The functional linear model with scalar response is a regression model where the predictor is a random function defined on some compact set of R R and the response is scalar. The response is modelled as Y=WðXÞ+e, where W is some linear continuous operator defined on the space of square integrable functions and valued in R R. The random input X is independent from the noise e. In this paper, we are interested in testing the null hypothesis of no effect, that is, the nullity of W restricted to the Hilbert space generated by the random variable X. We introduce two test statistics based on the norm of the empirical cross-covariance operator of (X; Y). The first test statistic relies on a v 2 approximation and we show the asymptotic normality of the second one under appropriate conditions on the covariance operator of X. The test procedures can be applied to check a given relationship between X and Y. The method is illustrated through a simulation study.

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