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