Approximate local isometries on spaces of absolutely continuous functions (Q2038620)

Let \(X,Y\) be compact sets of real numbers. Let \(AC(X)$ $(AC(Y))\) denote space of complex absolutely continous functions, equipped with the norm \(\|f\|_{\Sigma}= \|f\|_{\infty}+V(f)\), for \(f \in AC(X)\), where \(V(f)\) denotes the variation of~\(f\). A description of the surjective isometries between such spaces has long been known (see [\textit{V. D. Pathak}, Can. J. Math. 34, 298--306 (1982; Zbl 0464.46029)]). The authors first note that the group of isometries is not topologically reflexive and go on to provide a description of the objects in the topological closure of the isometry group (Theorem~2).