# Supremum

In

mathematics, a

**supremum** is a largest possible quantity (subject to given conditions), if such a thing exists; or otherwise, within a larger set of quantities, it is a minimal larger choice (if such exists). For example, under different tax systems there might be a 'largest' percentage

*tax rate that anyone would have to pay*; and this could mean that someone actually paid at that rate, or it might refer instead to the top rate that limits the percentage a very high earner might pay.

In analysis the **supremum** or **least upper bound** of a set *S* of real numbers is denoted by sup(*S*) and is defined to be the smallest real number that is greater than or equal to every number in *S*.
An important property of the real numbers is that every nonempty set of real numbers that is bounded above has a supremum. This is sometimes called the *supremum axiom* and expresses the completeness of the real numbers.
If in addition we define sup(*S*) = -∞ when *S* is empty, and sup(*S*) = +∞ when *S* is not bounded above then *every* set of real numbers has a supremum (see extended real number line).

Examples:

- sup {
*x* in **R** : 0 < x < 1 } = 1
- sup {
*x* in **R** : *x*^{2} < 2 } = √2
- sup { (-1)
^{n} - 1/*n* : *n* = 1, 2, 3, ...} = 1

Note that the supremum of

*S* does not have to belong to

*S* (like in these examples).
If the supremum value belongs to the set then we can say there is a largest element in the set.

In general, in order to show that sup(*S*) ≤ *A*, one only has to show that *x* ≤ *A* for all *x* in *S*. Showing that sup(*S*) ≥ *A* is a bit harder: for any ε > 0, you have to exhibit an element *x* in *S* with *x* ≥ *A* - ε.

In functional analysis, one often considers the **supremum norm** of a bounded function *f* : *X* `->` **R** (or **C**); it is defined as

- ||
*f*||_{∞} = sup { |*f*(*x*)| : *x* ∈ *X* }

and gives rise to several important Banach spaces.

*See also*: infimum or *greatest lower bound*, limit superior.

For subsets *S* of arbitrary partially ordered sets (*P*, <=), a **supremum** or **least upper bound** of *S* is an element *u* in *P* such that

*x* <= *u* for all *x* in *S*, and
- for any
*v* in *P* such that *x* <= *v* for all *x* in *S* it holds that *u* <= *v*.

It can easily be shown that, if *S* has a supremum, then the supremum is unique: if *u*_{1} and *u*_{2} are both suprema of *S* then it follows that *u*_{1} <= *u*_{2} and *u*_{2} <= *u*_{1}, and since <= is antisymmetric it follows that *u*_{1} = *u*_{2}.
In an arbitrary partially ordered set, there may exist subsets which don't have a supremum.
In a lattice every nonempty *finite* subset has a supremum, and in a complete lattice every subset has a supremum.

The difference between the supremum of a set and the maximum element of a set may not be immediately obvious. The difference is exemplified by the set of negative real numbers. This set has no maximum element; for every element of the set, there is another, larger element. For example, for any negative real number *x*, there is a negative real number *x/2*, which is greater. But, although the negative real numbers has no maximum, it *does* -- like all sets of real numbers -- have a supremum; namely, 0.

If, on the other hand, a set does contain a maximum element, then this maximum element is also the supremum of the set.

If a set *S* has the property that every nonempty subset of *S* has an upper bound also has a least upper bound, then *S* is said to have the *least upper bound property*. As noted above, the set **R** of all real numbers has the least upper bound property. Similarly, the set **Z** of integers has the least upper bound property; if *S* is a subset of **Z** and there is some number *n* such that every element *s* of *S* is less than or equal to *n*, then there is a least upper bound *u* for *S*, an integer that is an upper bound for *S* and is less than or equal to every other upper bound for *S*.

An example of a set that *lacks* the least upper bound property is **Q**, the set of rational numbers. Let *S* be the set of all rational numbers *q* such that *q*^{2} < 2. Then *S* has an upper bound (1000, for example, or 6) but no least upper bound. For suppose *p* is an upper bound for *S*, so *p*^{2} > 2. Then *q* = (2*p*+2)/(*p*+2) is also an upper bound for *S*, and *q* < *p*. (To see this, note that *q* = *p* - (*p*^{2} - 2)/(*p* + 2), and that *q*^{2} - 2 is positive.)

There is a corresponding 'greatest lower bound property'; an ordered set possesses the greatest lower bound property if and only if it also posseses the least upper bound property.