Some metric spaces enlarge familiar sets by adjoining ideal points. The extended real line includes $\infty$ and $-\infty$, while the extended complex plane includes a point at infinity. Ordinary formulas such as $|x-y|$ do not directly handle infinity, so bounded transforms and geometric constructions are used to define meaningful metrics. :::definition[Metric on the Extended Real Line by a Bounded Transform] Let $X=\mathbb{R}\cup\{\infty\}\cup\{-\infty\}$. Define $f:X\to\mathbb{R}$ by $$\begin{equation*} f(x)= \begin{cases} \dfrac{x}{1+|x|}, & x\in\mathbb{R},\\ 1, & x=\infty,\\ -1, & x=-\infty. \end{cases} \end{equation*}$$ For $x,y\in X$, define $d(x,y)=|f(x)-f(y)|$. Then $d$ is a metric on $X$. ::: :::proof Given that $d(x,y)=|f(x)-f(y)|$. To prove that $d$ is a metric on $X$. The first, third, and fourth metric axioms follow from the usual metric on $\mathbb{R}$. If $d(x,y)=0$, then $f(x)=f(y)$. Since $f$ is one-to-one on $X$, it follows that $x=y$. Conversely, if $x=y$, then $d(x,y)=0$. Hence $d$ is a metric on $X$. $\square$ ::: :::definition[Arctangent Metric on the Extended Real Line] Let $X=\mathbb{R}\cup\{\infty\}\cup\{-\infty\}$. Define $$\begin{equation*} d(x,y)=|\tan^{-1}x-\tan^{-1}y|, \end{equation*}$$ where $$\begin{equation*} \tan^{-1}(\infty)=\frac{\pi}{2},\qquad \tan^{-1}(-\infty)=-\frac{\pi}{2}. \end{equation*}$$ Then $d$ is a metric on $X$. ::: :::definition[Chordal Metric on the Extended Complex Plane] Let $X=\mathbb{C}\cup\{\infty\}$. For $z_1,z_2\in\mathbb{C}$, define $$\begin{equation*} d(z_1,z_2)=\frac{2|z_1-z_2|}{\sqrt{1+|z_1|^2}\sqrt{1+|z_2|^2}}. \end{equation*}$$ For $z\in\mathbb{C}$, define $$\begin{equation*} d(z,\infty)=\frac{2}{\sqrt{1+|z|^2}}, \end{equation*}$$ and let $d(\infty,\infty)=0$. This is the $chordal metric$. ::: :::theorem[Metric Property of the Chordal Metric] Let $X=\mathbb{C}\cup\{\infty\}$ and let $d$ be the chordal metric. Then $(X,d)$ is a metric space. ::: :::proof Given that $d$ is the chordal distance on $\mathbb{C}\cup\{\infty\}$. To prove that $d$ is a metric. The chordal distance is the Euclidean distance between corresponding points on the unit sphere under stereographic projection. Euclidean distance satisfies the metric axioms, and the correspondence is one-to-one after adjoining $\infty$. Therefore the chordal distance satisfies the metric axioms. Hence $(X,d)$ is a metric space. $\square$ ::: :::solved-problem Let $X=\mathbb{C}$ and define $$\begin{equation*} d(z_1,z_2)= \begin{cases} 0, & z_1=z_2,\\ |z_1|+|z_2|, & z_1\neq z_2. \end{cases} \end{equation*}$$ Show that $(X,d)$ is complete. ::: :::solution Let $\{z_n\}$ be a Cauchy sequence. If it is eventually constant, it converges. Otherwise the Cauchy condition forces the moduli of late distinct terms to be small, because $d(z_n,z_m)=|z_n|+|z_m|$ when $z_n\neq z_m$. Hence $|z_n|\to 0$, and then $d(z_n,0)\to 0$. Thus every Cauchy sequence converges in $X$. Hence $(X,d)$ is complete. $\square$ ::: :::exercise [1] State the bounded transform used on the extended real line. [2] Define the arctangent metric on the extended real line. [3] State $d(z,\infty)$ for the chordal metric. ::: :::answer [1] $f(x)=x/(1+|x|)$ for real $x$, $f(\infty)=1$, and $f(-\infty)=-1$. [2] $d(x,y)=|\tan^{-1}x-\tan^{-1}y|$ with endpoint values $\pi/2$ and $-\pi/2$. [3] $d(z,\infty)=2/\sqrt{1+|z|^2}$. ::: :::call-to-action[Continue to Sequence Spaces] subtitle: Next, study metrics whose points are scalar sequences. button: Next Lesson | /functional-analysis/metric-spaces/introduction-to-metric-spaces/metrics-on-sequence-spaces button-ghost: Previous Lesson | /functional-analysis/metric-spaces/introduction-to-metric-spaces/pseudometrics-and-quotient-metrics :::