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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.

Duality theory of vector-valued function spaces III

Nowak M.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

1998

[Z] 38

101-108

The paper is a continuation of [5] and [6], where the duality theory of Banach-space valued function spaces E(X) is developed. We deal with order continuous linear functionals and singular linear functionals on E(X). We show that if (E,||ˇ||E) is a Banach function space then the singular dual E(X)~s is equal to the topological dual of E(X) provided with the Minkowski functional P of the set W = {f