Approximation in Banach Spaces of vector-valued Lipschitz functions

Let (X, d) be a compact metric space, S* a complex Banach space and S its dual, and α ∈ (0, 1). Using S* -valued measure theory and duality, we give a criterion for the density of a subspace of lipα(X, S). In the case where X = [a, b], we conclude that Lip1(X, S) is dense in lipα(X, S). Also using Bochner spaces and duality we show that C1([a, b], S), the space of continuously differentiable S-valued functions on [a, b],is dense in lipα([a, b], S).