The Norm Metric of a Real Normed Space

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $(V,\lVert\cdot\rVert)$ be a real normed space, with zero vector $0_{V}$, differences $u-v$ and norm metric $d_{V}$. Sequences are as in Sequences (i). Convergence of a sequence in $V$ means convergence in the metric space $(V,d_{V})$, by (ii), and convergence of a sequence in $\mathbb{R}$ is as in Convergent Sequences of Real Numbers (i). In (iv), (v), (vi) and (vii), let $(u_{m})$ and $(v_{m})$ be sequences in $V$ converging to $u\in V$ and to $v\in V$, and let $(\lambda_{m})$ be a sequence in $\mathbb{R}$ converging to $\lambda\in\mathbb{R}$.

(i) Norm of the zero vector. $\lVert0_{V}\rVert=0$.

(ii) Norm metric. $d_{V}$ is a metric on $V$.

(iii) Reverse triangle inequality. For all $u,v\in V$,

$$\bigl|\,\lVert u\rVert-\lVert v\rVert\,\bigr|\le\lVert u-v\rVert .$$

(iv) Limit of a sum. The termwise sum $(u_{m}+v_{m})$ converges to $u+v$.

(v) Limit of a difference. The termwise difference $(u_{m}-v_{m})$ converges to $u-v$.

(vi) Limit of a scalar multiple. The termwise product $(\lambda_{m}u_{m})$ converges to $\lambda u$.

(vii) Limit of the norms. The sequence $(\lVert u_{m}\rVert)$ of termwise norms in $\mathbb{R}$ converges to $\lVert u\rVert$.