Lebesgue–Bochner spaces, decomposable sets and strong weakly compact generation

2015

Monatsh Math

Self Cite

We study strongly Asplund generated (SAG) and strongly conditionally weakly compactly generated (SCWCG) Banach spaces. These spaces are defined like the strongly weakly compactly generated (SWCG) Banach spaces of Schlüchtermann and Wheeler, but replacing weakly compact sets by Asplund sets and conditionally weakly compact sets, respectively. We show that every SAG space is SCWCG and that a Banach space is SWCG if and only if it is SAG/SCWCG and weakly sequentially complete. We also prove that the notions of SAG and SCWCG space coincide for Banach lattices. Some related results on Lebesgue-Bochner spaces are also given. We prove that if the norm of the Banach space X is weakly uniformly rotund (WUR) and μ is any probability measure, then L 1 (μ, X ) admits an equivalent norm which is WUR when restricted to any Asplund subspace of L 1 (μ, X ).

Section: Lebesgue-bochner Spacesmentioning

confidence: 99%

Strongly Asplund generated and strongly conditionally weakly compactly generated Banach spaces

Lajara

2015

Monatsh Math

Self Cite

“…The answer is affirmative for reflexive spaces and L 1 (µ) (for any finite measure µ), but the general case remains open. In [80] we gave a positive answer to the problem when restricted to weakly compact decomposable sets (e.g. sets of selectors of multi-functions).…”

Section: Problems In Lebesgue-bochner Spacesmentioning

confidence: 99%

Open problems in Banach spaces and measure theory

Quaestiones Mathematicae

2017

Self Cite

We collect several open questions in Banach spaces, mostly related to measure theoretic aspects of the theory. The problems are divided into five categories: miscellaneous problems in Banach spaces (non-separable L p spaces, compactness in Banach spaces, w * -null sequences in dual spaces), measurability in Banach spaces (Baire and Borel σ-algebras, measurable selectors), vector integration (Riemann, Pettis and McShane integrals), vector measures (range and associated L 1 spaces) and Lebesgue-Bochner spaces (topological and structural properties, scalar convergence).2000 Mathematics Subject Classification. Primary: 28B05, 46G10. Secondary: 28B20, 46B26, 46B50, 46E30, 46E40.

“…Our motivation to study δS-sets comes from several attempts to solve an open problem of Schlüchtermann and Wheeler [23] asking if the property of being strongly weakly compactly generated (SWCG for short) passes from X to L 1 (µ, X), see [16,17,20]. To be more precise we need a definition: Definition 1.3.…”

Section: Introductionmentioning

confidence: 99%

A class of weakly compact sets in Lebesgue–Bochner spaces

Topology and its Applications

2017

Self Cite

Abstract. Let X be a Banach space and µ a probability measure. A set K ⊆ L 1 (µ, X) is said to be a δS-set if it is uniformly integrable and for every δ > 0 there is a weakly compact set W ⊆ X such that µ(f −1 (W )) ≥ 1 − δ for every f ∈ K. This is a sufficient, but in general non necessary, condition for relative weak compactness in L 1 (µ, X). We say that X has property (δSµ) if every relatively weakly compact subset of L 1 (µ, X) is a δS-set. In this paper we study δS-sets and Banach spaces having property (δSµ). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property (δSµ) are provided. Special attention is paid to the relationship with strongly weakly compactly generated (SWCG) spaces. In particular, we show an example of a SWCG (in fact, separable Schur) space failing property (δSµ) when µ is the Lebesgue measure on [0, 1].