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 | ==Formal definition== | ||
Let ''X'' be a normed space with the [[norm]] <math>\|\cdot\|</math>. Then a set <math>A \subset X</math> is bounded if there exists a real number ''M>0 | |||
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>. | |||
==Theorems about bounded sets== | |||
[[ | Every bounded set of [[real number]]s has a [[supremum]] and an [[infimum]]. It follows that a [[monotonic sequence]] of real numbers that is bounded has a [[limit of a sequence|limit]]. A bounded sequence that is not monotonic does not necessarily have a limit, but it has a [[monotonic sequence|monotonic]] [[subsequence]], and this does have a limit (this is the [[Bolzano–Weierstrass theorem]]). | ||
[[Category: | The [[Heine–Borel theorem]] states that a subset of the [[Euclidean space]] '''R'''<sup>''n''</sup> is [[compact space|compact]] if and only if it is [[closed set|closed]] and bounded.[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 16:00, 20 July 2024
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 .
Theorems about bounded sets
Every bounded set of real numbers has a supremum and an infimum. It follows that a monotonic sequence of real numbers that is bounded has a limit. A bounded sequence that is not monotonic does not necessarily have a limit, but it has a monotonic subsequence, and this does have a limit (this is the Bolzano–Weierstrass theorem).
The Heine–Borel theorem states that a subset of the Euclidean space Rn is compact if and only if it is closed and bounded.