Metric transformations of the real line (Q1068361)

A metric transformation between two metric spaces is a function f such that \(d_ 2(f(x),f(y))=\rho (d_ 1(x,y))\) for some function \(\rho\) (\(\rho\) the so-called scale function of f). In this paper all metric transformations between \({\mathbb{R}}\) and \(E^ n\) are characterized. In a 1938 paper, von Neumann and Schönberg characterized all continuous metric transformations of \({\mathbb{R}}\) into Hilbert space. Thus the significant contribution of this paper is in characterizing the discontinuous transformations, of which there are many, from \({\mathbb{R}}\) into \(E^ n\).