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

TL;DR

This paper characterizes the second dual space of vector-valued little Lipschitz functions on compact metric spaces, showing it is isometrically isomorphic to a space of Lipschitz functions with values in the bidual space.

Contribution

It establishes an isometric isomorphism between the second dual of extit{lip} spaces and Lipschitz spaces with bidual target, extending duality theory for vector-valued Lipschitz functions.

Findings

The second dual space extit{lip}(X, E)^{**} is isometrically isomorphic to extit{Lip}(X, E^{**}) under certain conditions.

Explicit description of the isometric isomorphism between extit{lip}(X, E)^{**} and extit{Lip}(X, E^{**}).

Extension of duality results to 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 \(\Lip (X, E^{**})\).