  We give tight upper and lower bounds of the cardinality of the index sets of
certain hyperbolic crosses which reflect mixed Sobolev-Korobov-type smoothness
and mixed Sobolev-analytic-type smoothness in the infinite-dimensional case
where specific summability properties of the smoothness indices are fulfilled.
These estimates are then applied to the linear approximation of functions from
the associated spaces in terms of the $\varepsilon$-dimension of their unit
balls. Here, the approximation is based on linear information. Such function
spaces appear for example for the solution of parametric and stochastic PDEs.
The obtained upper and lower bounds of the approximation error as well as of
the associated $\varepsilon$-complexities are completely independent of any
dimension. Moreover, the rates are independent of the parameters which define
the smoothness properties of the infinite-variate parametric or stochastic part
of the solution. These parameters are only contained in the order constants.
This way, linear approximation theory becomes possible in the
infinite-dimensional case and corresponding infinite-dimensional problems get
tractable.
