View source for Translation-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.

====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>