Editing Metric space (section)

== History ==
[[Arthur Cayley]], in his article "On Distance", extended metric concepts beyond Euclidean geometry into domains bounded by a conic in a projective space. His [[distance]] was given by logarithm of a [[cross  ratio]].  Any projectivity leaving the conic stable also leaves the cross ratio constant, so isometries are implicit. This method provides models for [[elliptic geometry]] and [[hyperbolic geometry]], and [[Felix Klein]], in several publications, established the field of [[non-euclidean geometry]] through the use of the [[Cayley-Klein metric]].

The idea of an abstract space with metric properties was addressed in 1906 by [[René Maurice Fréchet]]<ref>{{cite journal |last1=Fréchet |first1=M. |title=Sur quelques points du calcul fonctionnel |journal=Rendiconti del Circolo Matematico di Palermo |date=December 1906 |volume=22 |issue=1 |pages=1–72 |doi=10.1007/BF03018603|s2cid=123251660 |url=https://zenodo.org/record/1428464 }}</ref> and the term ''metric space'' was coined by [[Felix Hausdorff]] in 1914.<ref>F. Hausdorff (1914) ''Grundzuge der Mengenlehre''</ref><ref>{{cite journal |last1=Blumberg |first1=Henry |title=Hausdorff's Grundzüge der Mengenlehre |journal=Bulletin of the American Mathematical Society |date=1927 |volume=6 |pages=778–781 |doi=10.1090/S0002-9904-1920-03378-1 |doi-access=free}}</ref><ref>Mohamed A. Khamsi & William A. Kirk (2001) ''Introduction to Metric Spaces and Fixed Point Theory'', page 14, [[John Wiley & Sons]]</ref>

Fréchet's work laid the foundation for understanding [[Convergence of random variables|convergence]], [[Continuity equation|continuity]], and other key concepts in non-geometric spaces. This allowed mathematicians to study functions and sequences in a broader and more flexible way. This was important for the growing field of functional analysis. Mathematicians like Hausdorff and [[Stefan Banach]] further refined and expanded the framework of metric spaces. Hausdorff introduced [[topological space]]s as a generalization of metric spaces. Banach's work in [[functional analysis]] heavily relied on the metric structure. Over time, metric spaces became a central part of [[History of mathematics|modern mathematics]]. They have influenced various fields including [[topology]], [[geometry]], and [[applied mathematics]]. Metric spaces continue to play a crucial role in the study of abstract mathematical concepts.