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Difference between revisions of "Quasi-norm"

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A non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q0766001.png" /> defined on a [[Linear space|linear space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q0766002.png" /> and satisfying the same axioms as a [[Norm|norm]] except for the triangle inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q0766003.png" />, which is replaced by the weaker requirement: There exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q0766004.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q0766005.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q0766006.png" />.
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A non-negative function $\|x\|$ defined on a [[Linear space|linear space]] $R$ and satisfying the same axioms as a [[Norm|norm]] except for the triangle inequality $\|x+y\|\leq\|x\|+\|y\|$, which is replaced by the weaker requirement: There exists a constant $c>0$ such that $\|x+y\|\leq c(\|x\|+\|y\|)$ for all $x,y\in R$.
  
  
  
 
====Comments====
 
====Comments====
The topology of a locally bounded topological vector space can be given by a quasi-norm. Conversely, a quasi-normed vector space is locally bounded. Here a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q0766007.png" /> in a topological vector space is bounded if for each open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q0766008.png" /> of zero there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q0766009.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q07660010.png" />, and a topological vector space is locally bounded if there is a bounded neighbourhood of zero. Given a circled bounded neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q07660011.png" /> of zero in a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q07660012.png" /> (a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q07660013.png" /> is circled if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q07660014.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q07660015.png" />), the Minkowski functional of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q07660016.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076600/q07660017.png" />. It is a quasi-norm.
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The topology of a locally bounded topological vector space can be given by a quasi-norm. Conversely, a quasi-normed vector space is locally bounded. Here a set $B$ in a topological vector space is bounded if for each open neighbourhood $U$ of zero there is a $\rho>0$ such that $B\subset\rho U$, and a topological vector space is locally bounded if there is a bounded neighbourhood of zero. Given a circled bounded neighbourhood $U$ of zero in a topological vector space $E$ (a set $M\subset E$ is circled if $\alpha M\subset M$ for all $|\alpha|\leq1$), the Minkowski functional of $U$ is defined by $q(x)=\inf_{x\in\alpha U}\alpha$. It is a quasi-norm.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Köthe,  "Topological vector spaces" , '''1''' , Springer  (1969)  pp. 159</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Köthe,  "Topological vector spaces" , '''1''' , Springer  (1969)  pp. 159</TD></TR></table>

Latest revision as of 11:23, 2 September 2014

A non-negative function $\|x\|$ defined on a linear space $R$ and satisfying the same axioms as a norm except for the triangle inequality $\|x+y\|\leq\|x\|+\|y\|$, which is replaced by the weaker requirement: There exists a constant $c>0$ such that $\|x+y\|\leq c(\|x\|+\|y\|)$ for all $x,y\in R$.


Comments

The topology of a locally bounded topological vector space can be given by a quasi-norm. Conversely, a quasi-normed vector space is locally bounded. Here a set $B$ in a topological vector space is bounded if for each open neighbourhood $U$ of zero there is a $\rho>0$ such that $B\subset\rho U$, and a topological vector space is locally bounded if there is a bounded neighbourhood of zero. Given a circled bounded neighbourhood $U$ of zero in a topological vector space $E$ (a set $M\subset E$ is circled if $\alpha M\subset M$ for all $|\alpha|\leq1$), the Minkowski functional of $U$ is defined by $q(x)=\inf_{x\in\alpha U}\alpha$. It is a quasi-norm.

References

[a1] G. Köthe, "Topological vector spaces" , 1 , Springer (1969) pp. 159
How to Cite This Entry:
Quasi-norm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-norm&oldid=17511
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article