I have started studying normed spaces. I wonder what's the need of defining notion of distance using norm function. For example , we know that $\mathbb{R}$ is a metric space with respect to usual metric defined by $d(x,y) = \mid x-y \mid$.
Now, I am studying $\mathbb{R}$ is a metric space with respect to metric induced by norm defined by $d(x,y) = \parallel x - y\parallel$.
Edit 1: I mean can't we simply study metric spaces using distance function which doesn't involve norms? Why we have introduced concept of norms?
I have no problem in understanding things related with norms. But this question is troubling me which might sound trivial.
Thanks for helping me.