Absolute value (object)

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For other meanings of absolute value see Absolute value (disambiguation)

Relation to norm

The reader will find this definition is very similar to that of a norm, a norm, denoted [ilmath]\Vert\cdot\Vert_v[/ilmath] instead is a map with the same properties on a vector space an absolute value is defined on a field instead. Note that all fields are vector spaces; so this really is little more than a special case.

Definition

Let [ilmath]F[/ilmath] be a field. An absolute value (AKA: real valuation or real valued valuation) on [ilmath]F[/ilmath] is a mapping, [ilmath]v[/ilmath], with the special notation defined as follows:

• [ilmath]\vert\cdot\vert_v:F\rightarrow\mathbb{R} [/ilmath] given by [ilmath]\vert\cdot\vert_v:x\mapsto \vert x\vert_v[/ilmath]

Such that:

1. [ilmath]\forall x\in F[\vert x\vert_v\ge 0][/ilmath]
2. [ilmath]\forall x\in F[(\vert x\vert_v\eq 0)\iff(x\eq 0)][/ilmath] where [ilmath]0[/ilmath] is the additive identity element of the field.
3. [ilmath]\forall x,y\in F[\vert xy\vert_v\eq \vert x\vert_v\vert y\vert_v][/ilmath]
4. [ilmath]\forall x,y\in F[\vert x+y\vert_v\le \vert x\vert_v+\vert y\vert_v][/ilmath]

We may omit the [ilmath]v[/ilmath] and just write [ilmath]\vert x\vert[/ilmath] or use something more meaningful than [ilmath]v[/ilmath] such as [ilmath]\infty[/ilmath] as in [ilmath]\vert\cdot\vert_\infty[/ilmath] to allow one to distinguish between various absolute values in play.

Trivial absolute value

The trivial absolute value is [ilmath]\vert\cdot\vert_T:F\rightarrow\mathbb{R} [/ilmath] (for any field [ilmath]F[/ilmath]) and acts as follows: [ilmath]\vert\cdot\vert_T:x\mapsto 1[/ilmath]. That is to say: [ilmath]\vert x\vert_T:\eq 1[/ilmath]