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For a general normed space $(V,\|{\cdot}\|)$ suppose that the vector space $V$ is over some field $K$ that is not $\Bbb Q$, $\Bbb R$, $\Bbb C$, $\Bbb Z_p$ or any of it expansions, then, in general, how we define the absolute value such that

$$\|\alpha x\|=|\alpha|\|x\|$$

where $\alpha\in K$? I suppose we will need to define some function of the kind

$$|{\cdot}|:K\to\Bbb R_{\ge 0}$$

I searched some information about the general topic of normed spaces but I dont found something about this, can you give me some text, link, bibliography or just enlighten this question? Thank you.

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The absolute value is a notion related to vector lattices (which has nothing to do with the notion of norm). Indeed, given a vector space $X$ with a partial order $\le$ such that finite suprema (and infima) exist, then the absolute value is defined by $$ |x|:=\sup\{x,-x\}. $$ for each $x \in X$. For a standard textbook, you can see Positive Operators.

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  • $\begingroup$ If I understand correctly you mean here that $X$ is at the same time the vector space and the field, right? $\endgroup$ – Masacroso Oct 18 '16 at 12:13
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    $\begingroup$ The definition of absolute value works for vector lattices; in your case, you need a partial order on the field for which finite suprema and infima exist (it can be shown that this is equivalent to the existence of the absolute value itself) $\endgroup$ – Paolo Leonetti Oct 18 '16 at 12:33

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