A Wide Class of Ultrabornological Spaces of Measurable Functions

Abstract

Our main result states that a bornological locally convex space having a suitable Boolean algebra of projections is ultrabornological. This general theorem, whose proof is a variation of the sliding-hump techniques used in [Dı́az et al., Arch. Math. (Basel)60 (1993), 73-78; Dı́az et al., Resultate Math.23 (1993), 242-250; Drewnowski el al., Proc. Amer. Math. Sec.114 (1992), 687-694; Drewnowski et al., Atti. Sem. Mat. Fis. Univ. Modena41 (1993), 317-329], is applied to prove that some non-complete normed spaces such as the spaces of Dunford, Pettis, or McShane integrable functions, as well as other interesting spaces of weakly or strongly measurable functions, are ultrabornological. We also give applications to vector-valued sequence spaces; in particular, we prove that ℓp{X} (1 ≤ p < ∞) is an ultrabornological DF-space when X is.