Editing Infimum and supremum (section)

== Infima and suprema of real numbers ==

In [[Mathematical analysis|analysis]], infima and suprema of subsets <math>S</math> of the [[real numbers]] are particularly important. For instance, the negative [[real number]]s do not have a greatest element, and their supremum is <math>0</math> (which is not a negative real number).<ref name=BabyRudin />
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>

===Properties===

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}} 

'''Set inclusion'''

If <math>A \subseteq B</math> are sets of real numbers then <math>\inf A \geq \inf B</math> (if <math>A = \varnothing</math> this reads as <math>\inf B \le \infty</math>) and <math>\sup A \leq \sup B.</math>

'''Image under functions'''
If <math>f \colon \mathbb{R} \to \mathbb{R}</math> is a nonincreasing function, then 
<math>f (\inf(S)) \le \inf (f (S))</math> and <math>\sup(f(S))</math>, where the image is defined as <math>f(S) \,\stackrel{\scriptscriptstyle\text{def}}{=}\, \{f(s) : s \in S\}.</math>

'''Identifying infima and suprema'''

If the infimum of <math>A</math> exists (that is, <math>\inf A</math> is a real number) and if <math>p</math> is any real number then <math>p = \inf A</math> if and only if <math>p</math> is a lower bound and for every <math>\epsilon > 0</math> there is an <math>a_\epsilon \in A</math> with <math>a_\epsilon < p + \epsilon.</math> 
Similarly, if <math>\sup A</math> is a real number and if <math>p</math> is any real number then <math>p = \sup A</math> if and only if <math>p</math> is an upper bound and if for every <math>\epsilon > 0</math> there is an <math>a_\epsilon \in A</math> with <math>a_\epsilon > p - \epsilon.</math>

'''Relation to limits of sequences'''

If <math>S \neq \varnothing</math> is any non-empty set of real numbers then there always exists a non-decreasing sequence <math>s_1 \leq s_2 \leq \cdots</math> in <math>S</math> such that <math>\lim_{n \to \infty} s_n = \sup S.</math> Similarly, there will exist a (possibly different) non-increasing sequence <math>s_1 \geq s_2 \geq \cdots</math> in <math>S</math> such that <math>\lim_{n \to \infty} s_n = \inf S.</math>
In particular, the infimum and supremum of a set belong to its [[Closure (topology)|closure]] if <math>\inf S \in \mathbb{R}</math> then <math>\inf S \in \bar{S}</math> and if  <math>\sup S \in \mathbb{R}</math> then <math>\sup S \in \bar{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 satisfy, if <math>A \ne \varnothing \ne B</math>
<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>

If <math>A</math> and <math>B</math> are nonempty sets of positive real numbers then <math>\inf (A \cdot B) = (\inf A) \cdot (\inf B)</math> and similarly for suprema <math>\sup (A \cdot B) = (\sup A) \cdot (\sup B).</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>

'''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 > 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 < 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>
In the case <math>r = 0</math>,
one has, if <math>A \ne \varnothing</math>
<math display=block>
  \inf (0 \cdot A) = 0 \quad \text{ and } \quad  \sup (0 \cdot A) = 0 
</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> 
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>\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>