  We prove the existence of Rado sets in the Banach space of continuous
functions on [0,1]. A countable dense set S is Rado if with probability 1, the
infinite geometric random graph on S, formed by probabilistically making
adjacent elements of S that are within unit distance of each other, is unique
up to isomorphism. We show that for a suitable measure which we construct,
almost all countable dense sets in the subspaces of piecewise linear functions
and of polynomials are Rado. Moreover, all graphs arising from such sets are of
a unique isomorphism type. For the subspace of Brownian motion paths, almost
all countable subsets are Rado (for a suitable measure) and the resulting
graphs are of a unique isomorphism type. We show that the graph arising from
piecewise linear functions and polynomials is not isomorphic to the graph
arising from Brownian motion paths. Moreover, these graphs are non-isomorphic
to graphs arising from Rado sets in $\mathbb{R}^n$, or the sequence spaces $c$
and $c_0$.
