  For statistical inference on an infinite-dimensional Hilbert space $\H $ with
no moment conditions we introduce a new class of energy distances on the space
of probability measures on $\H$. The proposed distances consist of the
integrated squared modulus of the corresponding difference of the
characteristic functionals with respect to a reference probability measure on
the Hilbert space. Necessary and sufficient conditions are established for the
reference probability measure to be {\em characteristic}, the property that
guarantees that the distance defines a metric on the space of probability
measures on $\H$. We also use these results to define new distance covariances,
which can be used to measure the dependence between the marginals of a two
dimensional distribution of $\H^2$ without existing moments.
  On the basis of the new distances we develop statistical inference for
Hilbert space valued data, which does not require any moment assumptions. As a
consequence, our methods are robust with respect to heavy tails in finite
dimensional data. In particular, we consider the problem of comparing the
distributions of two samples and the problem of testing for independence and
construct new minimax optimal tests for the corresponding hypotheses. We also
develop aggregated (with respect to the reference measure) procedures for power
enhancement and investigate the finite-sample properties by means of a
simulation study.
