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{{short description|Greatest lower bound and least upper bound}}
[[Image:Infimum illustration.svg|thumb|upright=1.2|A set <math>P</math> of real numbers (hollow and filled circles), a subset <math>S</math> of <math>P</math> (filled circles), and the infimum of <math>S.</math> Note that for
[[Image:Supremum illustration.svg|thumb|upright=1.2|A set <math>A</math> of real numbers (blue circles), a set of upper bounds of <math>A</math> (red diamond and circles), and the smallest such upper bound, that is, the supremum of <math>A</math> (red diamond).]]
In
The infimum is, in a precise sense, [[Duality (order theory)|dual]] to the concept of a supremum.
The concepts of infimum and supremum are close to [[minimum]] and [[maximum]], but are more useful in analysis because they better characterize special sets which may have {{em|no minimum or maximum}}.
▲The infimum is in a precise sense [[Duality (order theory)|dual]] to the concept of a supremum. Infima and suprema of [[real number]]s are common special cases that are important in [[Mathematical analysis|analysis]], and especially in [[Lebesgue integration]]. However, the general definitions remain valid in the more abstract setting of [[order theory]] where arbitrary partially ordered sets are considered.
▲The concepts of infimum and supremum are close to [[minimum]] and [[maximum]], but are more useful in analysis because they better characterize special sets which may have {{em|no minimum or maximum}}. For instance, the set of [[positive real numbers]] <math>\R^+</math> (not including <math>0</math>) does not have a minimum, because any given element of <math>\R^+</math> could simply be divided in half resulting in a smaller number that is still in <math>\R^+.</math> There is, however, exactly one infimum of the positive real numbers relative to the real numbers: <math>0,</math> which is smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound. An infimum of a set is always and only defined relative to a superset of the set in question. For example, there is no infimum of the positive real numbers inside the positive real numbers (as their own superset), nor any infimum of the positive real numbers inside the complex numbers with positive real part.
== Formal definition ==
[[File:Illustration of supremum.svg|thumb|upright=1.2|supremum = least upper bound]]
A {{em|lower bound}} of a subset <math>S</math> of a [[partially ordered set]] <math>(P, \leq)</math> is an element <math>
* <math>
A lower bound <math>a</math> of <math>S</math> is called an {{em|infimum}} (or {{em|greatest lower bound}}, or [[Join and meet|{{em|meet}}]]) of <math>S</math> if
* for all lower bounds <math>y</math> of <math>S</math> in <math>P,</math> <math>y \leq a</math> (<math>a</math> is larger than
Similarly, an {{em|upper bound}} of a subset <math>S</math> of a partially ordered set <math>(P, \leq)</math> is an element <math>
* <math>
An upper bound <math>b</math> of <math>S</math> is called a {{em|supremum}} (or {{em|least upper bound}}, or [[Join and meet|{{em|join}}]]) of <math>S</math> if
* for all upper bounds <math>z</math> of <math>S</math> in <math>P,</math> <math>z \geq b</math> (<math>b</math> is less than
== Existence and uniqueness ==
Infima and suprema do not necessarily exist. Existence of an infimum of a subset <math>S</math> of <math>P</math> can fail if <math>S</math> has no lower bound at all, or if the set of lower bounds does not contain a greatest element.
Consequently, partially ordered sets for which certain infima are known to exist become especially interesting. For instance, a [[Lattice (order)|lattice]] is a partially ordered set in which all {{em|nonempty finite}} subsets have both a supremum and an infimum, and a [[complete lattice]] is a partially ordered set in which {{em|all}} subsets have both a supremum and an infimum. More information on the various classes of partially ordered sets that arise from such considerations are found in the article on [[Completeness (order theory)|completeness properties]].
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The [[completeness of the real numbers]] implies (and is equivalent to) that any bounded nonempty subset <math>S</math> of the real numbers has an infimum and a supremum. If <math>S</math> is not bounded below, one often formally writes <math>\inf_{} S = -\infty.</math> If <math>S</math> is [[Empty set|empty]], one writes <math>\inf_{} S = +\infty.</math>
===
If <math>A</math> is any set of real numbers then <math>A \neq \varnothing</math> if and only if <math>\sup A \geq \inf A,</math> and otherwise <math>-\infty = \sup \varnothing < \inf \varnothing = \infty.</math>{{sfn|Rockafellar|Wets|2009|pp=1-2}}
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'''Relation to limits of sequences'''
If <math>
Expressing the infimum and supremum as a limit of a such a sequence allows theorems from various branches of mathematics to be applied. Consider for example the well-known fact from [[topology]] that if <math>f</math> is a [[Continuous function (topology)|continuous function]] and <math>s_1, s_2, \ldots</math> is a sequence of points in its domain that converges to a point <math>p,</math> then <math>f\left(s_1\right), f\left(s_2\right), \ldots</math> necessarily converges to <math>f(p).</math>
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<math display=block>f(\sup S) = f\left(\lim_{n \to \infty} s_n\right) = \lim_{n \to \infty} f\left(s_n\right),</math>
which (for instance) guarantees<ref group=note>Since <math>f\left(s_1\right), f\left(s_2\right), \ldots</math> is a sequence in <math>f(S)</math> that converges to <math>f(\sup S),</math> this guarantees that <math>f(\sup S)</math> belongs to the [[Closure (topology)|closure]] of <math>f(S).</math></ref> that <math>f(\sup S)</math> is an [[adherent point]] of the set <math>f(S) \,\stackrel{\scriptscriptstyle\text{def}}{=}\, \{f(s) : s \in S\}.</math>
If in addition to what has been assumed, the continuous function <math>f</math> is also an increasing or [[
This may be applied, for instance, to conclude that whenever <math>g</math> is a real (or [[Complex number|complex]]) valued function with domain <math>\Omega \neq \varnothing</math> whose [[sup norm]] <math>\|g\|_\infty \,\stackrel{\scriptscriptstyle\text{def}}{=}\, \sup_{x \in \Omega} |g(x)|</math> is finite, then for every non-negative real number <math>q,</math>
<math display=block>\|g\|_\infty^q ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left(\sup_{x \in \Omega} |g(x)|\right)^q = \sup_{x \in \Omega} \left(|g(x)|^q\right)</math>
since the map <math>f : [0, \infty) \to \R</math> defined by <math>f(x) = x^q</math> is a continuous non-decreasing function whose domain <math>[0, \infty)</math> always contains <math>S := \{|g(x)| : x \in \Omega\}</math> and <math>\sup S \,\stackrel{\scriptscriptstyle\text{def}}{=}\, \|g\|_\infty
Although this discussion focused on <math>\sup,</math> similar conclusions can be reached for <math>\inf</math> with appropriate changes (such as requiring that <math>f</math> be non-increasing rather than non-decreasing). Other [[Norm (mathematics)|norms]] defined in terms of <math>\sup</math> or <math>\inf</math> include the [[weak Lp space|weak <math>L^{p,w}</math> space]] norms (for <math>1 \leq p < \infty</math>), the norm on [[Lp space|Lebesgue space]] <math>L^\infty(\Omega, \mu),</math> and [[operator norm]]s. Monotone sequences in <math>S</math> that converge to <math>\sup S</math> (or to <math>\inf S</math>) can also be used to help prove many of the formula given below, since addition and multiplication of real numbers are continuous operations.
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For any set <math>S</math> that does not contain <math>0,</math> let
<math display=block>\frac{1}{S} ~:=\; \left\{\tfrac{1}{s} : s \in S\right\}.</math>
If <math>S \subseteq (0, \infty)</math> is non-empty then
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where this equation also holds when <math>\sup_{} S = \infty</math> if the definition <math>\frac{1}{\infty} := 0</math> is used.<ref group="note" name="DivisionByInfinityOr0">The definition <math>\tfrac{1}{\infty} := 0</math> is commonly used with the [[extended real number]]s; in fact, with this definition the equality <math>\tfrac{1}{\sup_{} S} = \inf_{} \tfrac{1}{S}</math> will also hold for any non-empty subset <math>S \subseteq (0, \infty].</math> However, the notation <math>\tfrac{1}{0}</math> is usually left undefined, which is why the equality <math>\tfrac{1}{\inf_{} S} = \sup_{} \tfrac{1}{S}</math> is given only for when <math>\inf_{} S > 0.</math></ref>
This equality may alternatively be written as
<math>\
Moreover, <math>\inf_{} S = 0</math> if and only if <math>\sup_{} \tfrac{1}{S} = \infty,</math> where if<ref group=note name="DivisionByInfinityOr0" /> <math>\inf_{} S > 0,</math> then <math>\tfrac{1}{\inf_{} S} = \sup_{} \tfrac{1}{S}.</math>
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* {{Rockafellar Wets Variational Analysis 2009 Springer}} <!-- {{sfn|Rockafellar|Wets|2009|p=}} -->
==
* {{springer|title=Upper and lower bounds|id=p/u095810}}
* {{MathWorld|Supremum|author=Breitenbach, Jerome R.|author2=Weisstein, Eric W.|name-list-style=amp}}
{{Lp spaces}}
[[Category:Order theory]]
[[Category:Superlatives]]
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