I've come across the following statement in literature (without proof or reference) about the flat norm of currents $$ F(T) = \sup \{ T(\omega) : \omega \in D^k(U), |\omega(x)| \leq 1, |d\omega(x)| \leq 1 \}: $$

The importance of the flat norm is due the fact that (at least in the space of normal currents with a bound on the mass of the current and on the mass of the boundary) it metrizes the weak* topology.

Is there a reference for this? If not, I would be happy about hints how one would one go about showing this. I have been looking into proofs which show that the Wasserstein-1 distance metrizes the weak*-topology of probability measures but they seem difficult to adapt to that case.