  In this paper we study the relationships between a reproducing kernel Hilbert
space, its multiplier algebra, and the geometry of the point set on which they
live. We introduce a variant of the Banach-Mazur distance suited for measuring
the distance between reproducing kernel Hilbert spaces, that quantifies how far
two spaces are from being isometrically isomorphic as reproducing kernel
Hilbert spaces. We introduce an analogous distance for multiplier algebras,
that quantifies how far two algebras are from being completely isometrically
isomorphic. We show that, in the setting of finite dimensional quotients of the
Drury-Arveson space, two spaces are "close" to one another if and only if their
multiplier algebras are "close", and that this happens if and only if the
underlying point-sets are "almost congruent", meaning that one of the sets is
very close to an image of the other under a biholomorphic automorphism of the
unit ball. These equivalences are obtained as corollaries of quantitative
estimates that we prove.
