  Kernel-based methods in Numerical Analysis have the advantage of yielding
optimal recovery processes in the "native" Hilbert space $\calh$ in which they
are reproducing. Continuous kernels on compact domains have an expansion into
eigenfunctions that are both $L_2$-orthonormal and orthogonal in $\calh$
(Mercer expansion). This paper examines the corresponding eigenspaces and
proves that they have optimality properties among all other subspaces of
$\calh$. These results have strong connections to $n$-widths in Approximation
Theory, and they establish that errors of optimal approximations are closely
related to the decay of the eigenvalues.
  Though the eigenspaces and eigenvalues are not readily available, they can be
well approximated using the standard $n$-dimensional subspaces spanned by
translates of the kernel with respect to $n$ nodes or centers. We give error
bounds for the numerical approximation of the eigensystem via such subspaces. A
series of examples shows that our numerical technique via a greedy point
selection strategy allows to calculate the eigensystems with good accuracy.
