The decision problem for normed spaces over any class of ordered fields

It is known that the theory of any class of normed spaces over the reals that includes all spaces of a given dimension d > 1 is undecidable, and indeed, admits a relative interpretation of second-order arithmetic. The notion of a normed space makes sense over any ordered field of scalars, but such a strong undecidability result cannot hold in the more general case. Nonetheless, we find that the theory of any class of normed spaces in the more general sense that includes all spaces of a given dimension d > 1 over some ordered field admits a relative interpretation of Robinson's theory Q and hence is undecidable.