Hello!(adsbygoogle = window.adsbygoogle || []).push({});

It is said that not every metric comes from a norm.

Consider for example a metric defined on all sequences of real numbers with the metric:

[tex]d(x,y):=\displaystyle\sum_{i=1}^{\infty}\frac{1}{2^i}\frac{|x_i-y_i|}{1+|x_i-y_i|}[/tex]

I can't grasp how can that be.

There is a proof, so could you please give me hints so that I can try and do the proof on my own.

At the moment I can only see that the whole thing is bounded by 1. What does it actually mean when we say "it does not come from a norm"?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Not every metric comes from a norm

Loading...

Similar Threads for every metric comes |
---|

I Surface Metric Computation |

I Metrics and topologies |

I Lie derivative of a metric determinant |

I Conformal Related metrics |

A On the dependence of the curvature tensor on the metric |

**Physics Forums | Science Articles, Homework Help, Discussion**