Editing Infimum and supremum (section)

{{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 [[Total order|totally ordered]] finite sets, the infimum and the [[Maximum_and_minimum#In_relation_to_sets|minimum]] are equal.]]
[[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 mathematics, the '''infimum''' (abbreviated '''inf'''; {{plural form}}: '''infima''') of a [[subset]] <math>S</math> of a [[partially ordered set]] <math>P</math> is the [[greatest element]] in <math>P</math> that is less than or equal to each element of <math>S,</math> if such an element exists.<ref name="BabyRudin">{{cite book|first=Walter|last=Rudin|author-link=Walter Rudin|title=Principles of Mathematical Analysis|publisher=McGraw-Hill|edition=3rd|year=1976|isbn=0-07-054235-X|chapter=Chapter 1 The Real and Complex Number Systems|format=print|page=[https://archive.org/details/principlesofmath00rudi/page/n15 4]|url=https://archive.org/details/principlesofmath00rudi|url-access=registration}}</ref> If the infimum of <math>S</math> exists, it is unique, and if ''b'' is a [[Upper and lower bounds|lower bound]] of <math>S</math>, then ''b'' is less than or equal to the infimum of <math>S</math>. Consequently, the term ''greatest lower bound'' (abbreviated as {{em|GLB}}) is also commonly used.<ref name="BabyRudin" /> The '''supremum''' (abbreviated '''sup'''; {{plural form}}: '''suprema''') of a subset <math>S</math> of a partially ordered set <math>P</math> is the [[least element]] in <math>P</math> that is greater than or equal to each element of <math>S,</math> if such an element exists.<ref name=BabyRudin /> If the supremum of <math>S</math> exists, it is unique, and if ''b'' is an [[Upper and lower bounds|upper bound]] of <math>S</math>, then the supremum of <math>S</math> is less than or equal to ''b''. Consequently, the supremum is also referred to as the ''least upper bound'' (or {{em|LUB}}).<ref name=BabyRudin />

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.