Definition:Infimum of Mapping/Real-Valued Function
This page is about Infimum in the context of Real-Valued Function. For other uses, see Infimum.
Definition
Let $f: S \to \R$ be a real-valued function.
Let $f$ be bounded below on $S$.
Definition 1
The infimum of $f$ on $S$ is defined by:
- $\ds \inf_{x \mathop \in S} \map f x = \inf f \sqbrk S$
where
Definition 2
The infimum of $f$ on $S$ is defined as $\ds \inf_{x \mathop \in S} \map f x := k \in \R$ such that:
- $(1): \quad \forall x \in S: k \le \map f x$
- $(2): \quad \forall \epsilon \in \R_{>0}: \exists x \in S: \map f x < k + \epsilon$
Also defined as
Some sources refer to the infimum as being the lower bound.
Using this convention, any element less than this is not considered to be a lower bound.
Also known as
Particularly in the field of analysis, the infimum of a set $T$ is often referred to as the greatest lower bound of $T$ and denoted $\map {\mathrm {glb} } T$ or $\map {\mathrm {g.l.b.} } T$.
Some sources refer to the infimum of a set as the infimum on a set.
Some sources refer to the infimum of a set as the meet of the set and use the notation $\bigwedge T$ or $\ds \bigwedge_{y \mathop \in T} y$.
Some sources introduce the notation $\ds \inf_{y \mathop \in T} y$, which may improve clarity in some circumstances.
Some older sources, applying the concept to a (strictly) decreasing real sequence, refer to an infimum as a lower limit.
Also see
- Continuum Property, which guarantees that this infimum always exists.
Linguistic Note
The plural of infimum is infima, although the (incorrect) form infimums can occasionally be found if you look hard enough.
Sources
- 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)