taking values in a general Banach space with norm .
Unlike the finite dimensional case, when we consider functions taking values in , the relatively compactness of a set does not come out as a consequence of the equiregulatedness of the set and the boundedness of the set , for each . In the following lines, we present an example, borrowed from [177] which illustrates this fact.
Example 1.17: Let be bounded. Suppose not relatively compact in . Thus, given an arbitrary , there is a sequence of functions in for which
for all and for some constant . Hence, the set
is bounded, once is bounded. Moreover, is equiregulated and is relatively compact in . On the other hand, is not relatively compact in .
At this point, it is important to say that, in order to guarantee that a set is relatively compact, one needs an additional condition. It is clear that, if one assumes that, for each , the set is relatively compact in , then becomes relatively compact in . This is precisely what the next result says, and we refer to it as the Arzelà–Ascoli theorem for Banach space‐valued regulated functions. Such important result can be found in [97] and [177] as well.
Theorem 1.18:Suppose is equiregulated and, for every , is relatively compact in . Then, is relatively compact in .
Proof. Take a sequence of functions . The set is equiregulated by hypothesis. Then, for every , there exists a division fulfilling
for every and , .
On the other hand, the sets and are relatively compact in for every , where . Thus, there is a subsequence of indexes
taking values in a general Banach space 