Metric Spaces and Positive Definite Functions

Introduction.Let Em denote the w-dimensional euclidean space and generally Emp the pseudo-euclidean space of m real variables with the distance function (\ Xl -X{ \" + + I Xm -Xni I")1'", p > 0.As p- we get the space E" with the distance function max,-=i,... ,m | x0).A general theorem of Banach and Mazur (1, p. 187) states that any separable metric space @ may be imbedded isometrically in the space C. Furthermore, as a special case of a well known theorem of Urysohn, any such space may be imbedded topologically in .Isometric imbeddability of in is, however, a much more restricted property of .The chief purpose of this paper is to point out the intimate relationship between the problem of isometric imbedding and the concept of positive definite functions, if this concept is properly enlarged.As a first approach to this connection we consider here isometric imbedding in Hubert space only.It turns out that the possibility of imbeddingj in is very easily expressible in terms of the elementary function e-'2 and the concept of positive definite functions (Theorem 1).The author's previous result (10) to the effect that (7), (0 is homogeneous of degree k if (txx, , txm) =lK(xx, , x") holds identically in the Xi and for />0.A continuous homogeneous function with the properties (10) must, unless it vanishes identically, have a positive degree k.