  We introduce two novel procedures to test the nullity of the slope function
in the functional linear model with real output. The test statistics combine
multiple testing ideas and random projections of the input data through
functional Principal Component Analysis. Interestingly, the procedures are
completely data-driven and do not require any prior knowledge on the smoothness
of the slope nor on the smoothness of the covariate functions. The levels and
powers against local alternatives are assessed in a nonasymptotic setting. This
allows us to prove that these procedures are minimax adaptive (up to an
unavoidable \log\log n multiplicative term) to the unknown regularity of the
slope. As a side result, the minimax separation distances of the slope are
derived for a large range of regularity classes. A numerical study illustrates
these theoretical results.
