View source for Translation-invariant metric

''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, [[#References|[a2]]]. Two metrics $\rho$, $\rho'$ on $X$ 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>