Let X be a subset of . We say that X is boundedPlanetmathPlanetmathPlanetmath when there exists a real number M such that |x|<M for all xX. When X is an interval, we speak of a bounded interval.

This can be generalized first to n. We say that Xn is bounded if there is a real number M such that x<M for all xX and is the Euclidean distance between x and y.

This condition is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the statement: There is a real number T such that x-y<T for all x,yX.

A further generalizationPlanetmathPlanetmath to any metric space V says that XV is bounded when there is a real number M such that d(x,y)<M for all x,yX, where d is the metric on V.

Title bounded
Canonical name Bounded1
Date of creation 2013-03-22 14:00:00
Last modified on 2013-03-22 14:00:00
Owner yark (2760)
Last modified by yark (2760)
Numerical id 11
Author yark (2760)
Entry type Definition
Classification msc 54E35
Related topic EuclideanDistance
Related topic MetricSpace
Defines bounded interval