  Hypothesis testing for the slope function in functional linear regression is
of both practical and theoretical interest. We develop a novel test for the
nullity of the slope function, where testing the slope function is transformed
into testing a high-dimensional vector based on functional principal component
analysis. This transformation fully circumvents ill-posedness in functional
linear regression, thereby enhancing numeric stability. The proposed method
leverages the technique of bootstrapping max statistics and exploits the
inherent variance decay property of functional data, improving the empirical
power of tests especially when the sample size is limited or the signal is
relatively weak. We establish validity and consistency of our proposed test
when the functional principal components are derived from data. Moreover, we
show that the test maintains its asymptotic validity and consistency, even when
including \emph{all} empirical functional principal components in our test
statistics. This sharply contrasts with the task of estimating the slope
function, which requires a delicate choice of the number (at most in the order
of $\sqrt n$) of functional principal components to ensure estimation
consistency. This distinction highlights an interesting difference between
estimation and statistical inference regarding the slope function in functional
linear regression. To the best of our knowledge, the proposed test is the first
of its kind to utilize all empirical functional principal components.
