Infimum and supremum
Greatest lower bound and least upper bound / From Wikipedia, the free encyclopedia
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] If the infimum of
exists, it is unique, and if b is a lower bound of
, then b is less than or equal to the infimum 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] If the supremum of
exists, it is unique, and if b is an upper bound of
, then the supremum of
is less than or equal to b. Consequently, the supremum is also referred to as the least upper bound (or LUB).[1]
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Infimum_illustration.svg/640px-Infimum_illustration.svg.png)
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Supremum_illustration.svg/320px-Supremum_illustration.svg.png)
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.