The infimum is the greatest lower bound of a set S, defined as a quantity m such that no member of the set is less than m, but if epsilon is any positive quantity, however small, there is always one member that is less than m+epsilon (Jeffreys and Jeffreys 1988). When it exists (which is not required by this definition, e.g., infR does not exist), the infimum is denoted infS or inf_(x in S)x. The infimum is implemented in the Wolfram Language as MinValue[f, constr, vars].

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

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

Whenever an infimum exists, its value is unique.

See also

Infimum Limit, Lower Bound, Supremum

Portions of this entry contributed by Jerome R. Breitenbach

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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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Cite this as:

Breitenbach, Jerome R. and Weisstein, Eric W. "Infimum." From MathWorld--A Wolfram Web Resource.

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