  Let $\mathcal{H}$ be a reproducing kernel Hilbert space of functions on a set
$X$. We study the problem of finding a minimal geodesic of the Grassmann
manifold of $\mathcal{H}$ that joins two subspaces consisting of functions
which vanish on given finite subsets of $X$. We establish a necessary and
sufficient condition for existence and uniqueness of geodesics, and we then
analyze it in examples. We discuss the relation of the geodesic distance with
other known metrics when the mentioned finite subsets are singletons. We find
estimates on the upper and lower eigenvalues of the unique self-adjoint
operators which define the minimal geodesics, which can be made more precise
when the underlying space is the Hardy space. Also for the Hardy space we
discuss the existence of geodesics joining subspaces of functions vanishing on
infinite subsets of the disk, and we investigate when the product of
projections onto this type of subspaces is compact.
