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

We use the notation and claims of Elementary Identities in a Real Inner Product Space, the Cauchy-Schwarz inequality The Cauchy-Schwarz Inequality in a Real Inner Product Space, and for real numbers the claims of Properties of the Absolute Value in an Ordered Field and Elementary Order Arithmetic in an Ordered Field. Vector identities such as $-(x-y)=y-x$, $(x+z)-(y+z)=x-y$ and $\lambda x-\lambda y=\lambda(x-y)$ follow from the axioms of Vector Space over a Field and Elementary Identities in a Vector Space and are used without further comment.

Claim 1. By Elementary Identities in a Real Inner Product Space §expansion, claim 3 of Properties of the Absolute Value in an Ordered Field and The Cauchy-Schwarz Inequality in a Real Inner Product Space,

the inequality by claim 5 of Elementary Arithmetic in an Ordered Field (multiplier $2\ge 0$) and the compatibility of the order with addition (an axiom of Ordered Field). Both $|x+y|$ and $|x|+|y|$ are nonnegative (the latter by claim 2 of Elementary Arithmetic in an Ordered Field), so $|x+y|\le|x|+|y|$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Applying this to $x$ and $-y$ and using $|-y|=|y|$ (Elementary Identities in a Real Inner Product Space §homogeneity) gives $|x-y|\le|x|+|y|$.

Claim 2. By claim 1, $|x|=|(x-y)+y|\le|x-y|+|y|$, so $|x|-|y|\le|x-y|$; exchanging $x$ and $y$ and using $|y-x|=|-(x-y)|=|x-y|$ gives $|y|-|x|\le|x-y|$, that is, $-|x-y|\le|x|-|y|$. By claim 6 of Properties of the Absolute Value in an Ordered Field, $\bigl||x|-|y|\bigr|\le|x-y|$.

Claim 3. We verify the four conditions of Metric Space. (1) $d(x,y)=|x-y|\ge 0$ by Real Inner Product Space §norm. (2) $d(x,y)=0$ if and only if $x-y=0_{E}$ (Elementary Identities in a Real Inner Product Space §vanishing), if and only if $x=y$ (add $y$ to both sides). (3) $d(y,x)=|y-x|=|-(x-y)|=|x-y|=d(x,y)$ by Elementary Identities in a Real Inner Product Space §homogeneity. (4) $d(x,z)=|(x-y)+(y-z)|\le|x-y|+|y-z|=d(x,y)+d(y,z)$ by claim 1. Finally $d(x+z,y+z)=|(x+z)-(y+z)|=|x-y|=d(x,y)$, and $d(\lambda x,\lambda y)=|\lambda(x-y)|=|\lambda|\,|x-y|$ by Elementary Identities in a Real Inner Product Space §homogeneity.

In the remaining claims, convergence of $(x_{m})$ to $x$ means, by Convergent Sequence in a Metric Space and claim 3, that for every real $\varepsilon>0$ there is $N\in\mathbb{N}$ with $|x_{m}-x|<\varepsilon$ for all $m\ge N$; and convergence of a real sequence $(a_{m})$ to $a$ means, by Limit of a Sequence of Real Numbers, the same with $|a_{m}-a|$ in place of $|x_{m}-x|$. Given two indices $N_{1},N_{2}$ we may always pass to $N=\max(N_{1},N_{2})$, using claim 1 of Elementary Properties of the Maximum of Two Elements, and for a given $\varepsilon>0$ the numbers $\varepsilon/2$ and, for $c>0$, $\varepsilon/c$ are positive by claims 7, 8 and 5 of Elementary Order Arithmetic in an Ordered Field.

Claim 4. Let $\varepsilon>0$ and choose $N$ such that $|x_{m}-x|<\varepsilon/2$ and $|y_{m}-y|<\varepsilon/2$ for $m\ge N$. For such $m$, by claim 1, $|(x_{m}+y_{m})-(x+y)|=|(x_{m}-x)+(y_{m}-y)|\le|x_{m}-x|+|y_{m}-y|<\varepsilon$, and $|(x_{m}-y_{m})-(x-y)|=|(x_{m}-x)-(y_{m}-y)|\le|x_{m}-x|+|y_{m}-y|<\varepsilon$. For the multiple: if $\lambda=0$ then $\lambda x_{m}=0_{E}=\lambda x$ for all $m$ and there is nothing to prove; otherwise $|\lambda|>0$, and choosing $N$ with $|x_{m}-x|<\varepsilon/|\lambda|$ for $m\ge N$ gives $|\lambda x_{m}-\lambda x|=|\lambda|\,|x_{m}-x|<\varepsilon$. For the last assertion, the convergent real sequence $(\lambda_{m})$ is bounded by claim 2 of Uniqueness of Limits and Boundedness of Convergent Real Sequences: there is a real $B_{0}$ with $|\lambda_{m}|\le B_{0}$ for all $m$; put $B=\max(B_{0},1)$, so that $0<1\le B$ (claim 6 of Elementary Order Arithmetic in an Ordered Field) and $|\lambda_{m}|\le B$ for all $m$ by claim 1 of Elementary Properties of the Maximum of Two Elements. Then

by claim 1 and Elementary Identities in a Real Inner Product Space §homogeneity. Choose $N$ such that $|x_{m}-x|<\varepsilon/(2B)$ and $|\lambda_{m}-\lambda|<\varepsilon/(2(|x|+1))$ for $m\ge N$; then the right side is less than $\varepsilon/2+\varepsilon/2=\varepsilon$ for such $m$, using $|x|\le|x|+1$ and claim 5 of Elementary Arithmetic in an Ordered Field.

Claim 5. Choose $N_{0}$ with $|y_{m}-y|<1$ for $m\ge N_{0}$; then $|y_{m}|\le|y_{m}-y|+|y|\le|y|+1$ for such $m$ by claim 1. By Elementary Identities in a Real Inner Product Space §bilinear, claim 5 of Properties of the Absolute Value in an Ordered Field and The Cauchy-Schwarz Inequality in a Real Inner Product Space,

for $m\ge N_{0}$. Given $\varepsilon>0$, choose $N\ge N_{0}$ such that $|x_{m}-x|<\varepsilon/(2(|y|+1))$ and $|y_{m}-y|<\varepsilon/(2(|x|+1))$ for $m\ge N$; then $|\langle x_{m},y_{m}\rangle-\langle x,y\rangle|<\varepsilon$ for such $m$. Taking the constant sequence $y_{m}=z$, which converges to $z$ since $d(z,z)=0<\varepsilon$ for every $\varepsilon>0$ (Convergent Sequence in a Metric Space), gives the particular case. Finally $\bigl||x_{m}|-|x|\bigr|\le|x_{m}-x|$ by claim 2, so $|x_{m}|\to|x|$ with the same $N$ as for $x_{m}\to x$.

Claim 6. Suppose $A$ is bounded in $(E,d)$: by Bounded Subset of a Metric Space there are $x_{0}\in E$ and a real $R_{0}>0$ with $|x_{0}-a|\le R_{0}$ for every $a\in A$. Then $|a|=|x_{0}-(x_{0}-a)|\le|x_{0}|+|x_{0}-a|\le|x_{0}|+R_{0}$ for every $a\in A$ by claim 1, so $R=|x_{0}|+R_{0}$ serves. Conversely suppose $|a|\le R$ for every $a\in A$. Put $R'=|R|+1>0$; then $d(0_{E},a)=|0_{E}-a|=|-a|=|a|\le R\le|R|\le R'$ for every $a\in A$, by claim 3 of Properties of the Absolute Value in an Ordered Field, so $A$ is bounded with the point $0_{E}$ and the radius $R'$.

Claim 7. By claim 2, $d_{\mathbb{R}}(|x|,|y|)=\bigl||x|-|y|\bigr|\le|x-y|=1\cdot d(x,y)$, and $1\ge 0$; so the norm is Lipschitz with constant $1$ in the sense of Lipschitz Map Between Metric Spaces. By Elementary Identities in a Real Inner Product Space §bilinear and The Cauchy-Schwarz Inequality in a Real Inner Product Space, $d_{\mathbb{R}}(\langle x,z\rangle,\langle y,z\rangle)=|\langle x,z\rangle-\langle y,z\rangle|=|\langle x-y,z\rangle|\le|x-y|\,|z|=|z|\,d(x,y)$, and $|z|\ge 0$; so $x\mapsto\langle x,z\rangle$ is Lipschitz with constant $|z|$.