Definition:Bounded Below Mapping

This page is about Bounded Below in the context of Mapping. For other uses, see Bounded Below.

Definition

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

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

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

That is, iff $f \sqbrk S = \set {\map f x: x \in S}$ is bounded below by $L$.

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

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:

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

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

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

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