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Difference between revisions of "Balanced set"

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A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015100/b0151001.png" /> in a real or complex vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015100/b0151002.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015100/b0151003.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015100/b0151004.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015100/b0151005.png" />. An example of a balanced set is given by the unit ball in a normed vector space and, generally, by a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015100/b0151006.png" /> of zero in a base of neighbourhoods of zero in a topological vector space. These neighbourhoods of zero are moreover absorbing, i.e. such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015100/b0151007.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015100/b0151008.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015100/b0151009.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015100/b01510010.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015100/b01510011.png" /> is a convex, absorbing and balanced set, then the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015100/b01510012.png" /> is a [[Semi-norm|semi-norm]], i.e. it has the properties
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A set $U$ in a real or complex vector space $X$ such that $x \in U$ and $|\lambda| \le 1$ imply $\lambda x \in U$. An example of a balanced set is given by the unit ball in a normed vector space and, generally, by a neighbourhood $U$ of zero in a base of neighbourhoods of zero in a topological vector space. These neighbourhoods of zero are moreover absorbing, i.e. such that for any $x \in X$ there exists an $\alpha > 0$ such that $x \in \lambda U$ for $|\lambda| \ge \alpha$. If $U$ is a convex, absorbing and balanced set, then the functional $p_U(x) = \inf\{|\lambda| : x \in \lambda U\}$ is a [[semi-norm]], i.e. it has the properties
<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/b015/b015100/b01510013.png" /></td> </tr></table>
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$$
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p_U(x+y) \le p_U(x) + p_U(y)\ ,\ \ p_U(\lambda x) = |\lambda| p(x) \ .
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$$
  
 
A balanced set is also called centred.
 
A balanced set is also called centred.
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====Comments====
 
====Comments====
The functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015100/b01510014.png" /> mentioned above is also called the Minkowski functional of the convex, absorbing and balanced set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015100/b01510015.png" />.
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The functional $p_U$ mentioned above is also called the Minkowski functional of the convex, absorbing and balanced set $U$.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)  pp. Chapt. 8, Sect. 4; 5</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)  pp. Chapt. 8, Sect. 4; 5</TD></TR></table>

Latest revision as of 20:23, 23 December 2014

A set $U$ in a real or complex vector space $X$ such that $x \in U$ and $|\lambda| \le 1$ imply $\lambda x \in U$. An example of a balanced set is given by the unit ball in a normed vector space and, generally, by a neighbourhood $U$ of zero in a base of neighbourhoods of zero in a topological vector space. These neighbourhoods of zero are moreover absorbing, i.e. such that for any $x \in X$ there exists an $\alpha > 0$ such that $x \in \lambda U$ for $|\lambda| \ge \alpha$. If $U$ is a convex, absorbing and balanced set, then the functional $p_U(x) = \inf\{|\lambda| : x \in \lambda U\}$ is a semi-norm, i.e. it has the properties $$ p_U(x+y) \le p_U(x) + p_U(y)\ ,\ \ p_U(\lambda x) = |\lambda| p(x) \ . $$

A balanced set is also called centred.

References

[1] L.V. Kantorovich, G.P. Akilov, "Functional analysis" , Pergamon (1982) (Translated from Russian)


Comments

The functional $p_U$ mentioned above is also called the Minkowski functional of the convex, absorbing and balanced set $U$.

References

[a1] K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5
How to Cite This Entry:
Balanced set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Balanced_set&oldid=18616
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article