A more abstract (and more general) version is the following.
Let and be totally bounded metric spaces and let be an uniformly equicontinuous family of continuous mappings from to . Then is totally bounded (with respect to the uniform convergence metric induced by ).
Notice that the first version is a consequence of the second. Recall, in fact, that a subset of a complete metric space is totally bounded if and only if its closure is compact (or sequentially compact). Hence is totally bounded and all the functions have image in a totally bounded set. Being totally bounded means that is sequentially compact and hence has a convergent subsequence.
|Date of creation||2013-03-22 12:41:00|
|Last modified on||2013-03-22 12:41:00|
|Last modified by||paolini (1187)|