## Types on stable Banach spaces

### Volume 157 / 1998

#### 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 $\widehat{X}$ is an ultrapower of X and B is a ball in $\widehat{X}$, 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.