Metric space

In mathematics, a metric space is, roughly speaking, a mathematical object which generalizes the notion of a Euclidean space $\mathbb {R} ^{n}$ (a $n$-tuple of real numbers) equipped with the usual Euclidean distance to more general classes of sets such as a set of functions. A metric space consists of two components, a set and a metric on that set. On a metric space, the metric replaces the Euclidean distance as a notion of "distance" between any pair of elements in its associated set (for example, as an abstract distance between two functions if the set consists of a class of functions) and induces a topology on the set called the metric topology.

Metric on a set

Let $X$ be an arbitrary set. A metric $d$ on $X$ is a function $d:X\times X\rightarrow \mathbb {R}$ with the following properties:

1. $d(x_{1},x_{2})\geq 0\quad \forall x_{1},x_{2}\in X$ (non-negativity)
2. $d(x_{2},x_{1})=d(x_{1},x_{2})\quad \forall x_{1},x_{2}\in X$ (symmetry)
3. $d(x_{1},x_{2})\leq d(x_{1},x_{3})+d(x_{3},x_{2})\quad \forall x_{1},x_{2},x_{3}\in X$(triangular inequality)
4. $d(x_{1},x_{2})=0$ if and only if $x_{1}=x_{2}$

Formal definition of metric space

A metric space is an ordered pair $(X,d)$ where $X$ is a set and $d$ is a metric on $X$.

For shorthand, a metric space $(X,d)$ is usually written simply as $X$ once the metric $d$ has been defined or is understood.

References

1. K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980