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Difference between revisions of "Translation-invariant metric"

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''invariant metric''
 
''invariant metric''
  
A [[Metric|metric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093830/t0938301.png" /> on a vector or linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093830/t0938302.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093830/t0938303.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093830/t0938304.png" />. A [[Norm|norm]] or an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093830/t0938305.png" />-norm, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093830/t0938306.png" /> (cf. (the editional comments to) [[Universal space|Universal space]] for a definition of this notion), defines a translation-invariant metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093830/t0938307.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093830/t0938308.png" /> is a metric linear space, i.e. a vector space with a metric such that addition and scalar multiplication are continuous, then there is an invariant metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093830/t0938309.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093830/t09383010.png" /> that is equivalent to the original one, [[#References|[a2]]]. Two metrics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093830/t09383011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093830/t09383012.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093830/t09383013.png" /> are equivalent if they induce the same topology.
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A [[metric]] $\rho$ on a vector or [[linear space]] $X$ such that $\rho(x+z,y+z) = \rho(x,y)$ for all $x,y,z \in X$. A [[norm]] or an $F$-norm, $\Vert \cdot \Vert$ (cf. (the editional comments to) [[Universal space]] for a definition of this notion), defines a translation-invariant metric $\rho(x,y) = \Vert x-y \Vert$. If $(X,\rho)$ is a metric linear space, i.e. a vector space with a metric such that addition and scalar multiplication are continuous, then there is an invariant metric $\rho'$ on $X$ that is equivalent to the original one, [[#References|[a2]]]. Two metrics $\rho$, $\rho'$ on $X$ are equivalent if they induce the same topology.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Rolewicz,  "Metric linear spaces" , Reidel  (1987)  pp. §1.1</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Kakutani,  "Über die Metrisation der topologischen Gruppen"  ''Proc. Imp. Acad. Tokyo'' , '''12'''  (1936)  pp. 82–84</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Rolewicz,  "Metric linear spaces" , Reidel  (1987)  pp. §1.1</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Kakutani,  "Über die Metrisation der topologischen Gruppen"  ''Proc. Imp. Acad. Tokyo'' , '''12'''  (1936)  pp. 82–84</TD></TR>
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</table>

Revision as of 18:23, 8 December 2014

invariant metric

A metric $\rho$ on a vector or linear space $X$ such that $\rho(x+z,y+z) = \rho(x,y)$ for all $x,y,z \in X$. A norm or an $F$-norm, $\Vert \cdot \Vert$ (cf. (the editional comments to) Universal space for a definition of this notion), defines a translation-invariant metric $\rho(x,y) = \Vert x-y \Vert$. If $(X,\rho)$ is a metric linear space, i.e. a vector space with a metric such that addition and scalar multiplication are continuous, then there is an invariant metric $\rho'$ on $X$ that is equivalent to the original one, [a2]. Two metrics $\rho$, $\rho'$ on $X$ are equivalent if they induce the same topology.

References

[a1] S. Rolewicz, "Metric linear spaces" , Reidel (1987) pp. §1.1
[a2] S. Kakutani, "Über die Metrisation der topologischen Gruppen" Proc. Imp. Acad. Tokyo , 12 (1936) pp. 82–84
How to Cite This Entry:
Translation-invariant metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Translation-invariant_metric&oldid=13810