Difference between revisions of "Bounded set"
From Encyclopedia of Mathematics
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| − | A bounded set in a metric space $X$ (with metric $\rho$) is a set $A$ whose diameter | + | A bounded set in a [[metric space]] $X$ (with metric $\rho$) is a set $A$ whose diameter |
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\delta(A) = \sup_{x,y \in A} \rho(x,y) | \delta(A) = \sup_{x,y \in A} \rho(x,y) | ||
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is finite. | is finite. | ||
| − | A bounded set in a topological vector space $E$ (over a field $k$) is a set $B$ which is absorbed by every neighbourhood $U$ of zero (i.e. there exists an $\alpha \in k$ such that $B \subseteq \alpha U$). | + | A bounded set in a [[topological vector space]] $E$ (over a field $k$) is a set $B$ which is absorbed by every neighbourhood $U$ of zero (i.e. there exists an $\alpha \in k$ such that $B \subseteq \alpha U$). |
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| + | A bounded set in a [[partially ordered set]] $P$ (with order $\le$) is a set $C$ for which there exist elements $u, v \in P$ such that $u \le x \le v$ for all $x \in C$. | ||
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| + | The three definitions coincide in the case of subsets of the real numbers. | ||
{{TEX|done}} | {{TEX|done}} | ||
Latest revision as of 11:33, 23 October 2016
A bounded set in a metric space $X$ (with metric $\rho$) is a set $A$ whose diameter $$ \delta(A) = \sup_{x,y \in A} \rho(x,y) $$ is finite.
A bounded set in a topological vector space $E$ (over a field $k$) is a set $B$ which is absorbed by every neighbourhood $U$ of zero (i.e. there exists an $\alpha \in k$ such that $B \subseteq \alpha U$).
A bounded set in a partially ordered set $P$ (with order $\le$) is a set $C$ for which there exist elements $u, v \in P$ such that $u \le x \le v$ for all $x \in C$.
The three definitions coincide in the case of subsets of the real numbers.
How to Cite This Entry:
Bounded set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bounded_set&oldid=39503
Bounded set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bounded_set&oldid=39503
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article