Characterizations of the Radon-nikodym Property in Terms of Inverse Limits

We show that a separable Banach space has the RadonNikodym Property (RNP) if and only if it is isomorphic to the limit of an inverse system, V1 ← V2 ← . . . ← Vk ← . . . , where the Vi’s are finite dimensional Banach spaces, and the bonding maps Vk−1 ← Vk are quotient maps. We also show that the inverse system can be chosen to be a good finite dimensional approximation (GFDA), a notion introduced in [CK06]. As a corollary, it follows that the differentiation and bi-Lipschitz non-embedding theorems in [CK06], which were proved for maps into GFDA targets, are optimal in the sense that they hold for RNP targets.