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This subsection brings a few results borrowed from [177]. In particular, Lemma 1.13 describes an interesting and useful property of equiregulated converging sequences of Banach space‐valued functions and it is used later in the proof of a version of Arzelà–Ascoli theorem for Banach space‐valued regulated functions.

Lemma 1.13: Let be a sequence of functions from to . If the sequence converges pointwisely to and is equiregulated, then it converges uniformly to .

Proof. By hypothesis, the sequence of functions is equiregulated. Then, Theorem 1.11 yields that, for every , there is a division for which

for every and , .

Take . Because the sequence converges pointwisely to , we have and also , for . Thus, for every , there is such that, whenever , we have and for

Take an arbitrary . Then, either for some , or for some . In the former case, The other case yields

Then, and, therefore, uniformly on .

Lemma 1.14: Let be a sequence in . The following assertions hold:

1 if the sequence of functions converges uniformly to as on , then , for , and , for ;

2 if the sequence of functions converges pointwisely to as on and , for , and , for , where , then the sequence converges uniformly to as .

Proof. We start by proving . By hypothesis, the sequence converges uniformly to . Then, Moore–Osgood theorem (see, e.g., [19]) implies

Therefore, , for . In a similar way, one can show that , for every .

Now, we prove . It suffices to show that is an equiregulated set. Indeed, by Lemma 1.13, converges uniformly to .

Since the function is regulated, its lateral limits exist. Then, for every and every , there is such that

for every .

But the hypotheses say that we can find such that, for , we have

When satisfies , we have, for every ,

On the other hand, when satisfies , we get, for ,

But this yields the fact that is an equiregulated sequence, and the proof is complete.

The next lemma guarantees that, if a sequence of functions is bounded by an equiregulated sequence of functions, then is also equiregulated.

Lemma 1.15: Let be a sequence of functions in . Suppose, for each , the function satisfies

(1.2)

for every , where for each and the sequence is equiregulated. Then, the sequence is equiregulated.

Proof. Let be given. Since the sequence is equiregulated, it follows from Theorem 1.11 that there is a division such that for every and , for . Thus, by (1.2), for every and every interval , with . Finally, Theorem 1.11 ensures the fact that the sequence is equiregulated.

A clear outcome of Lemmas 1.13 and 1.15 follows below:

Corollary 1.16: Let be a sequence of functions from to and suppose the function satisfies condition (1.2) for every and , where and the sequence is equiregulated. If the sequence converges pointwisely to a function , then it also converges uniformly to .