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I shall call these spaces "sequence spaces". While there exist well known separable Banach spaces that have subsets which are isometric to every separable metric space, I never heard of any separable sequence space which has this property. The space of all bounded infinite sequences of real numbers (with the "sup norm") appears to have this property but it is not separable. I would like to know whether every metric space belonging to one of the following classes can be isometrically embedded into some separable sequence space. The class of all countable compact metric spaces? The class of all countable, bounded and complete metric spaces? The class of all countable and complete metric spaces? Any information about this will be much appreciated.

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    $\begingroup$ You can represent every separable Banach space as a space of sequences. Do you want to restrict the class of sequence spaces, and, if so, how? $$$$ Related to your question is I. Aharoni's result that every separable metric space is Lipschitz equivalent to a subset of $c_0$. $$$$ Aharoni, Israel Every separable metric space is Lipschitz equivalent to a subset of c+0. Israel J. Math. 19 (1974), 284–291. $\endgroup$ Sep 29, 2013 at 18:23
  • $\begingroup$ Do you mean that every separable Banach space is isometric to some separable space of infinite sequences of real numbers? What "sequence space" would be isometric to the Banach space of all continuous functions on the unit interval-which has the "universal" property that I mentioned? $\endgroup$ Sep 30, 2013 at 18:30

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