On Certain Metric Spaces Arising From Euclidean Spaces by a Change of Metric and Their Imbedding in Hilbert Space

1. W. A. Wilson ([9])2 has recently investigated those metric spaces which arise from a metric space by taking as its new metric a suitable (one variable) function of the old one. He considered in particular the euclidean straight line R, whose metric 6 = PP' is changed to A = d(P, P') = PP'S and showed that this new metric space can be imbedded3 in Hilbert space A). Here the old metric 6 and the new metric A are connected by the relation A2 = 6. In an article soon to appear ([5]), John von Neumann and the author have determined all the functions f(b) such that if R, is provided with the new metric A, defined by A = f(b), 6 = PP', the new metric space thus arising shall be imbeddable in A. They are of the form