# All possible norms on a finite-dimensional vector space?

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.

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 $$\|\cdot\|_p$$.
• Your definition seems to be off: we need the smallest $\lambda \in (0,\infty)$ with $\lambda^{-1} v \in K$ (or equivalently: $v \in \lambda K$). Moreover, one should note that $K = \{v | \|v\|_K \le 1\}$ becomes the unit ball and $\|\cdot\|_K$ is called Minkowski functional of $K$.
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: $$\|x\|=\sqrt {|x_1+x_2|^{2}+|x_2|^{2}+\cdots |x_n|^{2}}$$.