Infimum (original) (raw)
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The infimum is the greatest lower bound of a set 
, defined as a quantity 
such that no member of the set is less than 
, but if 
is any positive quantity, however small, there is always one member that is less than 
(Jeffreys and Jeffreys 1988). When it exists (which is not required by this definition, e.g., 
does not exist), the infimum is denoted 
or 
. 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 
, the infimum 
exists (in 
) if and only if 
is bounded from below and nonempty.
More formally, the infimum 
for 
a (nonempty) subset of the affinely extended real numbers 
is the largest value 
such that for all 
we have 
. Using this definition, 
always exists and, in particular, 
.
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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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." ยง1.044 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. https://mathworld.wolfram.com/Infimum.html