Let $K =(K,| \cdot |)$ be a non-Archimedean valued field. Let $E$ be a $K$-vector space. A norm on $E$ is a map $||\cdot||:E\to[0,\infty)$ such that:
- $||x||=0$ if and only if $x=0$,
- $||\lambda x||=|\lambda|\,||x||$,
for all $x,y\in E$.
A function $||\cdot||:E\to[0,\infty)$ satisfying conditions 1, 2 and $||x+y||\leq ||x||+||y||$ is called an A-norm on E.
Clearly every norm is an A-norm, but not conversely. Why in the literature (van-Rooij, Schikhof, etc,) the study of Banach spaces and the development of non-Archimedean functional Analysis is done with norms instead of A-norms? Does it worth to generalize the results of norms to A-norms?