# Equicontinuity

From Encyclopedia of Mathematics

*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) |

#### Comments

#### References

[a1] | J.A. Dieudonné, "Foundations of modern analysis" , Acad. Press (1961) (Translated from French) |

**How to Cite This Entry:**

Equicontinuity. E.M. Semenov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Equicontinuity&oldid=17759

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098