All possible norms on a finite-dimensional vector space?

Question

Let $X=\mathbb K^n$, where $\mathbb K=\mathbb R$ or $\mathbb C$. I have seen proofs that the functions

$$\|x\|_p:=\sqrt[p]{\sum_i|x_i|^p},\qquad p\in[1,\infty]$$

are all norms. (The $p=\infty$ case must be interpreted as a limit $p\to\infty$, which turns out to be equivalent to $\max_i|x_i|$.) It is also straightforward to check that multiplying any of these functions by a positive constant results in a norm. That is,

$$\|x\|_{(p,\lambda)}:=\lambda\|x\|_p,\qquad p\in[1,\infty],\lambda>0$$

are norms. Are there any other norms?

By the way, I am aware that all norms on finite-dimensional vector spaces yield identical topologies, but this question is just about the norms.

Answer 1

Let $K$ be a compact and convex subset of $\Bbb R^n$ such that $0$ is an interior point of $K$ and that $v\in K\implies-v\in K$. Then you can define the norm $\|\cdot\|_K$ such that $\|0\|_K=0$ and that, if $v\ne0$, $\|v\|_K$ is the smallest $\lambda\in(0,\infty)$ such that $\lambda^{-1}v\in K$. With respect to this norm, $K$ is the unit ball. If, for instance $n=2$ and $K$ is an hexagon centered at the origin, $\|\cdot\|_K$ is norm which is different from any $\|\cdot\|_p$.

Answer 2

If $T:X \to X$ is any vector space isomorphism then $\|x\|=\|Tx\|_2$ (where $\|x\|_2$ is the usual norm gives you a new norm on $X$.

A specific example: $\|x\|=\sqrt {|x_1+x_2|^{2}+|x_2|^{2}+\cdots |x_n|^{2}}$.