A caveat on metrizing convergence in distribution on Hilbert spaces

We 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.