ArticleOriginal scientific text

Title

Duality and some topological properties of vector-valued function spaces

Authors

Abstract

Let E be an ideal of L0 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 L0(X) of μ-equivalence classes of all strongly Σ-measurable functions f:ΩX and consisting of all those fL0(X), for which the scalar function f~=f()X 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.

Keywords

vector-valued function spaces, locally solid topologies, KB-spaces, Levy topologies, Lebesgue topologies, order dual, order continuous dual, perfectness
Main language of publication
English
Published
2008
Published online
2017-12-19
Exact and natural sciences