Natural ultrabornological, non-complete, normed function spaces (Q1312346)

Using a method with a family of projections \((P_ \lambda)_{\lambda\in[a,b]}\) it is proved that certain subspaces of Fréchet spaces are ultrabornological. It turns out that many elementary spaces of classical analysis, which are non-complete subspaces of the space of continuous functions on \([a,b]\) with values in a Banach space, are ultrabornological. In particular, the space of Denjoy-Perron- Kurzweil-Henstock-integrable functions, under the Alexiewicz norm, is ultrabornological. These new examples of non-complete, normed ultrabornological spaces seem to be the simplest ones known so far.