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A class of weakly compact sets in Lebesgue–Bochner spaces
| Content Provider | Semantic Scholar |
|---|---|
| Author | Rodríguez, José Ángel |
| Copyright Year | 2016 |
| Abstract | Abstract Let X be a Banach space and μ a probability measure. A set K ⊆ L 1 ( μ , X ) is said to be a δ S -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 ( δ S μ ) if every relatively weakly compact subset of L 1 ( μ , X ) is a δ S -set. In this paper we study δ S -sets and Banach spaces having property ( δ S μ ). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property ( δ S μ ) 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 ( δ S μ ) when μ is the Lebesgue measure on [ 0 , 1 ] . |
| Starting Page | 16 |
| Ending Page | 28 |
| Page Count | 13 |
| File Format | PDF HTM / HTML |
| DOI | 10.1016/j.topol.2017.02.075 |
| Volume Number | 222 |
| Alternate Webpage(s) | https://webs.um.es/joserr/miwiki/lib/exe/fetch.php?cache=cache&id=papers&media=l1x_deltas_final.pdf |
| Alternate Webpage(s) | https://arxiv.org/pdf/1611.07199v1.pdf |
| Alternate Webpage(s) | https://doi.org/10.1016/j.topol.2017.02.075 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
