by a travel grant of the Institute of Mathematics (IEMath-GR) of the University of Granada, Spain. The research of Abraham Rueda Zoca was supported by Vicerrectorado de Investigación y Transferencia de la Universidad de Granada in the program ¿Contratos puente¿, by MICINN (Spain) Grant PGC2018-093794-B-I00 (MCIU, AEI, FEDER, UE), by Junta de Andalucía Grant A-FQM-484-UGR18 and by Junta de Andalucía Grant FQM-0185Aliaga, RJ.; Rueda Zoca, A. (2020). Points of differentiability of the norm in Lipschitz-free spaces. Journal of Mathematical Analysis and Applications. 489(2):1-17. https://doi.org/10.1016/j.jmaa.2020.124171117489