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It is well-known that, given a vector space with an inner product $\langle \cdot,\cdot\rangle$, there is an induced metric $d$ given by $$d(x,y)=\sqrt{\langle x-y,x-y\rangle}$$

Is it true that given a vector space with a metric $d$, there exists an inner product with the above property? I suspect it is not. What conditions are required so that this is true?

Specifically, I am interested in finding an inner product consistent with the one-point compactification topology of $\mathbb{R}^n$, so if there is some condition this space satisfies that is sufficient but not necessary, that is an adequate answer as well.

Duncan W
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    Here we talk about distance on a vector space that comes from a norm. Then a norm comes from an inner product iff it satisfies the "parallelogram law". – Yuval Dec 15 '20 at 22:12
  • And a metric on a vector space comes from a norm iff it is homogeneous and translation invariant. – Steve D Dec 15 '20 at 22:18
  • ah I was searching "metric" instead of "norm;" the latter is apparently how it is more commonly framed. That helps, thanks. – Duncan W Dec 15 '20 at 22:27

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