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I'm trying to figure out that which condition should be provided for a metric space to be normed also?

Sina
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  • Not exactly the same question, but related: http://math.stackexchange.com/questions/166380/not-every-metric-is-induced-from-a-norm – Martin Sleziak Oct 24 '14 at 08:45
  • This post has some examples: http://math.stackexchange.com/questions/423508/examples-of-metric-spaces-which-are-not-normed-linear-spaces – Martin Sleziak Oct 24 '14 at 08:46

1 Answers1

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When the metric is induced from a norm.

This kind of metric space $(X,d)$ must satisfy $$ d(x+a,y+a)=d(x,y)$$ $$ d(\alpha x,\alpha y)=|\alpha|d(x,y)$$ for all $x,y,a\in X$,and scalar $\alpha$.

And $X$ must be a vector space.

John
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