Definition:Lower Bound of Mapping/Real-Valued

This page is about Lower Bound of Real-Valued Function. For other uses, see Lower Bound.

Definition

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

Let $f$ be bounded below in $T$ by $L \in T$.

Then $L$ is a lower bound of $f$.

Lower Bound of Number

When considering the lower bound of a set of numbers, it is commonplace to ignore the set and instead refer just to the number itself.

Thus the construction:

The set of numbers which fulfil the propositional function $P \left({n}\right)$ is bounded below with the lower bound $N$

would be reported as:

The number $n$ such that $P \left({n}\right)$ has the lower bound $N$.

This construct obscures the details of what is actually being stated. Its use on $\mathsf{Pr} \infty \mathsf{fWiki}$ is considered an abuse of notation and so discouraged.

This also applies in the case where it is the lower bound of a mapping which is under discussion.

Also see

• Definition:Bounded Below Real-Valued Function

• Definition:Bounded Above Real-Valued Function
• Definition:Upper Bound of Real-Valued Function

• Definition:Bounded Real-Valued Function

• Definition:Lower Bound of Mapping
• Definition:Bounded Below Mapping

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 7.13$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): bound: 1. (of a function)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): bound: 1. (of a function)