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In this article, we present a cost-benefit analysis of the approximation intensor products of Hilbert spaces of Sobolev-analytic type. The Sobolev part isdefined on a finite dimensional domain, whereas the analytical space is definedon an infinite dimensional domain. As main mathematical tool, we use the$\varepsilon$-dimension of a subset in a Hilbert space. The$\varepsilon$-dimension gives the lowest number of linear information that isneeded to approximate an element from the set in the norm of the Hilbert spaceup to an accuracy $\varepsilon>0$. From a practical point of view this meansthat we a priori fix an accuracy and ask for the amount of information toachieve this accuracy. Such an analysis usually requires sharp estimates on thecardinality of certain index sets which are in our case infinite-dimensionalhyperbolic crosses. As main result, we obtain sharp bounds of the$\varepsilon$-dimension of the Sobolev-analytic-type function classes whichdepend only on the smoothness differences in the Sobolev spaces and thedimension of the finite dimensional domain where these spaces are defined. Thisimplies in particular that, up to constants, the costs of the infinitedimensional (analytical) approximation problem is dominated by thefinite-variate Sobolev approximation problem. We demonstrate this procedurewith an examples of functions spaces stemming from the regularity theory ofparametric partial differential equation.

$\varepsilon$-dimension in infinite dimensional hyperbolic cross approximation and application to parametric elliptic PDEs