On a Barrelled Space of Class ... and Measure Theory.
In this paper it is proved that if and are two sequences of infinite-dimensional Banach spaces then is not -complete. If and are also reflexive spaces there is on a separated locally convex topology , coarser than the initial one, such that is a bornological barrelled space which is not an inductive limit of Baire spaces. It is given also another results on -completeness and bornological spaces.
Let be a -algebra on a set . If belongs to let be the characteristic function of . Let be the linear space generated by endowed with the topology of the uniform convergence. It is proved in this paper that if is an increasing sequence of subspaces of covering it, there is a positive integer such that is a dense barrelled subspace of , and some new results in measure theory are deduced from this fact.
If is the topological product of a non-countable family of barrelled spaces of non-nulle dimension, there exists an infinite number of non-bornological barrelled subspaces of . The same result is obtained replacing “barrelled” by “quasi-barrelled”.