MathGloss

Let $F$ be a field. An absolute value on $F$ is a map $\vert \cdot\vert :F\to \mathbb R$ such that

1. $\vert x\vert \geq 0$ for all $x\in F$;
2. $\vert x\vert = 0$ if and only if $x=0$;
3. $\vert x+y\vert \leq \vert x\vert + \vert y\vert$ for all $x,y\in F$;
4. $\vert x\cdot y\vert = \vert x\vert \cdot \vert y\vert$ for all $x,y\in F$.

Together, $1$ and $2$ make up the condition that $\vert \cdot\vert$ be positive definite while $3$ is just the triangle inequality.

Wikidata ID: Q120812

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