Isometric representations of weighted spaces of little Lipschitz functions

Abstract

Given a compact pointed metric space X and a weight v on the complement of the diagonal set in $$X\times X$$ , we prove that the Banach space $$\mathrm {lip}_v(X)$$ of all weighted little Lipschitz scalar-valued functions on X vanishing at the basepoint, equipped with the weighted Lipschitz norm, embeds almost isometrically into $$c_0$$ . This result has many consequences on the structure of those Banach spaces and their duals. Moreover, we prove that this isomorphism can never be an isometric embedding whenever X is a $$\mathbb {T}$$ -balanced subset containing 0 and compact for some metrizable topology of a complex Banach space and v is a radial 0-weight.