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I am an undergraduate student and we have both linear algebra and metric space course in this semester.In linear algebra we studied norm and in metric spaces we studied metric.Now studying some examples I think every metric on vector space may define a norm and every norm can define a metric.I am sure about the last one but does every metric on vector space give rise to a norm?

Kishalay Sarkar
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    Norms can be used to define metrics. The most common way is by defining $d(x,y)=|x-y|$, which is called the induced metric. On the other hand, if $d$ is a metric on a vector space, there doesn't need to be a norm that satisfies the previous equation. Note that one necessary condition would be that $d(x+z,y+z)=|(x+z)-(y+z)|=|x-y|=d(x,y)$ implies that the metric needs to be translation invariant. So, any metric that is not translation invariant will not have a norm such that $d(x,y)=|x-y|$. For example, on $\mathbb{R}$ define $d(x,y)=|\arctan(x)-\arctan(y)|$. – OscarRascal Jan 23 '20 at 15:25
  • Google "metric which is not a norm" – Paul Jan 23 '20 at 15:25
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    Does this answer your question? Not every metric is induced from a norm – OscarRascal Jan 23 '20 at 15:29

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No, there are metrics that are not equivalent to a norm. A Fréchet space is a vector space with a translation-invariant metric that makes the space complete and locally convex. There are Fréchet spaces that do not have a norm equivalent to the metric. An example is $\mathbb R^\omega$, the space of all real-valued sequences, where you can take the metric to be $$d(x, y) = \sum_{n=1}^\infty 2^{-n} \frac{|x_n - y_n|}{1 + |x_n - y_n|}$$

Robert Israel
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