Difference between revisions of "Bounded set"
From Encyclopedia of Mathematics
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| − | A bounded set in a metric space | + | 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 $B \subseteq \alpha U$). | |
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Revision as of 11:28, 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 $B \subseteq \alpha U$).
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