Embedding quasi-metric spaces in Hilbert spaces (Q789668)

Let \((X,\delta)\) be a quasi-metric set. The author discusses the problem of embedding this set in some vector space equipped with a quadratic form i.e. if \(\delta\) is \(\mu\)-square summable with \(\mu\) being a finite positive measure on X, then \((X,\delta\),\(\mu)\) can be embedded almost everywhere in \(\ell^ 2\) (space of all square-summable sequences) and in this case the coordinates of \(x\in X\) in \(\ell^ 2\) can be determined explicitly as well as the dimension of \((X,\delta\),\(\mu)\). This generalizes a result of \textit{H. S. M. Coxeter} and \textit{J. A. Todd} [Proc. Camb. Philos. Soc. 30, 1-3 (1934; Zbl 0008.17002)]. The problem is: The quasi-metric set \((X,\delta\),\(\mu)\) can be embedded almost everywhere in \((\ell^ 2,q)\) (where the quadratic form q on \(\ell^ 2\) is defined by \(q(z,z)=\sum \epsilon_ i(z_ i)^ 2\) for all sequences \(z=(z_ i)^{\infty}\!_{i=1}\in \ell^ 2)\) and each \(x\in X\) is represented almost everywhere by the sequence \(x\in \ell^ 2\) where \(x=(| \lambda_ i|^{1/2}\phi_ i(x))^{\infty}\!_{i=1}.\) Necessary and sufficient conditions to embed \((X,\delta\),\(\mu)\) in Euclidean space or Hilbert space are also derived in the form of a theorem.