Strongly Asplund generated and strongly conditionally weakly compactly generated Banach spaces

Lajara, Sebastián; Rodríguez, José

doi:10.1007/s00605-015-0804-x

Cited by 4 publications

(7 citation statements)

References 21 publications

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“…Indeed, if E is any Banach space with a normalized 1-unconditional basis and separable dual, and (X m ) m∈N is a sequence of Banach spaces, then the space ( m∈N X m ) E fails property KM w if X m contains isomorphic copies of ℓ 1 for infinitely many m ∈ N (Theorem 3.23). This extends some previous results on property KM and subspaces of SWPG spaces obtained in [26] and [29]. As an application, we show that the Banach space of Batt and Hiermeyer [6, §3] fails property KM w (Corollary 3.28).…”

Section: Introductionsupporting

confidence: 86%

“…The following lemma exploits the argument used to get the weak sequential completeness of SWCG spaces in [41,Theorem 2.5]. A similar idea was used in the proof of the weak sequential completeness of Banach spaces having property KM (see [26,Theorem 2.20]) and also in [29,Lemma 2.3]. Lemma 3.5.…”

Section: Banach Spaces Having Property Km Wmentioning

confidence: 99%

“…The situation changes for ℓ p -sums when 1 < p < ∞. Indeed, if (X m ) m∈N is a sequence of Banach spaces and ( m∈N X m ) ℓp is SWPG (or just a subspace of a SWPG space), then X m contains no subspace isomorphic to ℓ 1 for all but finitely many m ∈ N, see [27, Theorem 4.5] (resp., [29,Theorem 2.6]). In particular, for 1 < p < ∞ the space ℓ p (ℓ 1 ) does not embed isomorphically into a SWPG space.…”

Section: Unconditional Sumsmentioning

confidence: 99%

“…Indeed, Mercourakis and Stamati (see [31, §3]) gave an example of a subspace of the SWCG space L 1 [0, 1] which is not SWCG. The same example works for the property of being SWPG, because a Banach space is SWCG if and only if it is weakly sequentially complete and SWPG (see [41,Theorem 2.5] and [29,Theorem 2.2]). In an attempt of characterizing subspaces of SWCG spaces, Kampoukos and Mercourakis [26] studied the following class of Banach spaces (called there spaces having "property (*)"): Definition 1.3.…”

Section: Introductionmentioning

confidence: 95%

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$\varepsilon$-weakly precompact sets in Banach spaces

Rodríguez¹

2021

Preprint

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A bounded subset M of a Banach space X is said to be ε-weakly precompact, for a given ε ≥ 0, if every sequenceIn this paper we discuss several aspects of ε-weakly precompact sets. On the one hand, we give quantitative versions of the following known results: (a) the absolutely convex hull of a weakly precompact set is weakly precompact (Stegall), and (b) for any probability measure µ, the set of all Bochner µ-integrable functions taking values in a weakly precompact subset of X is weakly precompact in L 1 (µ, X) (Bourgain, Maurey, Pisier). On the other hand, we introduce a relative of a Banach space property considered by Kampoukos and Mercourakis when studying subspaces of strongly weakly compactly generated spaces. We say that a Banach space X has property KM w if there is a family {M n,p : n, p ∈ N} of subsets of X such that: (i) M n,p is 1 p -weakly precompact for all n, p ∈ N, and (ii) for each weakly precompact set C ⊆ X and for each p ∈ N there is n ∈ N such that C ⊆ M n,p . All subspaces of strongly weakly precompactly generated spaces have property KM w . Among other things, we study the three-space problem and the stability under unconditional sums of property KM w .

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Section: Introductionsupporting

confidence: 86%

Section: Banach Spaces Having Property Km Wmentioning

confidence: 99%

Section: Unconditional Sumsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

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$\varepsilon$-weakly precompact sets in Banach spaces

Rodríguez¹

2021

Preprint

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“…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

Rodríguez

2017

Topology and its Applications

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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].

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On weak compactness in projective tensor products

Rodríguez

2022

The Quarterly Journal of Mathematics

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We study the property of being strongly weakly compactly generated (and some relatives) in projective tensor products of Banach spaces. Our main result is as follows. Let $1 < p,q<\infty$ be such that $1/p+1/q\geq 1$. Let X (resp., Y) be a Banach space with a countable unconditional finite-dimensional Schauder decomposition having a disjoint lower p-estimate (resp., q-estimate). If X and Y are strongly weakly compactly generated, then so is their projective tensor product $X {\widehat{\otimes}_\pi} Y$.

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Strongly Asplund generated and strongly conditionally weakly compactly generated Banach spaces

Cited by 4 publications

References 21 publications

$\varepsilon$-weakly precompact sets in Banach spaces

$\varepsilon$-weakly precompact sets in Banach spaces

A class of weakly compact sets in Lebesgue–Bochner spaces

On weak compactness in projective tensor products

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