Infimum and supremum

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 minimum are equal.]]

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; : 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. If the infimum of <math>S</math> exists, it is unique, and if b is a 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 ) is also commonly used. 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) 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 norms defined in terms of <math>\sup</math> or <math>\inf</math> include the weak <math>L^{p,w}</math> space norms (for <math>1 \leq p < \infty</math>), the norm on Lebesgue space <math>L^\infty(\Omega, \mu),</math> and operator norms. 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

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>

and

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:

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>

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

If <math>r > 0</math> then

while if <math>r < 0</math> then

In the case <math>r = 0</math>,

one has, if <math>A \ne \varnothing</math>

\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,

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

where this equation also holds when <math>\sup_{} S = \infty</math> if the definition <math>\frac{1}{\infty} := 0</math> is used.

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