Translation-invariant metric

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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.


[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
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