[efr-7O1A]

Let A be a real Banach algebra. Then a (real) Frechet module over A is a real vector space V equipped with a family of mappings \rho _i: V \to A^+ (where positivity is defined in the spectral sense), satisfying:

1. \rho _i(\alpha v) = \alpha \rho _i(v)
2. \rho _i(v + w) \leq \rho _i(v) + \rho _i(w). Again this is understood in the sense that \rho _i(v) + \rho _i(w)-\rho _i(v + w) has nonnegative spectrum.
3. If \rho _i(v)=0 for every i, then v = 0.

Given two Frechet modules, a module homomorphism f: V \to W is said to be bounded if, for every i, there exist K, \epsilon so that \rho _i^W(f(v)) \leq 1 if \rho _1^V(v), \dots ,\rho ^V_K(v) \leq \epsilon .

Note that in this definition we, a priori, allow \epsilon to be a generic element of A. Viewing A = C_\mathbb {R}(X) for some space X, this means the maps V_x \to W_x are each required to be bounded, but not uniformly bounded. But in fact since \epsilon must be strictly positive this requires it to be bounded away from 0, and so there is some uniform choice of \epsilon (say the infimum of the spectrum of the element \epsilon )