  We introduce an infinite-dimensional version of the Christoffel function,
where now (i) its argument lies in a Hilbert space of functions, and (ii) its
associated underlying measure is supported on a compact subset of the Hilbert
space. We show that it possesses the same crucial property as its
finite-dimensional version to identify the support of the measure (and so to
detect outliers). Indeed, the growth of its reciprocal with respect to its
degree is at least exponential outside the support of the measure and at most
polynomial inside. Moreover, for a fixed degree, its computation mimics that of
the finite-dimensional case, but now the entries of the moment matrix
associated with the measure are moments of moments. To illustrate the potential
of this new tool, we consider the following application. Given a data base of
registered reference trajectories, we consider the problem of detecting whether
a newly acquired trajectory is abnormal (or out of distribution) with respect
to the data base. As in the finite-dimensional case, we use the
infinite-dimensional Christoffel function as a score function to detect
outliers and abnormal trajectories. A few numerical examples are provided to
illustrate the theory.
