After the metric axioms, the first library of examples comes from Euclidean spaces. The Euclidean metric measures straight-line distance, the finite-dimensional $p$-metric generalizes it, the maximum metric measures the largest coordinate difference, and the discrete metric treats distinct points as equally separated. Each example reinforces the same four axioms in a different setting. :::definition[Euclidean Metric] Let $X=\mathbb{R}^n$. For $x=(x_1,x_2,\ldots,x_n)$ and $y=(y_1,y_2,\ldots,y_n)$, define $$\begin{equation*} d(x,y)=\left(\sum_{i=1}^{n}(x_i-y_i)^2\right)^{1/2}. \end{equation*}$$ Then $d$ is called the $Euclidean metric$ on $\mathbb{R}^n$. ::: :::definition[Finite-Dimensional p-Metric] Let $X=\mathbb{R}^n$ and let $p\geq 1$. Define $$\begin{equation*} d_p(x,y)=\left(\sum_{i=1}^{n}|x_i-y_i|^p\right)^{1/p}. \end{equation*}$$ Then $d_p$ is called the $finite-dimensional p-metric$ on $\mathbb{R}^n$. ::: :::theorem[Metric Property of the Finite-Dimensional p-Metric] Let $X=\mathbb{R}^n$ and let $p\geq 1$. Then $d_p$ defines a metric on $\mathbb{R}^n$. ::: :::proof Given that $X=\mathbb{R}^n$, $p\geq 1$, and $d_p$ is defined by the stated formula. To prove that $d_p$ is a metric. Non-negativity, identity of indiscernibles, and symmetry follow from the absolute value. Let $x,y,z\in\mathbb{R}^n$. By Minkowski's inequality, $$\begin{align*} \left(\sum_{i=1}^{n}|x_i-y_i|^p\right)^{1/p} &\leq \left(\sum_{i=1}^{n}|x_i-z_i|^p\right)^{1/p}+\left(\sum_{i=1}^{n}|z_i-y_i|^p\right)^{1/p}. \end{align*}$$ Therefore $d_p(x,y)\leq d_p(x,z)+d_p(z,y)$. Hence $d_p$ is a metric on $\mathbb{R}^n$. $\square$ ::: :::definition[Maximum Metric] Let $X=\mathbb{R}^n$. Define $$\begin{equation*} d_\infty(x,y)=\max_{1\leq i\leq n}|x_i-y_i|. \end{equation*}$$ Then $d_\infty$ is called the $maximum metric$ on $\mathbb{R}^n$. ::: :::theorem[Metric Property of the Maximum Metric] Let $X=\mathbb{R}^n$. Then $d_\infty$ defines a metric on $\mathbb{R}^n$. ::: :::proof Given that $d_\infty(x,y)=\max_{1\leq i\leq n}|x_i-y_i|$. To prove that $d_\infty$ is a metric. The first three axioms follow from the absolute value. For each $i$, $$\begin{equation*} |x_i-y_i|\leq |x_i-z_i|+|z_i-y_i|\leq d_\infty(x,z)+d_\infty(z,y). \end{equation*}$$ Taking maximum over $i$ gives $d_\infty(x,y)\leq d_\infty(x,z)+d_\infty(z,y)$. Hence $d_\infty$ is a metric. $\square$ ::: :::definition[Discrete Metric] Let $X$ be a nonempty set. Define $$\begin{equation*} d(x,y)= \begin{cases} 0, & x=y,\\ 1, & x\neq y. \end{cases} \end{equation*}$$ Then $d$ is called the $discrete metric$ on $X$. ::: :::example For $0<p<1$, the formula $$\begin{equation*} d_p(z,w)=\left(\sum_{k=1}^{n}|z_k-w_k|^p\right)^{1/p} \end{equation*}$$ need not define a metric on $\mathbb{C}^n$. Taking $z=(1,1,0,\ldots,0)$, $u=(0,1,0,\ldots,0)$, and $w=(0,0,0,\ldots,0)$ gives $d_p(z,w)=2^{1/p}$ and $d_p(z,u)+d_p(u,w)=2$. Since $2^{1/p}>2$, the triangle inequality fails. ::: :::solved-problem Show that $d(x,y)=\sqrt{|x-y|}$ defines a metric on $\mathbb{R}$. ::: :::solution Let $d(x,y)=\sqrt{|x-y|}$. The first three metric axioms follow from the usual absolute value. For $x,y,z\in\mathbb{R}$, $$\begin{equation*} |x-y|\leq |x-z|+|z-y|. \end{equation*}$$ Since the square-root function is increasing and $\sqrt{a+b}\leq\sqrt a+\sqrt b$ for $a,b\geq 0$, $$\begin{equation*} \sqrt{|x-y|}\leq\sqrt{|x-z|}+\sqrt{|z-y|}. \end{equation*}$$ Thus $d(x,y)\leq d(x,z)+d(z,y)$. Hence $d$ is a metric on $\mathbb{R}$. $\square$ ::: :::exercise [1] Write the Euclidean metric on $\mathbb{R}^3$. [2] Write the maximum metric on $\mathbb{R}^n$. [3] Explain why the discrete metric satisfies the triangle inequality. ::: :::answer [1] $d(x,y)=((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^{1/2}$. [2] $d_\infty(x,y)=\max_{1\leq i\leq n}|x_i-y_i|$. [3] If $x\neq y$, then at least one of $x\neq z$ or $z\neq y$ holds, so the right side is at least $1$. ::: :::call-to-action[Continue to Bounded Metrics] subtitle: Next, transform existing metrics into bounded metrics without losing the metric axioms. button: Next Lesson | /functional-analysis/metric-spaces/introduction-to-metric-spaces/bounded-metrics-from-existing-metrics button-ghost: Previous Lesson | /functional-analysis/metric-spaces/introduction-to-metric-spaces/metric-axioms-and-subspace-metrics :::