Information and metrics in Hilbert space

The concept of distance in Hilbert space is relevant in a variety of scenarios, in particular for investigating the quality of different approximations. In this work we study the relations between (i) statistical distances (SD) on a probability space, on the one hand, and (ii) different metrics on Hilbert space (MHS), on the other hand. As a result, we are able to establish some universal relations between SD and MHS and to apply them to one-dimensional problems.