  We consider a random geometric graph model, where pairs of vertices are
points in a metric space and edges are formed independently with fixed
probability $p$ between pairs within threshold distance $\delta $. A countable
dense set in a metric space is {\sl Rado} if this random model gives, with
probability 1, a graph that is unique up to isomorphism. In earlier work, the
first two authors proved that in finite dimensional spaces $\mathbb{R}^n$
equipped with the $\ell_{\infty}$ norm, all countable dense set satisfying a
mild non-integrality condition are Rado. In this paper, we extend this result
to infinite-dimensional spaces. If the underlying metric space is a separable
Banach space, then we show in some cases that we can almost surely recover the
Banach space from such a geometric random graph. More precisely, we show that
in the sequence spaces $c$ and $c_0$, for measures $\mu$ satisfying certain
conditions, $\mu^\N$-almost all countable sets are Rado. Moreover, with
probability 1, in $c$ as in $c_0$, all graphs obtained from the random
geometric model with a randomly chosen dense countable vertex set are
isomorphic to each other. Finally, we show that representatives of the
isomorphism classes obtained in this way from $c$ and $c_0$ are non-isomorphic
to each other, and also non-isomorphic to their counterparts obtained from
finite dimensional spaces.
