Definition:Absolute Value of Mapping

Definition
Let $D$ be an ordered integral domain, and let $\left\vert{\cdot}\right\vert_D$ be its absolute value.

Let $S$ be a set, and let $f: S \to D$ be a mapping.

Then the absolute value of $f$, denoted $\left\vert{f}\right\vert_D: S \to D$, is defined as:


 * $\forall s \in S: \left\vert{f}\right\vert_D \left({s}\right) := \left\vert{f \left({s}\right)}\right\vert_D$

where $\left\vert{f \left({s}\right)}\right\vert_D$ denotes the absolute value function on $D$.

Absolute value thence is an instance of a pointwise operation on a mapping.

Examples

 * Definition:Absolute Value of Real-Valued Function

Also see

 * Absolute Value of Extended Real-Valued Function, not an example as $\overline{\R}$ is not an ordered integral domain
 * Operation Induced on Set of Mappings