we call any element of Definition 1.25: Given a function
and the variation of
If
It is not difficult to prove that
Moreover (see [127, Corollary I.3.4]),
The space
for
Remark 1.26: Consider a BT
1 Let , or and . By the definition of the norm in , we haveThus, when we consider the BT , we write and instead of and , respectively.
2 Let , or and . By the Hahn–Banach Theorem, we haveand, hence,
Definition 1.27: Given
Such spaces are complete when endowed, respectively, with the norm given by the variation
where
