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

norm (see [5]) given by

      is noncomplete (see [24], for instance). The same applies to Banach space-valued functions. Let us denote by the space of all equivalence classes of functions , equipped with the Alexiewicz norm The space is noncomplete (see [129]). However, is ultrabornological (see [105]) and, therefore, barrelled. In particular, good functional analytic properties hold, such as the Banach–Steinhaus theorem and the Uniform Boundedness Principle (see, for instance, [142]). The same applies to the space of equivalence classes of functions , endowed with the Alexiewicz norm .

      The next example, borrowed from [73], exhibits a Cauchy sequence of Henstock integrable functions which is not convergent in the usual Alexiewicz norm, .

      Example 1.84: Consider functions , defined by , where , whenever , , and otherwise.

      Hence,

      Then,

      and, hence,

By induction, one can show that

      for every . Then,

      which goes to zero for sufficiently large , with . Thus, is a -Cauchy sequence. On the other hand,

      for every . Hence, there is no function