  In functional linear regression, the parameters estimation involves solving a
non necessarily well-posed problem and it has points of contact with a range of
methodologies, including statistical smoothing, deconvolution and projection on
finite-dimensional subspaces. We discuss the standard approach based explicitly
on functional principal components analysis, nevertheless the choice of the
number of basis components remains something subjective and not always properly
discussed and justified. In this work we discuss inferential properties of
least square estimation in this context with different choices of projection
subspaces, as well as we study asymptotic behaviour increasing the dimension of
subspaces.
