  In this paper, we reveal a new connection between approximation numbers of
periodic Sobolev type spaces, where the smoothness weights on the Fourier
coefficients are induced by a (quasi-)norm $\|\cdot\|$ on $\mathbb{R}^d$, and
entropy numbers of the embedding $\textrm{id}: \ell_{\|\cdot\|}^d \to
\ell_\infty^d$. This connection yields preasymptotic error bounds for
approximation numbers of isotropic Sobolev spaces, spaces of analytic
functions, and spaces of Gevrey type in $L_2$ and $H^1$, which find application
in the context of Galerkin methods. Moreover, we observe that approximation
numbers of certain Gevrey type spaces behave preasymptotically almost identical
to approximation numbers of spaces of dominating mixed smoothness. This
observation can be exploited, for instance, for Galerkin schemes for the
electronic Schr\"odinger equation, where mixed regularity is present.
