Metric spaces and positive definite functions

As poo we get the space Em with the distance function maxi-, ... I xi X. 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 summable to the pth 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. t L2 is equivalent to the real Hilbert space t. The spaces EmP, IP and LP are metric only if p > 1, but we shall consider them also for positive values of p O). A general theorem of Banach and Mazur ([1], p. 187) states that any separable metric space (5 may be imbedded isometrically in the space C. Furthermore, as a special case of a well known theorem of Urysohn, any such space (E may be imbedded topologically in t. Isometric imbeddability of (E in '& is, however, a much more restricted property of (B. 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 Hilbert space only. It turns out that the possibility of imbedding$ in 6& is very easily expressible in terms of the elementary function e-t2 and the concept of positive definite functions (Theorem 1). The author's previous result ([10]) to the effect that i(,y), (O <,y < 1), which is the space arising from 6& by raising its metric to a