A scalar $c_0$-approximation criterion and gap-coordinate preduals for Lipschitz-free spaces

We recover a scalar $c_0$-approximation criterion for Banach spaces canonically embedded into ${\mathrm{\ell }}^{\mathrm{\infty }}$ 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 ${\text{Lip}}_0(M)$. The main concrete application concerns countable proper subsets $K\subset [0,\mathrm{\infty })$ with the Euclidean metric. The connected components $(a_j,b_j)$ of the complement of $K$ give gap coordinates $${\mathrm{\Delta }}_{j}f=\frac{f(b_j)-f(a_j)}{b_j-a_j}\mathrm{.}$$ We prove that, in the infinite case, these coordinates identify ${\text{Lip}}_0(K)$ with ${\mathrm{\ell }}^{\mathrm{\infty }}$, that the corresponding $c_0$-subspace coincides with Dalet’s predual $S(K)$, and hence that $$S(K{)}^{\ast }\cong \text{F}(K),S(K{)}^{\ast \ast }\cong {\text{Lip}}_0(K)$$ 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.