  We propose a nonparametric two-sample test procedure based on Maximum Mean
Discrepancy (MMD) for testing the hypothesis that two samples of functions have
the same underlying distribution, using kernels defined on function spaces.
This construction is motivated by a scaling analysis of the efficiency of
MMD-based tests for datasets of increasing dimension. Theoretical properties of
kernels on function spaces and their associated MMD are established and
employed to ascertain the efficacy of the newly proposed test, as well as to
assess the effects of using functional reconstructions based on discretised
function samples. The theoretical results are demonstrated over a range of
synthetic and real world datasets.
