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
