Metric Spaces and Positive Definite
نویسنده
چکیده
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 ÍQ. 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
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