Weakly precompact subsets of L₁(μ,X)

Ioana Ghenciu

Displaying similar documents to “Weakly precompact subsets of L₁(μ,X)”

Completely Continuous operators

Ioana Ghenciu, Paul Lewis (2012)

Colloquium Mathematicae

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A Banach space X has the Dunford-Pettis property (DPP) provided that every weakly compact operator T from X to any Banach space Y is completely continuous (or a Dunford-Pettis operator). It is known that X has the DPP if and only if every weakly null sequence in X is a Dunford-Pettis subset of X. In this paper we give equivalent characterizations of Banach spaces X such that every weakly Cauchy sequence in X is a limited subset of X. We prove that every operator T: X → c₀ is completely...

On some new characterizations of weakly compact sets in Banach spaces

Lixin Cheng, Qingjin Cheng, Zhenghua Luo (2010)

Studia Mathematica

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We show several characterizations of weakly compact sets in Banach spaces. Given a bounded closed convex set C of a Banach space X, the following statements are equivalent: (i) C is weakly compact; (ii) C can be affinely uniformly embedded into a reflexive Banach space; (iii) there exists an equivalent norm on X which has the w2R-property on C; (iv) there is a continuous and w*-lower semicontinuous seminorm p on the dual X* with $p \geq s u p_{C}$ such that p² is everywhere Fréchet differentiable in...

A note on weakly $(μ, λ)$ -closed functions

Bishwambhar Roy (2013)

Mathematica Bohemica

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In this paper we introduce a new class of functions called weakly $(μ, λ)$ -closed functions with the help of generalized topology which was introduced by Á. Császár. Several characterizations and some basic properties of such functions are obtained. The connections between these functions and some other similar types of functions are given. Finally some comparisons between different weakly closed functions are discussed. This weakly $(μ, λ)$ -closed functions enable us to facilitate the formulation...

Dieudonné operators on the space of Bochner integrable functions

Marian Nowak (2011)

Banach Center Publications

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A bounded linear operator between Banach spaces is called a Dieudonné operator ( = weakly completely continuous operator) if it maps weakly Cauchy sequences to weakly convergent sequences. Let (Ω,Σ,μ) be a finite measure space, and let X and Y be Banach spaces. We study Dieudonné operators T: L¹(X) → Y. Let $i_{\infty} : L^{\infty} (X) \to L ¹ (X)$ stand for the canonical injection. We show that if X is almost reflexive and T: L¹(X) → Y is a Dieudonné operator, then $T \circ i_{\infty} : L^{\infty} (X) \to Y$ is a weakly compact operator. Moreover, we obtain that...

On the approximate solutions of linear equations in the $l_{2}$ -space

Rabindranath Sen (1965)

Rendiconti del Seminario Matematico della Università di Padova

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On projectional skeletons in Vašák spaces

Ondřej F. K. Kalenda (2017)

Commentationes Mathematicae Universitatis Carolinae

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We provide an alternative proof of the theorem saying that any Vašák (or, weakly countably determined) Banach space admits a full $1$ -projectional skeleton. The proof is done with the use of the method of elementary submodels and is comparably simple as the proof given by W. Kubiś (2009) in case of weakly compactly generated spaces.

Weakly compact sets in $L^{1} / H_{0}^{1}$

F. Delbaen (1977-1978)

Séminaire Analyse fonctionnelle (dit "Maurey-Schwartz")

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Weak vs. norm compactness in $L_{1}$ : the Bocce criterion

Maria Girardi (1991)

Studia Mathematica

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On weakly closed functions

N. Ergun, T. Noiri (1990)

Matematički Vesnik

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A new proof that every weakly compact operator with domain L(μ) is representable.

Diómedes Bárcenas (1991)

Extracta Mathematicae

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An alternative Dunford-Pettis Property

Walden Freedman (1997)

Studia Mathematica

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An alternative to the Dunford-Pettis Property, called the DP1-property, is introduced. Its relationship to the Dunford-Pettis Property and other related properties is examined. It is shown that $ℓ_{p}$ -direct sums of spaces with DP1 have DP1 if 1 ≤ p < ∞. It is also shown that for preduals of von Neumann algebras, DP1 is strictly weaker than the Dunford-Pettis Property, while for von Neumann algebras, the two properties are equivalent.

Open maps do not preserve Whyburn property

Franco Obersnel (2003)

Commentationes Mathematicae Universitatis Carolinae

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We show that a (weakly) Whyburn space $X$ may be mapped continuously via an open map $f$ onto a non (weakly) Whyburn space $Y$ . This fact may happen even between topological groups $X$ and $Y$ , $f$ a homomorphism, $X$ Whyburn and $Y$ not even weakly Whyburn.