  We propose an independence test for random variables valued into metric
spaces by using a test statistic obtained from appropriately centering and
rescaling the squared Hilbert-Schmidt norm of the usual empirical estimator of
normalized cross-covariance operator. We then get asymptotic normality of this
statistic under independence hypothesis, so leading to a new test for
independence of functional random variables. A simulation study that allows to
compare the proposed test to existing ones is provided.
