Metric Space

A metric space is a set s with a global distance function the metric g that for every two points x y in s gives the distance between them as a nonnegative real number g x y. Metric space is a concept from mathematics. Given a set of elments there is a function which takes two arguments which are elements of the set and returns a real number which is usually called distance d between those elements.

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D x y 0 if x y and d x y 1 otherwise.

Metric space. The discrete metric on the x is given by. The positive real numbers with distance function d x y log y x displaystyle d x y vert log. G x y 0 iff x y 2. Then this does define a metric in which no distinct pair of points are close.

A metric space must also satisfy 1. A metric space is a set equipped with a distance function which provides a measure of distance between any two points in the set. Metrics on spaces of functions. The function needs to be symmetric.

Any normed vector space is a. One represents a metric space s s with metric. A metric space is said to be complete if every sequence of points in which the terms are eventually pairwise arbitrarily close to each other a so called cauchy sequence converges to a point in the metric space. Examples of metric spaces the real numbers with the distance function d x y y x displaystyle d x y vert y x vert given by the.

D x y d y x. The distance function known as a metric must satisfy a collection of axioms. The fact that every pair is spread out is why this metric is called discrete.

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