Flat norm metrizes the weak* topology

Question

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.

Answer 1

The statement in the text cannot possibly be correct. The spaces in question are not metrisable in the weak $\star$ topology. This is a standard fact in locally convex space theory. There are, however, various versions which might be adequate for your purposes. Thus, I would conjecture that the weak $\star$ topology and your metric one agree on bounded sets with the strong topology and so all three have the same notion of convergence for sequences but this would require a proof. Your spaces are the duals of $LF$-spaces in the sense of L. Schwartz (sometimes called $DLF$-spaces) and as such projective limits of a sequence of nuclear $DF$-spaces (Grothendieck), albeit in a particularly nice manner (with partitions of unity---de Wilde). So rather complicated in structure, but not hopelessly so.