Does every norm induce a metric and vice versa?

Question

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?

Answer 1

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 $$d(x, y) = \sum_{n=1}^\infty 2^{-n} \frac{|x_n - y_n|}{1 + |x_n - y_n|}$$