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right-bracket comma upper X right-parenthesis"/> is

. Therefore,

is a Banach space when equipped with the usual supremum norm

See also [127, Theorem I.3.6].

denotes the Banach space of bounded functions from

, equipped with the supremum norm, then the inclusion

follows from Theorem 1.4, items (i) and (ii), taking the limit of step functions which are constant on each subinterval of continuity.

Recently, D. Franková established a fourth assertion equivalent to those assertions of Theorem 1.4 in the case where

. See [97, Theorem 2.3]. One can note, however, that such result also holds for any open set

. This is the content of the next lemma.

Lemma 1.5: Let and be a function. Then the assertions of Theorem 1.4 are also equivalent to the following assertion:

1 (iv) for every , there is a division such that

Proof. Note that condition (iii) from Theorem 1.4 implies condition (iv). Now, assume that condition (iv) holds. Given

, there is a division

such that

, for all

and

According to [97, Theorem 2.3], take

and consider a step function

given by

Hence,

The next result, borrowed from [7, Lemma 2.3], specifies the supremum of a function

Proposition 1.6: Let . Then where either , for some , or , for some , or for some .

Proof. Let

. Since

. By the definition of the supremum, for all

, one can choose

such that

which implies

Since

, there exists a subsequence

such that

Generalized Ordinary Differential Equations in Abstract Spaces and Applications. Группа авторов