Definition:Invariant Metric on Vector Space

Definition

Let $K$ be a field.

Let $X$ be a vector space over $K$.

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

We say that $d$ is invariant if and only if:

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

for each $x, y, z \in X$.

Also see

• Results about invariant metrics on vector spaces can be found here.

Sources

• 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.7$: Invariance