  We establish minimax convergence rates for classification of functional data
and for nonparametric regression with functional design variables. The optimal
rates are of logarithmic type under smoothness constraints on the functional
density and the regression mapping, respectively. These asymptotic properties
are attainable by conventional kernel procedures. The bandwidth selector does
not require knowledge of the smoothness level of the target mapping. In this
work, the functional data are considered as realisations of random variables
which take their values in a general Polish metric space. We impose certain
metric entropy constraints on this space; but no algebraic properties are
required.
