  We present a natural family of Hilbert function spaces on the d-dimensional
complex unit ball and classify which of them satisfy that subsets of the ball
yield isometrically isomorphic subspaces if and only if there is an analytic
automorphism of the ball taking one to the other. We also characterize pairs of
weighted Hardy spaces on the unit disk which are isomorphic via a composition
operator by a simple criterion on their respective sequences of weights.
