Distance graphs on normed function spaces

It is easy to see that the unit distance graphs on the classical real normed sequence and function spaces - L-p(N), L-p(R), 1 <= p <= infinity, c(0), C[0, 1], and C(R) boolean AND L-infinity(R), the continuous bounded functions from R = (-infinity, infinity) into itself - have infinite clique and chromatic numbers, because each graph contains a countably infinite clique. The question remains to determine exactly which infinite cardinals these numbers are. A related question is of interest as well: can the chromatic number be greater than the clique number?