Definition:Bounded Below Mapping

From ProofWiki
Jump to navigation Jump to search

This page is about mappings which are bounded below. For other uses, see Definition:Bounded Below.

Definition

Let $f: S \to T$ be a mapping whose codomain is an ordered set $\left({T, \preceq}\right)$.


Then $f$ is said to be bounded below (in $T \ $) by the lower bound $L$ if and only if:

$\forall x \in S: L \preceq f \left({x}\right)$


That is, iff $f \left({S}\right) = \left\{{f \left({x}\right): x \in S}\right\}$ is bounded below by $L$.


Real-Valued Function

The concept is usually encountered where $\left({T, \preceq}\right)$ is the set of real numbers under the usual ordering $\left({\R, \le}\right)$:


Let $f: S \to \R$ be a real-valued function.


Then $f$ is bounded below on $S$ by the lower bound $L$ if and only if:

$\forall x \in S: L \le \map f x$


Unbounded Below

Let $f: S \to T$ be a mapping whose codomain is an ordered set $\left({T, \preceq}\right)$.


Then $f$ is unbounded below (in $T \ $) if and only if there exists no $L \in S$ such that:

$\forall x \in S: L \preceq f \left({x}\right)$


Also see