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Not quite sure whether this question belongs here or on MESE. Anyway: Can anyone suggest a good example of a separable metric space that is neither normable nor a subset of normed space with the induced metric? Anything less trivial than a discrete metric space would be appreciated.

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All metric spaces are subspaces of a Banach space. See https://en.wikipedia.org/wiki/Kuratowski_embedding for instance.

If your metric space is separable you can even embed it in $\mathcal{C}([0.1],\mathbb{R})$ with the usual sup norm.

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  • $\begingroup$ I'm afraid I don't understand. In which sense is, say, the discrete metric space built upon the set $\{\{1\},\{2\},\{NYC\}\}$ a subspace of a Banach space? $\endgroup$ – Delio Mugnolo May 10 '16 at 11:56
  • $\begingroup$ It's homeomorphic to the subspace $\{1,2,3\}$ of the Banach space $\mathbb{R}$. $\endgroup$ – Captain Lama May 10 '16 at 11:58
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    $\begingroup$ @Delio: Your doubts are irrelevant, I’m afraid: Kuratowski’s theorem and the Banach-Mazur theorem dispose of them completely. $\endgroup$ – Brian M. Scott May 10 '16 at 14:02
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    $\begingroup$ @Delio: Since those spaces are not separable, they have nothing to do with your original question or with the statement in the second paragraph of this answer. $\endgroup$ – Brian M. Scott May 10 '16 at 14:35
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    $\begingroup$ @Delio: You’re asking for objects that don’t exist. $\endgroup$ – Brian M. Scott May 10 '16 at 14:36

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