Metric currents and polylipschitz forms

We construct, for a locally compact metric space $X$, a space of polylipschitz forms ${\bar{\mathrm{\Gamma }}}_c^{\ast }(X)$, which is a pre-dual for the space of metric currents of ${\mathcal{D}}_{\ast }(X)$ Ambrosio and Kirchheim. These polylipschitz forms may be seen as a substitute of differential forms in the metric setting.