A remark on M. M. Day’s characterization of inner-product spaces and a conjecture of L. M. Blumenthal
I. J. Schoenberg · Proceedings of the American Mathematical Society · 1952
1. A space of elements a, b, ■ • • , with a distance function ab is said to be semi-metric provided ab = ba>0 if a^b, and aa = 0. A reallinear space of elements/, g, ■ • ■ is said to be semi-normed provided a function 11/11 is defined in S having the usual properties of a norm with the exception of the inequality ||/+g|| =||/||+||g||> which is not assumed. Evidently ||/—g is a semimetric in the sense of the first definition. A semimetric space is called ptolemaic provided that among the distances between any four points a, b, c, d Ptolemy's inequality