In $\mathbb R^n$ one can define either distance or norm: distance is a nonnegative function $d(u,v)\ge 0$ for any vectors $u$ and $v$; while norm is a nonnegative function $p(v)\ge 0$ to measure a vector $v$ from the origin.
It seems space with a norm defined is “better” to one with only distance is defined , because a norm $p(v)$ defined actually means for any $u$, $d(u, u+v) = p(v)$, so with a distance the gap introduced by $v$ is “local”, while with a norm it’s “global”.
May I ask what are the other conclusions one can get if a space with distance is “upgraded” to have a norm?
Best Answer
You’re going the wrong direction. As you’ve noted, a norm defines a metric, so anything you can do with a metric space, you can do with a normed vector space on the derived metric.
Finite normed spaces are pretty straightforward and are all basically equivalent upon an appropriate transformation.
Metrics can be pretty complicated, and can be used to define the topology of a manifold.
(https://en.m.wikipedia.org/wiki/File:Mathematical_Spaces.png)