Distance vs norm 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?
 A: Generally speaking, an inner product induces a norm and a norm induces a metric.
To be precise, let us consider a vector space $V$ over the field $\mathbf{F}$. We say the application $\langle\cdot,\cdot\rangle:V\times V\to\textbf{F}$ is an inner product iff, for every $x\in V$, $y\in V$, $z\in V$ and $\alpha\in\mathbf{F}$, we have that

*

*$\langle \alpha x,y\rangle = \alpha\langle x,y\rangle$,

*$\langle x + y,z\rangle = \langle x,z\rangle + \langle y,z\rangle$,

*$\langle x,y\rangle = \overline{\langle y,x\rangle}$,

*$\langle x,x\rangle\geq 0$, where the equality holds iff $x = 0$.

Based on it, we can prove that $\|x\| = \sqrt{\langle x,x\rangle}$ is a norm indeed.
When we are talking about norms, we refer to an application $\|\cdot\|:V\to\mathbb{R}_{\geq0}$ such that for every $x\in V$, $y\in V$ and $\alpha\in\mathbf{F}$ one has that

*

*$\|x\| = 0$ iff $x = 0$,

*$\|\alpha x\| = |\alpha|\|x\|$,

*$\|x + y\| \leq \|x\| + \|y\|$
Based on such definition, we can prove that $d(x,y) = \|x - y\|$ is a distance indeed.
To be precise, when we are talking about distance, it suffices to take a non-empty set $X$ and a function $d_{X}:X\times X\to\mathbb{R}$ s.t. for every $x\in X$, $y\in X$ and $z\in X$, we have that

*

*$d(x,y)\geq 0$, where $d(x,y) = 0$ iff $x = y$,

*$d(x,y) = d(y,x)$,

*$d(x,z) \leq d(x,y) + d(y,z)$
Hopefully this helps!
A: 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)
