On integration in banach spaces and total sets

Abstract

Let X be a Banach space and Γ ⊆ X * a total linear subspace. We study the concept of Γ-integrability for X-valued functions f defined on a complete probability space, i.e. an analogue of Pettis integrability by dealing only with the compositions x * , f for x * ∈ Γ. We show that Γ-integrability and Pettis integrability are equivalent whenever X has Plichko's property (D ′ ) (meaning that every w * -sequentially closed subspace of X * is w * -closed). This property is enjoyed by many Banach spaces including all spaces with w * -angelic dual as well as all spaces which are w * -sequentially dense in their bidual. A particular case of special interest arises when considering Γ = T * (Y * ) for some injective operator T : X → Y . Within this framework, we show that if T : X → Y is a semi-embedding, X has property (D ′ ) and Y has the Radon-Nikodým property, then X has the weak Radon-Nikodým property. This extends earlier results by Delbaen (for separable X) and Diestel and Uhl (for weakly K-analytic X).2010 Mathematics Subject Classification. 46B22, 46G10.

Completeness in the Mackey topology by norming subspaces

“…This research line was motivated initially by Kunze's paper [11] on vector integration (cf. [1,15]). Bonet and Cascales [3] exploited some results of [7] to prove that if X contains a subspace isomorphic to ℓ 1 (c), then there is a norming and norm-closed subspace Y ⊂ X * for which (X, µ(X, Y )) is not complete.…”