Definition:Absolute Value of Mapping/Extended Real-Valued Function

Definition
Let $S$ be a set, and let $f: S \to \overline{\R}$ be an extended real-valued function.

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


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

where $\left\vert{f \left({s}\right)}\right\vert$ denotes the extended absolute value function on $\overline{\R}$.

Absolute value thence is an instance of a pointwise operation on extended real-valued functions.

Since extended absolute value coincides on $\R$ with the standard ordering, this definition incorporates the definition for real-valued functions.

Also see

 * Pointwise Operation on Extended Real-Valued Functions for more operations on extended real-valued functions
 * Absolute Value of Mapping for the absolute value of more general mappings
 * Absolute Value of Real-Valued Function