  We consider $\mathbb{L}_2$-approximation of elements of a Hermite space of
analytic functions over $\mathbb{R}^s$. The Hermite space is a weighted
reproducing kernel Hilbert space of real valued functions for which the Hermite
coefficients decay exponentially fast. The weights are defined in terms of two
sequences $\boldsymbol{a} = \{a_j\}$ and $\boldsymbol{b} = \{b_j\}$ of positive
real numbers. We study the $n$th minimal worst-case error $e(n,{\rm
APP}_s;\Lambda^{{\rm std}})$ of all algorithms that use $n$ information
evaluations from the class $\Lambda^{{\rm std}}$ which only allows function
evaluations to be used.
  We study (uniform) exponential convergence of the $n$th minimal worst-case
error, which means that $e(n,{\rm APP}_s; \Lambda^{{\rm std}})$ converges to
zero exponentially fast with increasing $n$. Furthermore, we consider how the
error depends on the dimension $s$. To this end, we study the minimal number of
information evaluations needed to compute an $\varepsilon$-approximation by
considering several notions of tractability which are defined with respect to
$s$ and $\log \varepsilon^{-1}$. We derive necessary and sufficient conditions
on the sequences $\boldsymbol{a}$ and $\boldsymbol{b}$ for obtaining
exponential error convergence, and also for obtaining the various notions of
tractability. It turns out that the conditions on the weight sequences are
almost the same as for the information class $\Lambda^{{\rm all}}$ which uses
all linear functionals. The results are also constructive as the considered
algorithms are based on tensor products of Gauss-Hermite rules for multivariate
integration. The obtained results are compared with the analogous results for
integration in the same Hermite space. This allows us to give a new sufficient
condition for EC-weak tractability for integration.
