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Topological properties of the complex vector lattice of bounded finitely additive mesures

Nowak M.

Commentationes Mathematicae

2013

Vol. 53, No. 2

271--282

Let Σ be a σ-algebra of subsets of a non-empty set Ω. Let ba(Σ) be the complex vector lattice of bounded finitely additive measures μ:Σ→C. We study locally solid topologies on ba(Σ). We develop the duality theory of ba(Σ), provided with locally convex-solid topologies.

Duality and some topological properties of vector-valued function spaces

Feledziak K.

Commentationes Mathematicae

2008

Vol. 48, [Z] 1

23-43

Let E be an ideal of L^0 over σ-finite measure space (Ω,Σ μ) and let (X, || X) be a real Banach space. Let E(X) be a subspace of the space L^0(X) of μ-equivalence classes of all strongly Σ-measurable functions f : Ω → X and consisting of all those f ε L^0(X), for which the scalar function [...] belongs to E. Let E be equipped with a Hausdorff locally convex-solid topology ξ and let ξ stand for the topology on E(X) associated with ξ. We examine the relationship between the properties of the space (E(X), ξ) and the properties of both the spaces (E, ξ) and (X, ||X). In particular, it is proved that E(X) (embedded in a natural way) is an order closed ideal of its bidual iff E is an order closed ideal of its bidual and X is reflexive. As an application, we obtain that E(X) is perfect iff E is perfect and X is reflexive.

Uniformly ž-Continuous Topologies on Orlicz-Bochner Spaces

Feledziak K.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2006

Vol. 46, [Z] 1

93-111

We examine the topological properties of Orlicz-Bochner spaces L^[fi](X) (over a -finite measure space [...], where ' is an Orlicz function (not necessarily convex) and X is a real Banach space. We continue the study of some class of locally convex topologies on L^[fi](X), called uniformly ž-continuous topologies. In particular, the generalized mixed topology [...] (in the sense of Turpin) is considered.

Weak sequential completeness and compactness in topological function spaces

Nowak M.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2004

Vol. 44, nr spec.

155-165

Let (E, r) be a Hausdorff locally convex-solid function space (over a cr-finite measure space) and let E* stand for its topological dual. It is proved that the space (E, r) is weakly sequentially complete if and only if r is a c-Lebesgue and cr-Levy topology. In particular, a characterization of weak sequential completeness of Or-licz spaces L* in terms of Orlicz functions is given. Moreover, it is proved that the Eberlein-Smulian type theorem remains valid for a locally convex space (E, o~(E, E*)). A characterization of conditional and relative weak compactness in (E, r) is obtained.