On the second dual space of Banach space of vector-valued little Lipschitz functions

Abstract: Let (X) be a compact metric space and (E) be a Banach space. (\lip (X, E)) denotes the Banach space of all (E)-valued little Lipschitz functions on (X). We show that (\lip (X, E)^{**}) is isometrically isomorphic to Banach space of (E^{**})-valued Lipschitz functions (\Lip(X, E^{**})) under several conditions. Moreover, we describe the isometric isomorphism from (\lip (X, E)^{**}) to (\Lip (X, E^{**})).