We study integration and $L^2$-approximation of functions of infinitely many variables in the following setting: The underlying function space is the countably infinite tensor product of univariate Hermite spaces and the probability measure is the corresponding product of the standard normal distribution. The maximal domain of the functions from this tensor product space is necessarily a proper subset of the sequence space $\mathbb{R}^\mathbb{N}$. We establish upper and lower bounds for the minimal worst case errors under general assumptions; these bounds do match for tensor products of well-studied Hermite spaces of functions with finite or with infinite smoothness. In the proofs we employ embedding results, and the upper bounds are attained constructively with the help of multivariate decomposition methods.

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