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.

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.

Edit:

1. $U \subset R^n$, bounded open set
2. $D^k(U) = \{ \omega : U \to \Lambda^k R^n : \text{compactly supported and infinitely differentiable \}}$
3. $D_k(U) = D^k(U)'$ is the topological dual space (currents)
4. $d : D^k(U) \to D^{k+1}(U)$ is the exterior derivative

The mathematical context is, that $k$-currents $T \in D_k(U)$ provide a generalized notion of $k$-dimensional oriented surface in $R^n$, and the flat norm can be used to get a notion of distance between currents. If it metrizes the weak* topology, it means it is fundamental in some sense, similarly to the Wasserstein distances of probability measures.