The Norm Metric of a Real Normed Space

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $(V,\lVert\cdot\rVert)$ be a real normed space, with differences $u-v$.

(i) Norm metric. The norm metric of $(V,\lVert\cdot\rVert)$ is the function $d_{V}:V\times V\to\mathbb{R}$ given by

$$d_{V}(u,v)=\lVert u-v\rVert .$$