Bounded set: Difference between revisions
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In [[mathematics]], a '''bounded set''' is any [[set|subset]] of a [[normed space]] whose elements all have norms which are bounded from above by a fixed positive real constant. In other words, all its elements are uniformly bounded in magnitude. | In [[mathematics]], a '''bounded set''' is any [[set|subset]] of a [[normed space]] whose elements all have norms which are bounded from above by a fixed positive real constant. In other words, all its elements are uniformly bounded in magnitude. | ||
==Formal definition== | ==Formal definition== | ||
Let ''X'' be a normed space with the [[norm (mathematics)|norm]] <math>\|\cdot\|</math>. Then a set <math>A \subset X</math> is bounded if there exists a real number ''M>0'' such that <math>\|x\|\leq M</math> for all <math>x \in A</math>. | Let ''X'' be a normed space with the [[norm (mathematics)|norm]] <math>\|\cdot\|</math>. Then a set <math>A \subset X</math> is bounded if there exists a real number ''M>0'' such that <math>\|x\|\leq M</math> for all <math>x \in A</math>. | ||
Revision as of 12:20, 27 January 2008
In mathematics, a bounded set is any subset of a normed space whose elements all have norms which are bounded from above by a fixed positive real constant. In other words, all its elements are uniformly bounded in magnitude.
Formal definition
Let X be a normed space with the norm . Then a set is bounded if there exists a real number M>0 such that for all .
