Compacidade relativa de conjuntos no espaço de funções continuas com peso

Abstract

The Ascoli-Arzelà theorems o er us the criteria (the necessary and su cient ) of relative compactness of sets in the metric space of continuous functions de ned in compact metric spaces, and also, in a more practical case , of sets in the Banach space C[a, b] of real continuous functions de ned in the segment [a; b] , endowed with the max -norm. A set in C[a, b] is relatively compact if and only if it is equicontinuous and equilimited (the equilimitation condition can also be replaced by the weaker condition limiting the set at a point). In this work, we generalize the classical Ascoli-Arzelà criteria and their consequences to the space C α [a, b] of continuous functions with weight α . In this way, we establish new criteria and some su cient conditions for relative compactness of sets in the space C α [a, b] .