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Let $X_1,X_2, \ldots$ be independent and identically distributed randomelements taking values in a separable Hilbert space $\mathbb{H}$. Withapplications for functional data in mind, $\mathbb{H}$ may be regarded as aspace of square-integrable functions, defined on a compact interval. We proposeand study a novel test of the hypothesis $H_0$ that $X_1$ has some unspecifiednon-degenerate Gaussian distribution. The test statistic$T_n=T_n(X_1,\ldots,X_n)$ is based on a measure of deviation between theempirical characteristic functional of $X_1,\ldots,X_n$ and the characteristicfunctional of a suitable Gaussian random element of $\mathbb{H}$. We derive theasymptotic distribution of $T_n$ as $n \to \infty$ under $H_0$ and provide aconsistent bootstrap approximation thereof. Moreover, we obtain an almost surelimit of $T_n$ as well as a normal limit distribution of $T_n$ underalternatives to Gaussianity. Simulations show that the new test is competitivewith respect to the hitherto few competitors available.

A test for Gaussianity in Hilbert spaces via the empirical characteristic functional