The linear compactness theorem is a variant of the compactness theorem holding for linear formulas. We show that the linear fragment of continuous logic is maximal with respect to the linear compactness theorem and the linear elementary chain property. We also characterize linear formulas as those preserved by the ultramean construction.

We prove Robinson consistency theorem as well as Craig, Lyndon and Herbrand interpolation theorems in linear continuous logic.

The ultraproduct construction is generalized to p‐ultramean constructions () by replacing ultrafilters with finitely additive measures. These constructions correspond to the linear fragments of continuous logic and are very close to the constructions in real analysis. A powermean variant of the Keisler‐Shelah isomorphism theorem is proved for. It is then proved that ‐sentences (and their approximations) are exactly those sentences of continuous logic which are preserved by such constructions. Some other applications are also given.