infimum
The infimum of a set is the greatest lower bound of and is denoted .
Let be a set with a partial order , and let . For any , is a lower bound of if for any . The infimum of , denoted , is the greatest such lower bound; that is, if is a lower bound of , then .
Note that it is not necessarily the case that . Suppose ; then , but .
Also note that a set does not necessarily have an infimum. See the attachments to this entry for examples.
Title | infimum |
Canonical name | Infimum |
Date of creation | 2013-03-22 11:48:09 |
Last modified on | 2013-03-22 11:48:09 |
Owner | vampyr (22) |
Last modified by | vampyr (22) |
Numerical id | 11 |
Author | vampyr (22) |
Entry type | Definition |
Classification | msc 06A06 |
Classification | msc 03D20 |
Related topic | Supremum |
Related topic | LebesgueOuterMeasure |
Related topic | MinimalAndMaximalNumber |
Related topic | InfimumAndSupremumForRealNumbers |
Related topic | NondecreasingSequenceWithUpperBound |