proof of Ascoli-Arzelà theorem

Given ϵ>0 we aim at finding a 4ϵ-net in F i.e. a finite setMathworldPlanetmath of points Fϵ such that


(see the definition of totally boundedPlanetmathPlanetmath). Let δ>0 be given with respect to ϵ in the definition of equi-continuity (see uniformly equicontinuous) of F. Let Xδ be a δ-latticeMathworldPlanetmath in X and Yϵ be a ϵ-lattice in Y. Let now YϵXδ be the set of functions from Xδ to Yϵ and define GϵYϵXδ by


Since YϵXδ is a finite set, Gϵ is finite too: say Gϵ={g1,,gN}. Then define FϵF, Fϵ={f1,,fN} where fk:XY is a function in F such that d(fk(x),gk(x))<ϵ for all xXδ (the existence of such a function is guaranteed by the definition of Gϵ).

We now will prove that Fϵ is a 4ϵ-lattice in F. Given fF choose gYϵXδ such that for all xXδ it holds d(f(x),g(x))<ϵ (this is possible as for all xXδ there exists yYϵ with d(f(x),y)<ϵ). We conclude that gGϵ and hence g=gk for some k{1,,N}. Notice also that for all xXδ we have d(f(x),fk(x))d(f(x),gk(x))+d(gk(x),fk(x))<2ϵ.

Given any xX we know that there exists xδXδ such that d(x,xδ)<δ. So, by equicontinuity of F,

Title proof of Ascoli-Arzelà theorem
Canonical name ProofOfAscoliArzelaTheorem
Date of creation 2013-03-22 13:16:19
Last modified on 2013-03-22 13:16:19
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 12
Author paolini (1187)
Entry type Proof
Classification msc 46E15