If one defines on a $\mathbb{R},\mathbb{C}$-vector space a norm this gives rise to a metric. Why are particularly mappings that satisfy the norm axioms so important that in every book for beginners on linear algebra/functional analysis norms are studied ?
Aren't there also other functions that always give rise to a metric, that are worth studying?
What are the properties that a norm-induced metric has, that makes it so special (except being translation-invariant, $d(x+z,y+z)=d(x,y)$, and compatible with scalar multiplication, $d(\lambda x, \lambda y)= |\lambda | d(x,y)$; because I imagine that there would be also other mappings defined on the vector space that give, by some other rule of definition, rise to a translation-invariant,scalar multiplication compatible metric) ?
(This question was similar but not really what I was looking for - in case someone would want to redirect me to that question)