Absolute value (object)

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Needed for work with series weirdly enough!

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 $\Vert\cdot\Vert_v$ 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 $F$ be a field. An absolute value (AKA: real valuation or real valued valuation) on $F$ is a mapping, $v$, with the special notation defined as follows:

• $\vert\cdot\vert_v:F\rightarrow\mathbb{R}$ given by $\vert\cdot\vert_v:x\mapsto \vert x\vert_v$

Such that:

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

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

Trivial absolute value

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

See also

• Norm - denoted $\Vert\cdot\Vert_v:V\rightarrow\mathbb{R}$ for a vector space $V$ with almost exactly the same set of properties (3 requires modification)