  We consider computing eigenspaces of an elliptic self-adjoint operator
depending on a countable number of parameters in an affine fashion. The
eigenspaces of interest are assumed to be isolated in the sense that the
corresponding eigenvalues are separated from the rest of the spectrum for all
values of the parameters. We show that such eigenspaces can in fact be extended
to complex-analytic functions of the parameters and quantify this analytic
dependence in way that leads to convergence of sparse polynomial
approximations. A stochastic collocation method on an anisoptropic sparse grid
in the parameter domain is proposed for computing a basis for the eigenspace of
interest. The convergence of this method is verified in a series of numerical
examples based on the eigenvalue problem of a stochastic diffusion operator.
