doi:10.1016/j.jmaa.2023.127253engGarcía-Lirola, Luis C.Grelier, GuillaumeLipschitz-free spaces, ultraproducts, and finite representability of metric spacesART-2023-133614We study several properties and applications of the ultrapower of a metric space M. We prove that the Lipschitz-free space is finitely representable in . We also characterize the metric spaces that are finitely Lipschitz representable in a Banach space as those that biLipschitz embed into an ultrapower of the Banach space. Thanks to this link, we obtain that if M is finitely Lipschitz representable in a Banach space X, then is finitely representable in . We apply these results to the study of cotype in Lipschitz-free spaces and the stability of Lipschitz-free spaces and spaces of Lipschitz functions under ultraproducts.2023http://zaguan.unizar.es/record/12628910.1016/j.jmaa.2023.127253http://zaguan.unizar.es/record/126289oai:zaguan.unizar.es:126289info:eu-repo/grantAgreement/ES/AEI-FEDER/ MTM2017-83262-C2-2-Pinfo:eu-repo/grantAgreement/ES/DGA/E48-20Rinfo:eu-repo/grantAgreement/ES/MICINN-AEI-FEDER/PID2021-122126NB-C32JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 526, 2 (2023), 127253 [14 pp.]byhttp://creativecommons.org/licenses/by/3.0/es/info:eu-repo/semantics/openAccess