Miguel Florencio

Barrelled subspaces of spaces with subseries decompositions or Boolean rings of projections

The research presented in this paper started by extending a theorem of Swetits [18]about barrelledness of subspaces of metrizable AK-spaces to general AK-spaces of scalar sequences. The extension reads as follows. (1) A subspace λ 0 of a barrelled AK-space λ such that λ 0 ⊃ φ is barrelled if and only if its dual is weak* sequentially complete. If in addition λ 0 is monotone, then it is barrelled if and only if equals the Kothe dual of λ 0 . As an easy consequence of this extension, we obtained the following result of Elstrodt and Roelcke [8, Corollary 3.4]. (2) If λ is a barrelled monotone AK-space, then also its subspace ℒ(λ), consisting of all sequences in λ with zero-density support, is barrelled .

An Abstract Banach-Steinhaus Theorem and Applications to Function Spaces

We give an abstract Banach-Steinhaus theorem for locally convex spaces having suitable algebras of linear projections modelled on a σ-finite measure space. This theorem is applied to deduce barrelledness results for the space L∞ (μ, E) of essentially bounded and μ-measurable functions from a Radon measure space (Ω, σ, μ) into a locally convex space E and also for B (μ, E), the closure of the space of simple functions. Sample: if μ is atomless, then B (μ, E) is barrelled if and only if E is quasi-barrelled and E′(β (E′, E)) has the property (B) of Pietsch.

Some new classes of Banach-Mackey spaces

Some weakenings of property (K) of Antosik for locally convex spaces are introduced: local property (K) and, for spaces with Schauder-type decompositions, two variants of property (K) defined in terms of block- and tail-sequences. It is shown that if a space enjoys any of these new properties, then it is Banach-Mackey. An application to the barrelledness of the spaces of Pettis integrable functions is given, and examples are provided to distinguish between the various K-type properties.

Inductive limits of spaces of vector- valued integrable functions

For an order-continuous Banach function space Λ and a separated inductive limit E:= indnEn, we prove that indn A {En} is a topological subspace of Λ {E}; moreover, both spaces coincide if the inductive limit is hyperstrict. As a consequence, we deduce that if E is an LF-space, then Lp {E} is barrelled for 1 ≤ p ≤ ∞.

COMPLEMENTED COPIES OF c 0 IN THE SPACE OF PETTIS INTEGRABLE FUNCTIONS

Abstract Let X be a Banach space containing a copy of c0, then the space of Pettis integrable functions defined from any perfect atomless measure space to X, contains a complemented copy of c0.

Barrelled function spaces

Abstract Let ( X , σ, μ) be a finite measure space, A a complete, solid lattice of scalar functions defined on X , and E a normed space. We study barrelledness properties of the space A(E) of strongly measurable functions f : X → E such that ‖ f (·)‖ ∈ A.

Barrelledness and bornological conditions on spaces of vector-valued μ-simple functions

Let S(μ, E) be the space of (classes of μ-a.e. equal) simple functions defined on a (non-trivial) measure space with values in a locally convex space E. The following results hold: S(μ,E) is quasi-barrelled (resp. bornological) if and only if E is quasi-barrelled (resp. bornological) and E′(β(E′,E)) has the property (B) of Pietsch; S(μ, E) is barrelled if and only if S(μ,K) is barrelled and E is barrelled and nuclear; S(μ, E) is never ultrabornological; and S(μ, E) is a DF-space if and only if E is a DF-space.

Duals of vector-valued Köthe function spaces

Let Λ be a perfect Kothe function space in the sense of Dieudonne, and Λ × its Kothe-dual. Let E be a normed space. Then the topological dual of the space Λ( E ) of Λ-Bochner integrable functions equals the corresponding Λ × ( E ′) if and only if E ′ has the Radon–Nikodým property. We also give some results concerning barrelledness for spaces of this kind.

Köthe echelon spaces à la Dieudonné

Abstract Let ( g n ) be a sequence of locally integrable functions defined on a Radon measure space. The echelon space associated to ( g n ) was defined by J. Dieudonne as the Kothe-dual of ( g n ), i.e. the space Λ of all locally integrable functions f such that all the integrals ∫ | f · g n | are finite. Denote by Λ x the Kothe-dual of Λ. We prove that Λ(β(Λ,Λ x )) is a Frechet space with dual Λ x . This result gives its correct sense to a wrong affirmation of J. Dieudonne and validates those instances where it has been used. As a tool to prove this result, we study the problem of when the strong dual of a perfect space coincides with its Kothe-dual and give some necessary and sufficient conditions.

Barrelledness of Generalized Sums of Normed Spaces

AbstractLet (Ei)i∈I be a family of normed spaces and λ a space of scalar generalized sequences. The λ-sum of the family (Ei)i∈I of spaces is