The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity

Let $\mathbb{R}$ be the ordered field of real numbers, with the notation of that item, and let $E$ be a real inner product space with inner product $\langle\cdot,\cdot\rangle$, norm $|\cdot|$, distance $d$ and zero vector $0_{E}$, and let $\mathbb{N}$ be the set of natural numbers. For a real number $s$, $|s|$ is its absolute value, and $d_{\mathbb{R}}(s,t)=|s-t|$ is the metric on $\mathbb{R}$ of The Absolute Value Metric on the Real Numbers. Convergence of a sequence in $E$ means convergence in the metric space $(E,d)$ of claim 3 below, and convergence of a sequence of real numbers is as in Limit of a Sequence of Real Numbers; claims 1, 2 and 3 do not involve convergence, and claims 4 to 7 presuppose claim 3. Then for all $x,y,z\in E$ and all $\lambda\in\mathbb{R}$ the following hold.

1. (Triangle inequality)(i) $|x+y|\le|x|+|y|$ and $|x-y|\le|x|+|y|$.

2. (Reverse triangle inequality)(ii) $\bigl|\,|x|-|y|\,\bigr|\le|x-y|$.

3. (Metric)(iii) $d$ is a metric on $E$, so that $(E,d)$ is a metric space; moreover $d(x+z,y+z)=d(x,y)$ and $d(\lambda x,\lambda y)=|\lambda|\,d(x,y)$.

4. (Algebra of limits)(iv) If $(x_{m})_{m\in\mathbb{N}}$ and $(y_{m})_{m\in\mathbb{N}}$ are sequences in $E$ converging to $x$ and to $y$, then $(x_{m}+y_{m})$ converges to $x+y$, $(x_{m}-y_{m})$ converges to $x-y$ and $(\lambda x_{m})$ converges to $\lambda x$; and if $(\lambda_{m})_{m\in\mathbb{N}}$ is a sequence of real numbers converging to $\lambda$, then $(\lambda_{m}x_{m})$ converges to $\lambda x$.

5. (Continuity of the inner product and norm)(v) If $(x_{m})_{m\in\mathbb{N}}$ and $(y_{m})_{m\in\mathbb{N}}$ are sequences in $E$ converging to $x$ and to $y$, then the sequences of real numbers $(\langle x_{m},y_{m}\rangle)$ and $(|x_{m}|)$ converge to $\langle x,y\rangle$ and to $|x|$; in particular $(\langle x_{m},z\rangle)$ converges to $\langle x,z\rangle$.

6. (Bounded sets)(vi) A subset $A\subseteq E$ is bounded in $(E,d)$ if and only if there is a real number $R$ with $|a|\le R$ for every $a\in A$.

7. (Lipschitz maps)(vii) The map $x\mapsto|x|$ is Lipschitz with constant $1$ from $(E,d)$ to $(\mathbb{R},d_{\mathbb{R}})$, and the map $x\mapsto\langle x,z\rangle$ is Lipschitz with constant $|z|$ from $(E,d)$ to $(\mathbb{R},d_{\mathbb{R}})$.