# Flat norm metrizes the weak* topology

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.

• Those who know the subject probably understand your notations, but for the rest of us it would help if you specified who $T$, $U$ and $D^k (U)$ are, and what the mathematical context is. Also, you should reconsider the tags: what is the connection between your question and (geometric) measure theory? Some editor also suggested the tag "metric geometry", but this too doesn't seem relevant to your question. – Alex M. Dec 30 '18 at 15:48
• Currents are one of the most fundamental and central objects of study in geometric measure theory, so I think the tag is very relevant. – golden-rabbit Dec 30 '18 at 16:21
• When I see "currents", I think of deRham's theory about currents on differential manifolds - this is why the connection with metric geometry was not clear to me. – Alex M. Dec 30 '18 at 17:35
• Is this not a consequence of Corollary 7.3 in the paper "Normal and Integral Currents" by Federer and Fleming? (Though there is the additional assumption that all the currents have support in a fixed compact set) – rozu Jan 5 at 9:19
• Yes, I think this is what I need -- thank you! – golden-rabbit Jan 5 at 13:25

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.
• I agree that the weak* topology of currents should not be metrizable, but the statement in the text is about "normal currents". These are currents with finite mass and also finite mass of the boundary. Normal currents can be represented as finite Radon measures with values in $\Lambda_k R^n$ and hence are dual to the separable normed space of continuous differential forms (with sup-norm topology), and hence the weak* topology on a bounded set of normal currents should be metrizable. But I don't know how to prove it for the specific metric at hand. – golden-rabbit Dec 30 '18 at 18:52