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

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--Perneck\'a--Petitjean--Proch\'azka; 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.