Tensor norms on ordered normed spaces, polarization constants, and exchangeable distributions

We define new norms for symmetric tensors over ordered normed spaces; these norms are defined by considering linear combinations of tensor products or powers of positive elements only. Relations between the different norms are studied. The results are applied to the problem of representing a finitely exchangeable distribution as a mixture of powers, i.e, mixture of distributions of i.i.d. sequences, using a signed mixing measure.