Bassetti, FedericoFedericoBassettiBourguin, SolesneSolesneBourguinCampese, SimonSimonCampesePeccati, GiovanniGiovanniPeccati2026-02-132026-02-132026-02-03Statistics and Probability Letters 233: 110671 (2026)https://hdl.handle.net/11420/61558We consider Sobolev-type distances on probability measures over separable Hilbert spaces involving the Schatten-p norms, which include as special cases a distance first introduced by Bourguin and Campese (2020) when p=2, and a distance introduced by Giné and Leon (1980) when p=∞. Our analysis shows that, unless p=∞, these distances fail to metrize convergence in distribution in infinite dimensions. This clarifies several inconsistencies and misconceptions in the recent literature that arose from confusion between different types of distances.en1879-21032026ElsevierConvergence in distributionHilbert spacesProbabilistic distancesNatural Sciences and Mathematics::519: Applied Mathematics, ProbabilitiesJournal Article10.1016/j.spl.2026.110671Journal Article