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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 Banachspace of \(E^{**}\)-valued Lipschitz functions \(\Lip(X, E^{**})\) underseveral conditions. Moreover, we describe the isometric isomorphism from \(\lip(X, E)^{**}\) to \(\Lip (X, E^{**})\).

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