Metric Spaces and Positive Definite Functions

Abstract

1. 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 | x< -as/ |.Let, furthermore, lp stand for the space of real sequences with the series of pth powers of the absolute values convergent.Similarly let Lp denote the space of real measurable functions in the interval (0, 1) which are oummable to the ^th power, while C shall mean the space of real continuous functions in the same interval.In all these spaces a distance function is assumed to be defined as usual, f L2 is equivalent to the real Hubert space Q.The spaces Emp, lp, and Lp are metric only if ^^1, but we shall consider them also for positive values of ^<1.Finally, if is a (not necessarily metric) space with the distance function PP', we shall denote by (7) the new space which arises by changing the distance function from PP' to PP'y, (y >0).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<7<1), which is the space arising from by raising its metric to a * Presented to the Society, December 29, 1937; received by the editors December 14, 1937.t See, for example, Banach [l], pp.11-12.The numbers in square brackets refer to the list of references at the end of the paper.% Here and below the word "imbedding" stands for "isometrical imbedding."* For a discussion and consequences of Schur's result see also Plya and Szeg [9], pp.106-107, 307-308.* We say that <i> is homogeneous of degree k if <t>(txx, , txm) =lK<t>(xx, , x") holds identically in the Xi and for />0.A continuous homogeneous function <j> with the properties (10) must, unless it vanishes identically, have a positive degree k.