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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 otherhand have been considered in tractability analysis of multivariate problems. Inthe present paper we study countably infinite tensor products for both types ofspaces. We show that the incomplete tensor product in the sense of von Neumannmay be identified with an RKHS whose domain is a proper subset of the sequencespace $\mathbb{R}^\mathbb{N}$. Moreover, we show that each tensor product ofspaces of Gaussian kernels having square-summable shape parameters isisometrically isomorphic to a tensor product of Hermite spaces; thecorresponding isomorphism is given explicitly, respects point evaluations, andis also an $L^2$-isometry. This result directly transfers to the case of finitetensor products. Furthermore, we provide regularity results for Hermite spacesof functions of a single variable.

Countable Tensor Products of Hermite Spaces and Spaces of Gaussian Kernels