  Functional linear regression has recently attracted considerable interest.
Many works focus on asymptotic inference. In this paper we consider in a non
asymptotic framework a simple estimation procedure based on functional
Principal Regression. It revolves in the minimization of a least square
contrast coupled with a classical projection on the space spanned by the m
first empirical eigenvectors of the covariance operator of the functional
sample. The novelty of our approach is to select automatically the crucial
dimension m by minimization of a penalized least square contrast. Our method is
based on model selection tools. Yet, since this kind of methods consists
usually in projecting onto known non-random spaces, we need to adapt it to
empirical eigenbasis made of data-dependent - hence random - vectors. The
resulting estimator is fully adaptive and is shown to verify an oracle
inequality for the risk associated to the prediction error and to attain
optimal minimax rates of convergence over a certain class of ellipsoids. Our
strategy of model selection is finally compared numerically with
cross-validation.
