A note on weakly Lindelöf determined Banach spaces

Carrere, Alberto; Montesinos, Vicente

doi:10.1007/s10587-009-0055-x

Cited by 9 publications

(8 citation statements)

References 11 publications

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“…For (1) we apply Theorem 12 for n = 0 as a Banach space is WLD if and only if it admits a linearly dense set such that every functional is countably supported by it (Theorem 7 of [9]).…”

Section: Positive Resultsmentioning

confidence: 99%

“…By L p ({0, 1} κ ) for p ∈ [1, ∞] and κ a cardinal we mean L p (µ), where µ is the homogeneous probability product measure on {0, 1} κ . The class of WLD (weakly Lindelöf determined) Banach spaces has many nice characterizations, the most convenient for this paper is the one as the class of Banach spaces X which admit a linearly dense set D ⊆ X such that {d ∈ D : x * (d) = 0} is countable for each x * ∈ X * ( [9]). X is a Grothendieck Banach space if and only in X * weakly * convergent sequences coincide with weakly convergent sequences.…”

Section: Preliminariesmentioning

confidence: 99%

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On the existence of overcomplete sets in some classical nonseparable Banach spaces

Koszmider¹

2020

Preprint

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For a Banach space X its subset Y ⊆ X is called overcomplete if |Y | = dens(X) and Z is linearly dense in X for every Z ⊆ Y with |Z| = |Y |. In the context of nonseparable Banach spaces this notion was introduced recently by T. Russo and J. Somaglia but overcomplete sets have been considered in separable Banach spaces since the 1950ties.We prove some absolute and consistency results concerning the existence and the nonexistence of overcomplete sets in some classical nonseparable Banach spaces. For example: c 0 (ω 1 ), C([0, ω 1 ]), L 1 ({0, 1} ω 1 ), ℓp(ω 1 ), Lp({0, 1} ω 1 ) for p ∈ (1, ∞) or in general WLD Banach spaces of density ω 1 admit overcomplete sets (in ZFC). The spaces ℓ∞, ℓ∞/c 0 , L∞({0, 1} κ ), C({0, 1} κ ) or in general superspaces of ℓ 1 (κ) of density κ for any cardinal κ of uncountable cofinality do not admit overcomplete sets (in ZFC). Whether the Johnson-Lindenstrauss space generated in ℓ∞ by c 0 and the characteristic functions of elements of an almost disjoint family of subsets of N of cardinality ω 1 admits an overcomplete set is undecidable. The same refers to all nonseparable Banach spaces with the dual balls of density ω 1 which are separable in the weak * topology. The results proved refer to wider classes of Banach spaces but several natural open questions remain open.

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“…For (1) we apply Theorem 12 for n = 0 as a Banach space is WLD if and only if it admits a linearly dense set such that every functional is countably supported by it (Theorem 7 of [9]).…”

Section: Positive Resultsmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

On the existence of overcomplete sets in some classical nonseparable Banach spaces

Koszmider¹

2020

Preprint

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“…then such a system is called fundamental. For the definition and various characterizations of WLD spaces see [13].…”

Section: Preliminariesmentioning

confidence: 99%

On coverings of Banach spaces and their subsets by hyperplanes

Głodkowski¹,

Koszmider²

2021

Preprint

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Given a Banach space we consider the σ-ideal of all of its subsets which are covered by countably many hyperplanes and investigate its standard cardinal characteristics as the additivity, the covering number, the uniformity, the cofinality. We determine their values for separable Banach spaces, and approximate them for nonseparable Banach spaces. The remaining questions reduce to deciding if the following can be proved in ZFC for every nonseparable Banach space X: (1) X can be covered by ω 1 -many of its hyperplanes; (2) All subsets of X of cardinalities less than cf([dens(X)] ω ) can be covered by countably many hyperplanes. We prove (1) and ( 2) for all Banach spaces in many well-investigated classes and that they are consistent with any possible size of the continuum. ( 1) is related to the problem whether every compact Hausdorff space which has small diagonal is metrizable and (2) to large cardinals.

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“…The notion of CSCP was introduced in [14] to study the structure of certain quotients of Banach spaces, although it was refered to as the Controlled Separable Projection Property. There is a wide class of Banach spaces which have this property, in fact, after the characterization of the Weakly Lindelöf Determined spaces (W LD) given in [9], it can be deduced that the class of spaces having the CSCP contains the W LD. Since we showed in [6] that the space C[0, ω 1 ] has the CSCP , it follows that the inclusion between those two classes is strict.…”

Section: Introductionmentioning

confidence: 99%

The controlled separable complementation property and monolithic compacta

Ferrer¹

2014

Banach J. Math. Anal.

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For a compact K, a necessary condition for C(K) to have the Controlled Separable Complementation Property is that K be monolithic. In this paper, we prove that when K contains no copy of [0, ω ω ] and the set of points which admit a countable neighborhood base is a cofinite subset of K, then monolithicity of K is sufficient for C(K) to enjoy the Controlled Separable Complementation Property. We also show that, for this type of compacta K, the space C(K) is separably extensible.

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A note on weakly Lindelöf determined Banach spaces

Cited by 9 publications

References 11 publications

On the existence of overcomplete sets in some classical nonseparable Banach spaces

On the existence of overcomplete sets in some classical nonseparable Banach spaces

On coverings of Banach spaces and their subsets by hyperplanes

The controlled separable complementation property and monolithic compacta

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