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{{short description|Greatest lower bound and least upper bound}}
[[Image:Infimum illustration.svg|thumb|
[[Image:Supremum illustration.svg|thumb|
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: <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.
== 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}}
The following formulas depend on a notation that conveniently generalizes arithmetic operations on sets: Let the sets <math>A, B \subseteq \R,</math> and scalar <math>r \in \R.</math> Define▼
* If <math>\varnothing \neq S \subseteq \R</math> then there exists a sequence <math>s_{\bull} = \left(s_n\right)_{n=1}^{\infty}</math> in <math>S</math> such that <math>\lim_{n \to \infty} s_n = \sup S.</math> Similarly, there will exist a (possibly different) sequence <math>s_{\bull}</math> in <math>S</math> such that <math>\lim_{n \to \infty} s_n = \inf S.</math> Consequently, if the limit <math>\lim_{n \to \infty} s_n = \sup S</math> is a real number and if <math>f : \R \to X</math> is a continuous function, then <math>f\left(\sup S\right)</math> is necessarily an [[adherent point]] of <math>f(S).</math> ▼
'''Identifying infima and suprema'''
'''Relation to limits of sequences'''
* If <math>A</math> and <math>B</math> are nonempty sets of positive real numbers then <math>\inf (A \cdot B) = \left(\inf A\right) \cdot \left(\inf B\right)</math> and similarly for suprema <math>\sup (A \cdot B) = \left(\sup A\right) \cdot \left(\sup B\right).</math><ref name="zakon">{{cite book|title=Mathematical Analysis I|first=Elias|last=Zakon|pages=39–42|publisher=Trillia Group|date=2004|url=http://www.trillia.com/zakon-analysisI.html}}</ref>▼
* If <math>S \subseteq (0, \infty)</math> is non-empty and if <math>\frac{1}{S} := \left\{ \frac{1}{s} : s \in S \right\},</math> then <math>\frac{1}{\sup_{} S} = \inf_{} \frac{1}{S}</math> 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>\frac{1}{\infty} := 0</math> is commonly used with the [[extended real number]]s; in fact, with this definition the equality <math>\frac{1}{\sup_{} S} = \inf_{} \frac{1}{S}</math> will also hold for any non-empty subset <math>S \subseteq (0, \infty].</math> However, the notation <math>\frac{1}{0}</math> is usually left undefined, which is why the equality <math>\frac{1}{\inf_{} S} = \sup_{} \frac{1}{S}</math> is given only for when <math>\inf_{} S > 0.</math></ref> This equality may alternatively be written as <math>\frac{1}{\displaystyle\sup_{s \in S} s} = \inf_{s \in S} \frac{1}{s}.</math> Moreover, <math>\inf_{} S = 0</math> if and only if <math>\sup_{} \frac{1}{S} = \infty,</math> where if<ref group=note name="DivisionByInfinityOr0" /> <math>\inf_{} S > 0,</math> then <math>\frac{1}{\inf_{} S} = \sup_{} \frac{1}{S}.</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>
It implies that if <math>\lim_{n \to \infty} s_n = \sup S</math> is a real number (where all <math>s_1, s_2, \ldots</math> are in <math>S</math>) and if <math>f</math> is a continuous function whose domain contains <math>S</math> and <math>\sup S,</math> then
<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 [[non-decreasing function]], then it is even possible to conclude that <math>\sup f(S) = f(\sup S).</math>
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.</math>
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.
===Arithmetic operations on sets===
▲The following formulas depend on a notation that conveniently generalizes arithmetic operations on sets
Throughout, <math>A, B \subseteq \R</math> are sets of real numbers.
'''Sum of sets'''
The [[Minkowski sum]] of two sets <math>A</math> and <math>B</math> of real numbers is the set
<math display=block>A + B ~:=~ \{a + b : a \in A, b \in B\}</math>
consisting of all possible arithmetic sums of pairs of numbers, one from each set. The infimum and supremum of the Minkowski sum satisfies
<math display=block>\inf (A + B) = (\inf A) + (\inf B)</math>
and
<math display=block>\sup (A + B) = (\sup A) + (\sup B).</math>
'''Product of sets'''
The multiplication of two sets <math>A</math> and <math>B</math> of real numbers is defined similarly to their Minkowski sum:
<math display=block>A \cdot B ~:=~ \{a \cdot b : a \in A, b \in B\}.</math>
▲
'''Scalar product of a set'''
The product of a real number <math>r</math> and a set <math>B</math> of real numbers is the set
<math display=block>r B ~:=~ \{r \cdot b : b \in B\}.</math>
If <math>r \geq 0</math> then
<math display=block>\inf (r \cdot A) = r (\inf A) \quad \text{ and } \quad \sup (r \cdot A) = r (\sup A),</math>
while if <math>r \leq 0</math> then
<math display=block>\inf (r \cdot A) = r (\sup A) \quad \text{ and } \quad \sup (r \cdot A) = r (\inf A).</math>
Using <math>r = -1</math> and the notation <math display=inline>-A := (-1) A = \{- a : a \in A\},</math> it follows that
<math display=block>\inf (- A) = - \sup A \quad \text{ and } \quad \sup (- A) = - \inf A.</math>
'''Multiplicative inverse of a set'''
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
<math display=block>\frac{1}{\sup_{} S} ~=~ \inf_{} \frac{1}{S}</math>
▲
This equality may alternatively be written as
<math>\frac{1}{\displaystyle\sup_{s \in S} s} = \inf_{s \in S} \tfrac{1}{s}.</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>
== Duality ==
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The supremum of a subset <math>S</math> of <math>(\N, \mid\,)</math> where <math>\,\mid\,</math> denotes "[[Divisor|divides]]", is the [[lowest common multiple]] of the elements of <math>S.</math>
The supremum of a
== See also ==
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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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