What do we measure using norm?

Question

Over the course of my studies I have acquired quite rigid notion of distance (probably due to my interaction with Physics). So when studying vector spaces, more specifically $\mathbb {R}^n$, I can't seem to understand the use of norms other than the norm induced by the dot product, this includes norms induced by inner products other than the dot product, other valid norms, etc. I would like soften this rigid notion of distance, using proper motivation for what norm represents and if it is distance then how so?

Answer 1

Topologically speaking, any two norms on a finite-dimensional space are equivalent in that they induce the same topology. So, if you only care about topological properties, you may choose whatever norm you want. The Euclidean norm is often annoying during calculations, and some norms like the maximum norm are much more convenient.

However, things start to change when you step into infinite-dimensional spaces: different norms may induce different topologies. Let $C[0,1]$ be the space of continuous functions on $[0,1]$, $C[0,1]$ is complete with the sup-norm, but not complete in the $L^2$ norm: its completion is the Lebesgue space $L^2[0,1]$.