Positive definite symmetric functions on linear spaces

If Φ is a positive definite function on a real linear space E of infinite dimension and Φ enjoys certain symmetry conditions we are able to show that Φ is expressible as a certain Laplace-Stieltjes transform. Conversely, if Φ is given by such a transform we can often show that Φ is positive definite on E. In particular, our results apply to the L^p spaces, 0 < p < ∞, as well as to other Orlicz spaces. We also are able to show that the only positive definite continuous sup-norm symmetric functions on C(T), the space of bounded real continuous functions on T, are constants whenever C(T) contains a sequence of functions with sup-norm one and disjoint support. Finally, we apply these ideas to obtain a result on radial exponentially convex functions on a Hilbert space.