Norms and Real Normed Spaces

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $V$, with its addition and scalar multiplication, be a real vector space, with zero vector $0_{V}$, and let $\lVert\cdot\rVert:V\to\mathbb{R}$, $v\mapsto\lVert v\rVert$, be a function. The conditions below are required for all $u,v\in V$ and all $\lambda\in\mathbb{R}$.

(i) Nonnegativity. $\lVert v\rVert\ge0$.

(ii) Definiteness. If $\lVert v\rVert=0$, then $v=0_{V}$.

(iii) Absolute homogeneity. $\lVert\lambda v\rVert=|\lambda|\,\lVert v\rVert$.

(iv) Triangle inequality. $\lVert u+v\rVert\le\lVert u\rVert+\lVert v\rVert$.

(v) Norm. $\lVert\cdot\rVert$ is a norm on $V$ if it satisfies (i), (ii), (iii) and (iv).

(vi) Real normed space. A real normed space $(V,\lVert\cdot\rVert)$ is a real vector space $V$ together with a norm $\lVert\cdot\rVert$ on $V$.