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We study integration and $L^2$-approximation of functions of infinitely manyvariables in the following setting: The underlying function space is thecountably infinite tensor product of univariate Hermite spaces and theprobability measure is the corresponding product of the standard normaldistribution. The maximal domain of the functions from this tensor productspace is necessarily a proper subset of the sequence space$\mathbb{R}^\mathbb{N}$. We establish upper and lower bounds for the minimalworst case errors under general assumptions; these bounds do match for tensorproducts of well-studied Hermite spaces of functions with finite or withinfinite smoothness. In the proofs we employ embedding results, and the upperbounds are attained constructively with the help of multivariate decompositionmethods.

Infinite-dimensional integration and $L^2$-approximation on Hermite spaces