  This paper is concerned with testing normality in a Hilbert space based on
the maximum mean discrepancy. Specifically, we discuss the behavior of the test
from two standpoints: asymptotics and practical aspects. Asymptotic normality
of the test under a fixed alternative hypothesis is developed, which implies
that the test has consistency. Asymptotic distribution of the test under a
sequence of local alternatives is also derived, from which asymptotic null
distribution of the test is obtained. A concrete expression for the integral
kernel associated with the null distribution is derived under the use of the
Gaussian kernel, allowing the implementation of a reliable approximation of the
null distribution. Simulations and applications to real data sets are reported
with emphasis on high-dimension low-sample size cases.
