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Difference between revisions of "Bounded set"

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A bounded set in a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017430/b0174301.png" /> (with metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017430/b0174302.png" />) is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017430/b0174303.png" /> whose diameter
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A bounded set in a metric space $X$ (with metric $\rho$) is a set $A$ whose diameter
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$$
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\delta(A) = \sup_{x,y \in A} \rho(x,y)
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$$
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is finite.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017430/b0174304.png" /></td> </tr></table>
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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 $B \subseteq \alpha U$).
 
 
is finite.
 
  
A bounded set in a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017430/b0174305.png" /> (over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017430/b0174306.png" />) is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017430/b0174307.png" /> which is absorbed by every neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017430/b0174308.png" /> of zero (i.e. there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017430/b0174309.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017430/b01743010.png" />).
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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=14421
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article