proof of Ascoli-Arzelà theorem
Given we aim at finding a -net in i.e. a finite set of points such that
(see the definition of totally bounded). Let be given with respect to in the definition of equi-continuity (see uniformly equicontinuous) of . Let be a -lattice in and be a -lattice in . Let now be the set of functions from to and define by
Since is a finite set, is finite too: say . Then define , where is a function in such that for all (the existence of such a function is guaranteed by the definition of ).
We now will prove that is a -lattice in . Given choose such that for all it holds (this is possible as for all there exists with ). We conclude that and hence for some . Notice also that for all we have .
Given any we know that there exists such that . So, by equicontinuity of ,
|Title||proof of Ascoli-Arzelà theorem|
|Date of creation||2013-03-22 13:16:19|
|Last modified on||2013-03-22 13:16:19|
|Last modified by||paolini (1187)|