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Supremum

The supremum is the least upper bound of a set S, defined as a quantity M such that no member of the set exceeds M, but if epsilon is any positive quantity, however small, there is a member that exceeds M-epsilon (Jeffreys and Jeffreys 1988). When it exists (which is not required by this definition, e.g., supR does not exist), it is denoted sup_(x in S)x (or sometimes simply sup_(S) for short). The supremum is implemented in the Wolfram Language as MaxValue[f, constr, vars].

More formally, the supremum sup_(x in S)x for S a (nonempty) subset of the affinely extended real numbers R^_=R union {+/-infty} is the smallest value y in R^_ such that for all x in S we have x<=y. Using this definition, sup_(x in S)x always exists and, in particular, supR=infty.

Whenever a supremum exists, its value is unique. On the real line, the supremum of a set is the same as the supremum of its set closure.

Consider the real numbers with their usual order. Then for any set M subset= R, the supremum supM exists (in R) if and only if M is bounded from above and nonempty.

See also

Infimum, Limit, Supremum Limit, Upper Bound

Portions of this entry contributed by Jerome R. Breitenbach

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References

Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991.Jeffreys, H. and Jeffreys, B. S. "Upper and Lower Bounds." §1.044 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 13, 1988.Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, p. 6, 1996.Royden, H. L. Real Analysis, 3rd ed. New York: Macmillan, p. 31, 1988.Rudin, W. Real and Complex Analysis, 3rd ed. New York: McGraw-Hill, p. 7, 1987.

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Supremum

Cite this as:

Breitenbach, Jerome R. and Weisstein, Eric W. "Supremum." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Supremum.html

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