Metric axioms give the formal language in which distance is studied without depending on coordinates, algebra, or geometry. A metric space begins with a nonempty set and a rule that measures distance between two points. The rule must separate distinct points, be symmetric, and obey the triangle inequality. This lesson introduces metric spaces, distance functions, and the subspace metric inherited by every nonempty subset of a metric space. :::definition[Metric Space] Let $X$ be a nonempty set. A function $d:X\times X\to\mathbb{R}$ is called a $metric$ on $X$ if the following conditions hold for all $x,y,z\in X$: (i) $d(x,y)\geq 0$. (ii) $d(x,y)=0$ if and only if $x=y$. (iii) $d(x,y)=d(y,x)$. (iv) $d(x,y)\leq d(x,z)+d(z,y)$. The pair $(X,d)$ is called a $metric space$. ::: :::definition[Distance Function] Let $X$ be a nonempty set. A metric $d$ on $X$ is also called a $distance function$ on $X$. The value $d(x,y)$ represents the distance between the points $x$ and $y$. ::: :::definition[Subspace Metric] Let $(X,d)$ be a metric space and let $Y$ be a nonempty subset of $X$. Define $d_Y:Y\times Y\to\mathbb{R}$ by $$\begin{equation*} d_Y(y_1,y_2)=d(y_1,y_2),\qquad y_1,y_2\in Y. \end{equation*}$$ Then $d_Y$ is called the $metric induced$ by $d$ on $Y$. ::: :::example Let $X=\mathbb{R}$ and define $d(x,y)=|x-y|$ for $x,y\in\mathbb{R}$. Then $d$ is a metric on $\mathbb{R}$. ::: :::proof Given that $X=\mathbb{R}$ and $d(x,y)=|x-y|$. To prove that $d$ is a metric on $\mathbb{R}$. For all $x,y\in\mathbb{R}$, $$\begin{equation*} d(x,y)=|x-y|\geq 0. \end{equation*}$$ Also, $$\begin{align*} d(x,y)=0 &\iff |x-y|=0\\ &\iff x-y=0\\ &\iff x=y. \end{align*}$$ Moreover, $$\begin{equation*} d(x,y)=|x-y|=|y-x|=d(y,x). \end{equation*}$$ For $x,y,z\in\mathbb{R}$, $$\begin{align*} d(x,y)&=|x-y|\\ &=|(x-z)+(z-y)|\\ &\leq |x-z|+|z-y|\\ &=d(x,z)+d(z,y). \end{align*}$$ Hence $d$ is a metric on $\mathbb{R}$. $\square$ ::: :::theorem[Weaker Triangle Condition] Let $X$ be a nonempty set and let $d:X\times X\to\mathbb{R}$ satisfy non-negativity, identity of indiscernibles, and symmetry. Suppose that $$\begin{equation*} d(x,z)\leq d(x,y)+d(y,z) \end{equation*}$$ whenever $x,y,z$ are distinct points of $X$. Then $d$ satisfies the triangle inequality for all $x,y,z\in X$. ::: :::proof Given that the triangle inequality holds whenever $x,y,z$ are distinct. To prove that it holds for all $x,y,z\in X$. If $x,y,z$ are distinct, the assertion is the hypothesis. If $x=y$, then $d(x,z)=d(y,z)=d(x,y)+d(y,z)$ because $d(x,y)=0$. If $y=z$, then $d(x,z)=d(x,y)=d(x,y)+d(y,z)$ because $d(y,z)=0$. If $x=z$, then $d(x,z)=0\leq d(x,y)+d(y,z)$ by non-negativity. Hence the triangle inequality holds for all $x,y,z\in X$. $\square$ ::: :::exercise [1] State the four metric axioms. [2] Define the subspace metric induced on a nonempty subset. [3] Verify the triangle inequality for $d(x,y)=|x-y|$ on $\mathbb{R}$. ::: :::answer [1] Non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. [2] If $Y\subseteq X$ is nonempty, then $d_Y(y_1,y_2)=d(y_1,y_2)$ for $y_1,y_2\in Y$. [3] Use $x-y=(x-z)+(z-y)$ and the usual triangle inequality for absolute value. ::: :::faq[Frequently Asked Questions] Q: Does every formula for distance define a metric? A: No. It must satisfy all four metric axioms. Q: Which axiom usually requires the most work? A: The triangle inequality usually requires the main verification. Q: Does every nonempty subset of a metric space inherit a metric? A: Yes. It inherits the restriction of the original metric. ::: :::call-to-action[Continue to Standard Metrics] subtitle: Practise the metric axioms by studying Euclidean, maximum, finite-dimensional p, and discrete metrics. button: Next Lesson | /functional-analysis/metric-spaces/introduction-to-metric-spaces/standard-metrics-on-euclidean-spaces button-ghost: Section Overview | /functional-analysis/metric-spaces/introduction-to-metric-spaces :::