Real Banach 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.

(i) Real Banach space. $(V,\lVert\cdot\rVert)$ is a real Banach space if the metric space $(V,d_{V})$ is complete, where $d_{V}$ is the norm metric of $(V,\lVert\cdot\rVert)$, a metric on $V$ by The Norm Metric of a Real Normed Space (ii).