Types on stable Banach spaces

Abstract

We prove a geometric characterization of Banach space stability. We show that a Banach space X is stable if and only if the following condition holds. Whenever X is an ultrapower of X and B is a ball in X , the intersection B ∩ X can be uniformly approximated by finite unions and intersections of balls in X ; furthermore, the radius of these balls can be taken arbitrarily close to the radius of B , and the norm of their centers arbitrarily close to the norm of the center of B . The preceding condition can be rephrased without any reference to ultrapowers, in the language of types, as follows. Whenever τ is a type of X , the set τ−1[0, r] can be uniformly approximated by finite unions and intersections of balls in X ; furthermore, the radius of these balls can be taken arbitrarily close to r , and the norm of their centers arbitrarily close to τ (0). We also provide a geometric characterization of the real-valued functions which satisfy the above condition.