Equicontinuity

of a set of functions

An idea closely connected with the concept of compactness of a set of continuous functions. Let $X$ and $Y$ be compact metric spaces and let $C(X,Y)$ be the set of continuous mappings of $X$ into $Y$. A set $D\subset C(X,Y)$ is called equicontinuous if for any $\epsilon>0$ there is a $\delta>0$ such that $\rho_X(x_1,x_2)\leq\delta$ implies $\rho_Y(f(x_1),f(x_2))\leq\epsilon$ for all $x_1,x_2\in X$, $f\in D$. Equicontinuity of $D$ is equivalent to the relative compactness of $D$ in $C(X,Y)$, equipped with the metric

$$\rho(f,g)=\max_{x\in X}\rho_Y(f(x),g(x));$$

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)

References

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