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 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 introduce the notation $\ds \inf_{y \mathop \in S} y$, 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 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.
Linguistic Note
The plural of infimum is infima, although the (incorrect) form infimums can occasionally be found if you look hard enough.
Also see
- Continuum Property, which guarantees that this infimum always exists.
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)