of a set of functions
An idea closely connected with the concept of compactness of a set of continuous functions. Let and be compact metric spaces and let be the set of continuous mappings of into . A set is called equicontinuous if for any there is a such that implies for all , . Equicontinuity of is equivalent to the relative compactness of in , equipped with the metric
this is the content of the Arzelà–Ascoli theorem. The idea of equicontinuity can be transferred to uniform spaces.
|||A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)|
|||R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)|
|[a1]||J.A. Dieudonné, "Foundations of modern analysis" , Acad. Press (1961) (Translated from French)|
Equicontinuity. E.M. Semenov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Equicontinuity&oldid=17759