  In recent years finite tensor products of reproducing kernel Hilbert spaces
(RKHSs) of Gaussian kernels on the one hand and of Hermite spaces on the other
hand have been considered in tractability analysis of multivariate problems. In
the present paper we study countably infinite tensor products for both types of
spaces. We show that the incomplete tensor product in the sense of von Neumann
may be identified with an RKHS whose domain is a proper subset of the sequence
space $\mathbb{R}^\mathbb{N}$. Moreover, we show that each tensor product of
spaces of Gaussian kernels having square-summable shape parameters is
isometrically isomorphic to a tensor product of Hermite spaces; the
corresponding isomorphism is given explicitly, respects point evaluations, and
is also an $L^2$-isometry. This result directly transfers to the case of finite
tensor products. Furthermore, we provide regularity results for Hermite spaces
of functions of a single variable.
