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.