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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:

  1. $||x||=0$ if and only if $x=0$,
  2. $||\lambda x||=|\lambda|\,||x||$,
  3. $||x+y||\leq\max\{||x||,||y||\}$,

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?

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I guess that putting an archimedean norm on a vector space over a nonarchimedean field gives just an uncorrelated product of something archimedean with something nonarchimedean. Number theorists sometimes look at all places of $\mathbb{Q}$ at once: all p-adic valuations and the archimedean valuation. I don't see a reason why one should want to look at just two places. I do not expect it would generate any new theory.

Anyway, the reason why people in nonarchimedean functional analysis work with nonarchimedean norms is that the vector spaces over a nonarchimedean field which turn up naturally, come with a nonarchimedean norm. Like finite dimensional vector spaces or spaces of functions.

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