# Equicontinuity

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.

#### References

 [1] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) [2] R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)