Markuševič Bases and Corson Compacta in Duality

Vanderwerff, Jon D.; Whitfield, J. H.; Zizler, Václav

doi:10.4153/cjm-1994-007-5

Cited by 20 publications

(12 citation statements)

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“…But C(K) is not separable, as K is uncountable and scattered and therefore non metrizable. It follows that C(K) is not weakly Lindelöf determined (WLD in short): see Theorem 5.37 and Proposition 5.40 in [21], or see [35]. We conclude by recalling that c 0 (Γ) is always WLD and that being WLD is stable by passing to subspaces or quotients (see also [21] and references therein).…”

Section: Examplesmentioning

confidence: 80%

Asymptotic smoothness in Banach spaces, three space properties and applications

Causey,

Fovelle,

Lancien

2021

Preprint

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We study four asymptotic smoothness properties of Banach spaces, denoted Tp, Ap, Np and Pp. We complete their description by proving the missing renorming theorem for Ap. We prove that asymptotic uniform flattenability (property T∞) and summable Szlenk index (property A∞) are three space properties. Combined with the positive results of the first named author, Draga, and Kochanek, and with the counterexamples we provide, this completely solves the three space problem for this family of properties. We also derive from our characterizations of Ap and Np in terms of equivalent renormings, new coarse Lipschitz rigidity results for these classes.

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Section: Examplesmentioning

confidence: 80%

Asymptotic smoothness in Banach spaces, three space properties and applications

Causey,

Fovelle,

Lancien

2021

Preprint

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“…It has been proved by Valdivia ([53], or see [54]) that every WLD-space has a fundamental biorthogonal system; hence we see (as in [24,Example 2]) that every half-pcc WLD-space is separable.…”

Section: Corollary ([24]) If a Half-pcc Banach Space Has A Fundamentmentioning

confidence: 94%

Chain conditions and weak topologies

Dow

Junnila

Pelant

2009

Topology and its Applications

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We study conditions on Banach spaces close to separability. We say that a topological space is pcc if every point-finite family of open subsets of the space is countable. For a Banach space E, we say that E is weakly pcc if E, equipped with the weak topology, is pcc, and we also consider a weaker property: we say that E is half-pcc if every point-finite family consisting of half-spaces of E is countable. We show that E is half-pcc if, and only if, every bounded linear map E → c 0 (ω 1 ) has separable range. We exhibit a variety of mild conditions which imply separability of a half-pcc Banach space. For a Banach space C (K ), we also consider the pcc-property of the topology of pointwise convergence, and we note that the space C p (K ) may be pcc even when C (K ) fails to be weakly pcc. We note that this does not happen when K is scattered, and we provide the following example: -There exists a non-metrizable scattered compact Hausdorff space K with C (K ) weakly pcc. Lemma. Let Γ be a set, and let S be a subset of c 0 (Γ ) such that S is pcc in the relative weak topology. Then S is separable.Proof. It suffices to show that the set Γ = {γ ∈ Γ : y γ = 0 for some y ∈ S} is countable. Assume that Γ is uncountable.Then there exists r > 0 such that the set Γ = {γ ∈ Γ : |y γ | > r for some y ∈ S} is uncountable. For every γ ∈ Γ , let G γ = {y ∈ S: |y γ | > r}, and note that G γ is a non-empty relatively weakly open subset of S. For every y ∈ S, the set {γ ∈ Γ : y ∈ G γ } is finite. It follows that {G γ : γ ∈ Γ } is an uncountable point-finite family of relatively weakly open subsets of S, and this is a contradiction. 2 Proposition. The following are equivalent for a Banach space E:A. E is weakly pcc.B. For any Γ , every weak-to-weak continuous mapping E → c 0 (Γ ) has separable range. C. Every weak-to-weak continuous mapping E → c 0 (ω 1 ) has separable range.Proof. The implication A ⇒ B follows by Lemmas 1.2(a) and 1.4. C ⇒ A: Assume that E is not weakly pcc. Then there exists a point-finite family {U α : α ∈ ω 1 } consisting of distinct weakly open subsets of E. For every α ∈ ω 1 , there exists a non-constant, weakly continuous function f α : E → I such that

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“…[23,Theorem 5.37]). The following result might be compared with that of Vanderwerff, Whitfield and Zizler that a Banach space having a Markushevich basis and Corson's property (C) is WLD (see [36,Theorem 3.3], cf. [23, Theorem 5.37]).…”

Section: Preliminaries On Scalar Measurability and Total Setsmentioning

confidence: 99%

On integration in banach spaces and total sets

Rodríguez

2019

Quaestiones Mathematicae

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Let X be a Banach space and Γ ⊆ X * a total linear subspace. We study the concept of Γ-integrability for X-valued functions f defined on a complete probability space, i.e. an analogue of Pettis integrability by dealing only with the compositions x * , f for x * ∈ Γ. We show that Γ-integrability and Pettis integrability are equivalent whenever X has Plichko's property (D ′ ) (meaning that every w * -sequentially closed subspace of X * is w * -closed). This property is enjoyed by many Banach spaces including all spaces with w * -angelic dual as well as all spaces which are w * -sequentially dense in their bidual. A particular case of special interest arises when considering Γ = T * (Y * ) for some injective operator T : X → Y . Within this framework, we show that if T : X → Y is a semi-embedding, X has property (D ′ ) and Y has the Radon-Nikodým property, then X has the weak Radon-Nikodým property. This extends earlier results by Delbaen (for separable X) and Diestel and Uhl (for weakly K-analytic X).2010 Mathematics Subject Classification. 46B22, 46G10.

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Markuševič Bases and Corson Compacta in Duality

Abstract: We characterize Banach spaces that admit Markuševič bases with various properties connected with weak countable determining or weak Lindelöf determining of spaces or with various norming properties.

Cited by 20 publications

References 12 publications

Asymptotic smoothness in Banach spaces, three space properties and applications

Asymptotic smoothness in Banach spaces, three space properties and applications

Chain conditions and weak topologies

On integration in banach spaces and total sets

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