Definition:Translation-Invariant Metric

Definition

Let $X$ be a vector space.

Let $d$ be a metric on $X$.

We say that $d$ is translation-invariant if and only if for each $x, y, z \in X$ we have:

$\map d {x + z, y + z} = \map d {x, y}$

Also see

• Metric Induced by Norm is Translation-Invariant

Sources

• 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.6$: Topological vector spaces