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Aug 2 at 7:31 vote accept Qingping Zeng
Jul 30 at 1:32 history edited David Gao CC BY-SA 4.0
added 9 characters in body
Jul 30 at 1:30 comment added David Gao Since $\varphi_x$ may be chosen of norm $\frac{1}{d(x,M)}$, this also implies the free algebra construction I proposed in comments also works, in that $E$ actually naturally isometrically embeds into the free algebra proposed there.
Jul 30 at 1:23 history answered David Gao CC BY-SA 4.0