# Separable metric spaces that are not normable

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.

If your metric space is separable you can even embed it in $\mathcal{C}([0.1],\mathbb{R})$ with the usual sup norm.
• 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? – Delio Mugnolo May 10 '16 at 11:56
• It's homeomorphic to the subspace $\{1,2,3\}$ of the Banach space $\mathbb{R}$. – Captain Lama May 10 '16 at 11:58