Supremum: Encyclopedia II - Supremum - Supremum of a set of real numbers

Supremum - Supremum of a set of real numbers

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 its completeness: every nonempty set of real numbers that is bounded above has a supremum. 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:

The supremum of S may or may not belong to S. In particular, note the third example where the supremum of a set of rationals is irrational (which means that the rationals are incomplete). However, if the supremum value belongs to the set then it is the greatest element in the set. The term maximal element is also synonymous as long as one deals with real numbers or any other totally ordered set.

Since sup(S) is the least upper bound, to show that sup(S) ≤ a, one only has to show that a itself is an upper bound for S, i.e. one only has to show that x ≤ a for all x in S. Showing that sup(S) ≥ a is a bit harder: for any b < a, we must find an x in S with x ≥ b.

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

and gives rise to several important Banach spaces.

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

Supremum - Approximation property

Let S be a nonempty set of real numbers with a supremum, say b = sup S. Then for every a < b there is some x in S such that

Proof:

First of all, for all x in S. If we had for every x in S, then a would be an upper bound for S smaller than the least upper bound. Therefore x > a for at least one x in S.

Supremum - Additive property

Given nonempy subsets A and B of R, let C denote the set

If each of A and B has a supremum, then C has a supremum and sup C = sup A + sup B.

proof:

Let a = sup A, b = sup B. If then z = x + y, where ,,so . Hence a + b is an upper bound for C, so C has a supremum, say c = sup C, and . We show next that . Choose any z > 0. By the approximation property, there is an x in A and a y in B such that a − z < x and b − z < y. Adding these inequalities we find . Thus, a + b < c + 2z for every z > 0 so .

Supremum - Comparison property

Given nonempty subsets S and T of R such that for every s in S and t in T. If T has a supremum then S has a supremum and .

proof:

Let c = sup T. For for every s in S and t in T, S is bounded above, thus S has a supremum. Let d = sup S. By the approximation property, there is an s in S such that d − z < s for any z > 0. Therefore d − z < . Because this holds for all z > 0, this implies that .

Lemma: Given real numbers a and b such that a < b + z for every z > 0. Then

Adapted from the Wikipedia article "Supremum of a set of real numbers", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki