On strongly WCG Banach spaces

Schlüchtermann, Georg; Wheeler, Robert F.

doi:10.1007/bf01159786

Cited by 43 publications

(53 citation statements)

References 17 publications

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“…The general question is obviously equivalent to: if µ is a σ -finite measure, does L 1 (µ)-by-reflexive imply WCG? This is interesting, because both reflexive spaces and such L 1 (µ) spaces are strongly WCG in the sense of [50]. It seems to be unknown whether being strongly WCG is a 3-space property.…”

Section: Duality and New Stability Resultsmentioning

confidence: 99%

Twisted properties of banach spaces

Castillo¹,

Gonzáles²,

Plichko³

et al. 2001

MATH. SCAND.

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If P, Q are two linear topological properties, say that a Banach space X has the property P-by-Q (or is a P-by-Q space) if X has a subspace Y with property P such that the corresponding quotient X/Y has property Q. The choices P, Q ∈ {separable, reflexive} lead naturally to some new results and new proofs of old results concerning weakly compactly generated Banach spaces. For example, every extension of a subspace of L 1 (0, 1) by a WCG space is WCG. They also give a simple new example of a Banach space property which is not a 3-space property but whose dual is a 3-space property.

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Section: Duality and New Stability Resultsmentioning

confidence: 99%

Twisted properties of banach spaces

Castillo¹,

Gonzáles²,

Plichko³

et al. 2001

MATH. SCAND.

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“…Following [41], a Banach space Y is called strongly weakly compactly generated (shortly SWCG) if there is a weakly compact set K ⊂ Y such that for every weakly compact set L ⊂ Y and every ε > 0 there is n ∈ N such that L ⊂ nK + εB Y (in this case, we say that K strongly generates Y ). Every SWCG space is both WCG and WSC [41] (cf.…”

Section: When Is L 1 (M) a Swcg Space?mentioning

confidence: 99%

Compactness in L¹of a vector measure

et al. 2014

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Abstract. We study compactness and related topological properties in the space L 1 (m) of a Banach space valued measure m when the natural topologies associated to the convergence of the vector valued integrals are considered. The resulting topological spaces are shown to be angelic and the relationship of compactness and equi-integrability is explored. A natural norming subset of the dual unit ball of L 1 (m) appears in our discussion and we study when it is a boundary. The (almost) complete continuity of the integration operator is analyzed in relation with the positive Schur property of L 1 (m). The strong weakly compact generation of L 1 (m) is discussed as well.

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“…X is strongly WCG (SWCG) if there is a sequence (K,,) of weakly compact subsets of X such that for each weakly compact subset H of X and each >0, there is an hen such that H C K, + B [9]. As noted in [9], restricting H to norm compact sets in the above definition gives a definition of WCG that is equivalent to the one above.…”

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confidence: 99%

“…We will see that each of these properties may be expressed as a property of the family of norm compact subsets of x. Our goal here is to introduce the weakly compact version of the WCD property, the SWCD property, and to examine its relationship to the strong WCG property of Schltichtermann and Wheeler [9].…”

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confidence: 99%