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Abstract:We recover a scalar $c_0$-approximation criterion for Banach spaces canonically embedded into $\ell^\infty$ by a countable norming family. It identifies the dual of the associated $c_0$-subspace with the prescribed atomic predual precisely when bounded coordinatewise approximation by $c_0$-elements is available. The formulation is a self-contained scalar version of the atomic predual and two-stars framework of D'Onofrio--Greco--Perfekt--Sbordone--Schiattarella.
We then apply the criterion to Lipschitz-free spaces. The weak-star density of Lipschitz functions with bounded support is due to Aliaga--Pernecká--Petitjean--Procházka; in proper spaces this yields a canonical quotient from the bidual of the norm-closed span of compactly supported Lipschitz functions onto $\Lip_0(M)$.
The main concrete application concerns countable proper subsets $K\subset [0,\infty)$ with the Euclidean metric. The connected components $(a_j,b_j)$ of the complement of $K$ give gap coordinates \[
\Delta_j f=\frac{f(b_j)-f(a_j)}{b_j-a_j}. \] We prove that, in the infinite case, these coordinates identify $\Lip_0(K)$ with $\ell^\infty$, that the corresponding $c_0$-subspace coincides with Dalet's predual $S(K)$, and hence that \[
S(K)^*\cong \F(K),\qquad S(K)^{**}\cong \Lip_0(K) \] canonically and isometrically. This gives a coordinate $c_0$ realization, for a non-discrete class with accumulation points, of the preduality covered abstractly by Dalet's theorem for countable proper metric spaces. No new proof of Dalet's theorem in full generality is claimed.

Submission history

From: Luigi D'Onofrio [view email]
[v1] Sun, 21 Jun 2026 16:27:18 UTC (12 KB)