Infimum and supremum

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In mathematics, the infimum (abbreviated inf; plural infima) of a subset of a partially ordered set is the greatest element in that is less than or equal to each element of if such an element exists.[1] In other words, it is the greatest element of that is lower or equal to the lowest element of . Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used.[1] The supremum (abbreviated sup; plural suprema) of a subset of a partially ordered set is the least element in that is greater than or equal to each element of if such an element exists.[1] In other words, it is the least element of that is greater than or equal to the greatest element of . Consequently, the supremum is also referred to as the least upper bound (or LUB).[1]

A set of real numbers (hollow and filled circles), a subset of (filled circles), and the infimum of Note that for totally ordered finite sets, the infimum and the minimum are equal.
A set of real numbers (blue circles), a set of upper bounds of (red diamond and circles), and the smallest such upper bound, that is, the supremum of (red diamond).

The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in 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 no minimum or maximum. For instance, the set of positive real numbers (not including ) does not have a minimum, because any given element of could simply be divided in half resulting in a smaller number that is still in There is, however, exactly one infimum of the positive real numbers relative to the real numbers: 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 Edit

supremum = least upper bound

A lower bound of a subset   of a partially ordered set   is an element   of   such that

  •   for all  

A lower bound   of   is called an infimum (or greatest lower bound, or meet) of   if

  • for all lower bounds   of   in     (  is larger than any other lower bound).

Similarly, an upper bound of a subset   of a partially ordered set   is an element   of   such that

  •   for all  

An upper bound   of   is called a supremum (or least upper bound, or join) of   if

  • for all upper bounds   of   in     (  is less than any other upper bound).

Existence and uniqueness Edit

Infima and suprema do not necessarily exist. Existence of an infimum of a subset   of   can fail if   has no lower bound at all, or if the set of lower bounds does not contain a greatest element. However, if an infimum or supremum does exist, it is unique.

Consequently, partially ordered sets for which certain infima are known to exist become especially interesting. For instance, a lattice is a partially ordered set in which all nonempty finite subsets have both a supremum and an infimum, and a complete lattice is a partially ordered set in which 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 properties.

If the supremum of a subset   exists, it is unique. If   contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to   (or does not exist in  ). Likewise, if the infimum exists, it is unique. If   contains a least element, then that element is the infimum; otherwise, the infimum does not belong to   (or does not exist in  ).

Relation to maximum and minimum elements Edit

The infimum of a subset   of a partially ordered set   assuming it exists, does not necessarily belong to   If it does, it is a minimum or least element of   Similarly, if the supremum of   belongs to   it is a maximum or greatest element of  

For example, consider the set of negative real numbers (excluding zero). This set has no greatest element, since for every element of the set, there is another, larger, element. For instance, for any negative real number   there is another negative real number   which is greater. On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set. Hence,   is the least upper bound of the negative reals, so the supremum is 0. This set has a supremum but no greatest element.

However, the definition of maximal and minimal elements is more general. In particular, a set can have many maximal and minimal elements, whereas infima and suprema are unique.

Whereas maxima and minima must be members of the subset that is under consideration, the infimum and supremum of a subset need not be members of that subset themselves.

Minimal upper bounds Edit

Finally, a partially ordered set may have many minimal upper bounds without having a least upper bound. Minimal upper bounds are those upper bounds for which there is no strictly smaller element that also is an upper bound. This does not say that each minimal upper bound is smaller than all other upper bounds, it merely is not greater. The distinction between "minimal" and "least" is only possible when the given order is not a total one. In a totally ordered set, like the real numbers, the concepts are the same.

As an example, let   be the set of all finite subsets of natural numbers and consider the partially ordered set obtained by taking all sets from   together with the set of integers   and the set of positive real numbers   ordered by subset inclusion as above. Then clearly both   and   are greater than all finite sets of natural numbers. Yet, neither is   smaller than   nor is the converse true: both sets are minimal upper bounds but none is a supremum.

Least-upper-bound property Edit

The least-upper-bound property is an example of the aforementioned completeness properties which is typical for the set of real numbers. This property is sometimes called Dedekind completeness.

If an ordered set   has the property that every nonempty subset of   having an upper bound also has a least upper bound, then   is said to have the least-upper-bound property. As noted above, the set   of all real numbers has the least-upper-bound property. Similarly, the set   of integers has the least-upper-bound property; if   is a nonempty subset of   and there is some number   such that every element   of   is less than or equal to   then there is a least upper bound   for   an integer that is an upper bound for   and is less than or equal to every other upper bound for   A well-ordered set also has the least-upper-bound property, and the empty subset has also a least upper bound: the minimum of the whole set.

An example of a set that lacks the least-upper-bound property is   the set of rational numbers. Let   be the set of all rational numbers   such that   Then   has an upper bound (  for example, or  ) but no least upper bound in  : If we suppose   is the least upper bound, a contradiction is immediately deduced because between any two reals   and   (including   and  ) there exists some rational   which itself would have to be the least upper bound (if  ) or a member of   greater than   (if  ). Another example is the hyperreals; there is no least upper bound of the set of positive infinitesimals.

There is a corresponding greatest-lower-bound property; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least-upper-bound of the set.

If in a partially ordered set   every bounded subset has a supremum, this applies also, for any set   in the function space containing all functions from   to   where   if and only if   for all   For example, it applies for real functions, and, since these can be considered special cases of functions, for real  -tuples and sequences of real numbers.

The least-upper-bound property is an indicator of the suprema.

Infima and suprema of real numbers Edit

In analysis, infima and suprema of subsets   of the real numbers are particularly important. For instance, the negative real numbers do not have a greatest element, and their supremum is   (which is not a negative real number).[1] The completeness of the real numbers implies (and is equivalent to) that any bounded nonempty subset   of the real numbers has an infimum and a supremum. If   is not bounded below, one often formally writes   If   is empty, one writes  

Properties Edit

If   is any set of real numbers then   if and only if   and otherwise  [2]

If   are sets of real numbers then   (unless  ) and  

Identifying infima and suprema

If the infimum of   exists (that is,   is a real number) and if   is any real number then   if and only if   is a lower bound and for every   there is an   with   Similarly, if   is a real number and if   is any real number then   if and only if   is an upper bound and if for every   there is an   with  

Relation to limits of sequences

If   is any non-empty set of real numbers then there always exists a non-decreasing sequence   in   such that   Similarly, there will exist a (possibly different) non-increasing sequence   in   such that  

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   is a continuous function and   is a sequence of points in its domain that converges to a point   then   necessarily converges to   It implies that if   is a real number (where all   are in  ) and if   is a continuous function whose domain contains   and   then

which (for instance) guarantees[note 1] that   is an adherent point of the set   If in addition to what has been assumed, the continuous function   is also an increasing or non-decreasing function, then it is even possible to conclude that   This may be applied, for instance, to conclude that whenever   is a real (or complex) valued function with domain   whose sup norm   is finite, then for every non-negative real number  
since the map   defined by   is a continuous non-decreasing function whose domain   always contains   and  

Although this discussion focused on   similar conclusions can be reached for   with appropriate changes (such as requiring that   be non-increasing rather than non-decreasing). Other norms defined in terms of   or   include the weak   space norms (for  ), the norm on Lebesgue space   and operator norms. Monotone sequences in   that converge to   (or to  ) 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 Edit

The following formulas depend on a notation that conveniently generalizes arithmetic operations on sets. Throughout,   are sets of real numbers.

Sum of sets

The Minkowski sum of two sets   and   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 satisfies

Product of sets

The multiplication of two sets   and   of real numbers is defined similarly to their Minkowski sum:


If   and   are nonempty sets of positive real numbers then   and similarly for suprema  [3]

Scalar product of a set

The product of a real number   and a set   of real numbers is the set


If   then

while if   then
Using   and the notation   it follows that

Multiplicative inverse of a set

For any set   that does not contain   let


If   is non-empty then

where this equation also holds when   if the definition   is used.[note 2] This equality may alternatively be written as   Moreover,   if and only if   where if[note 2]   then  

Duality Edit

If one denotes by   the partially-ordered set   with the opposite order relation; that is, for all   declare:

then infimum of a subset   in   equals the supremum of   in   and vice versa.

For subsets of the real numbers, another kind of duality holds:   where  

Examples Edit

Infima Edit

  • The infimum of the set of numbers   is   The number   is a lower bound, but not the greatest lower bound, and hence not the infimum.
  • More generally, if a set has a smallest element, then the smallest element is the infimum for the set. In this case, it is also called the minimum of the set.
  • If   is a decreasing sequence with limit   then  

Suprema Edit

  • The supremum of the set of numbers   is   The number   is an upper bound, but it is not the least upper bound, and hence is not the supremum.

In the last example, the supremum of a set of rationals is irrational, which means that the rationals are incomplete.

One basic property of the supremum is

for any functionals   and  

The supremum of a subset   of   where   denotes "divides", is the lowest common multiple of the elements of  

The supremum of a set   containing subsets of some set   is the union of the subsets when considering the partially ordered set  , where   is the power set of   and   is subset.

See also Edit

Notes Edit

  1. ^ Since   is a sequence in   that converges to   this guarantees that   belongs to the closure of  
  2. ^ a b The definition   is commonly used with the extended real numbers; in fact, with this definition the equality   will also hold for any non-empty subset   However, the notation   is usually left undefined, which is why the equality   is given only for when  

References Edit

  1. ^ a b c d e Rudin, Walter (1976). ""Chapter 1 The Real and Complex Number Systems"". Principles of Mathematical Analysis (print) (3rd ed.). McGraw-Hill. p. 4. ISBN 0-07-054235-X.
  2. ^ Rockafellar & Wets 2009, pp. 1–2.
  3. ^ Zakon, Elias (2004). Mathematical Analysis I. Trillia Group. pp. 39–42.

External links Edit