Displaying similar documents to “Variational Henstock integrability of Banach space valued functions”

Some remarks on descriptive characterizations of the strong McShane integral

Sokol Bush Kaliaj (2019)

Mathematica Bohemica

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We present the full descriptive characterizations of the strong McShane integral (or the variational McShane integral) of a Banach space valued function f : W → X defined on a non-degenerate closed subinterval W of ℝ m in terms of strong absolute continuity or, equivalently, in terms of McShane variational measure V ℳ F generated by the primitive F : ℐ W → X of f , where ℐ W is the family of all closed non-degenerate subintervals of W .

On coincidence of Pettis and McShane integrability

Marián J. Fabián (2015)

Czechoslovak Mathematical Journal

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R. Deville and J. Rodríguez proved that, for every Hilbert generated space X , every Pettis integrable function f : [ 0 , 1 ] → X is McShane integrable. R. Avilés, G. Plebanek, and J. Rodríguez constructed a weakly compactly generated Banach space X and a scalarly null (hence Pettis integrable) function from [ 0 , 1 ] into X , which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from [ 0 , 1 ] (mostly) into C ( K ) spaces. We focus in more detail on...

The topology of the space of ℋ𝒦 integrable functions in ℝ n

Varayu Boonpogkrong (2025)

Czechoslovak Mathematical Journal

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It is known that there is no natural Banach norm on the space ℋ𝒦 of n -dimensional Henstock-Kurzweil integrable functions on [ a , b ] . We show that the ℋ𝒦 space is the uncountable union of Fréchet spaces ℋ𝒦 ( X ) . On each ℋ𝒦 ( X ) space, an F -norm ∥ · ∥ X is defined. A ∥ · ∥ X -convergent sequence is equivalent to a control-convergent sequence. Furthermore, an F -norm is also defined for a ∥ · ∥ X -continuous linear operator. Hence, many important results in functional analysis hold for the ℋ𝒦 ( X ) space. It is well-known that every...

Compact operators and integral equations in the ℋ𝒦 space

Varayu Boonpogkrong (2022)

Czechoslovak Mathematical Journal

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The space ℋ𝒦 of Henstock-Kurzweil integrable functions on [ a , b ] is the uncountable union of Fréchet spaces ℋ𝒦 ( X ) . In this paper, on each Fréchet space ℋ𝒦 ( X ) , an F -norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the ℋ𝒦 ( X ) space. It is known that every control-convergent sequence in the ℋ𝒦 space always belongs to a ℋ𝒦 ( X ) space for some X . We illustrate how...

Limited p -converging operators and relation with some geometric properties of Banach spaces

Mohammad B. Dehghani, Seyed M. Moshtaghioun (2021)

Commentationes Mathematicae Universitatis Carolinae

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By using the concepts of limited p -converging operators between two Banach spaces X and Y , L p -sets and L p -limited sets in Banach spaces, we obtain some characterizations of these concepts relative to some well-known geometric properties of Banach spaces, such as * -Dunford–Pettis property of order p and Pelczyński’s property of order p , 1 ≤ p < ∞ .

A note on Dunford-Pettis like properties and complemented spaces of operators

Ioana Ghenciu (2018)

Commentationes Mathematicae Universitatis Carolinae

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Equivalent formulations of the Dunford-Pettis property of order p ( D P P p ), 1 < p < ∞ , are studied. Let L ( X , Y ) , W ( X , Y ) , K ( X , Y ) , U ( X , Y ) , and C p ( X , Y ) denote respectively the sets of all bounded linear, weakly compact, compact, unconditionally converging, and p -convergent operators from X to Y . Classical results of Kalton are used to study the complementability of the spaces W ( X , Y ) and K ( X , Y ) in the space C p ( X , Y ) , and of C p ( X , Y ) in U ( X , Y ) and L ( X , Y ) .

Property ( 𝐰𝐋 ) and the reciprocal Dunford-Pettis property in projective tensor products

Ioana Ghenciu (2015)

Commentationes Mathematicae Universitatis Carolinae

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A Banach space X has the reciprocal Dunford-Pettis property ( R D P P ) if every completely continuous operator T from X to any Banach space Y is weakly compact. A Banach space X has the R D P P (resp. property ( w L ) ) if every L -subset of X * is relatively weakly compact (resp. weakly precompact). We prove that the projective tensor product X ⊗ π Y has property ( w L ) when X has the R D P P , Y has property ( w L ) , and L ( X , Y * ) = K ( X , Y * ) .

The weak Gelfand-Phillips property in spaces of compact operators

Ioana Ghenciu (2017)

Commentationes Mathematicae Universitatis Carolinae

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For Banach spaces X and Y , let K w * ( X * , Y ) denote the space of all w * - w continuous compact operators from X * to Y endowed with the operator norm. A Banach space X has the w G P property if every Grothendieck subset of X is relatively weakly compact. In this paper we study Banach spaces with property w G P . We investigate whether the spaces K w * ( X * , Y ) and X ⊗ ϵ Y have the w G P property, when X and Y have the w G P property.

Almost demi Dunford--Pettis operators on Banach lattices

Hedi Benkhaled (2023)

Commentationes Mathematicae Universitatis Carolinae

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We introduce new concept of almost demi Dunford–Pettis operators. Let E be a Banach lattice. An operator T from E into E is said to be almost demi Dunford–Pettis if, for every sequence { x n } in E + such that x n → 0 in σ ( E , E ' ) and ∥ x n - T x n ∥ → 0 as n → ∞ , we have ∥ x n ∥ → 0 as n → ∞ . In addition, we study some properties of this class of operators and its relationships with others known operators.

Some theorems of Korovkin type

Tomoko Hachiro, Takateru Okayasu (2003)

Studia Mathematica

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We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, C ℝ ( X ) ) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., C ℝ ( Y ) ), and ϕ ∞ a linear isometry from M into C(Y) (resp., C ℝ ( Y ) ). We show, under the assumption that Π N ⊂ Π T , where Π N is...

ℓ ∞ -sums and the Banach space ℓ ∞ / c ₀

Christina Brech, Piotr Koszmider (2014)

Fundamenta Mathematicae

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This paper is concerned with the isomorphic structure of the Banach space ℓ ∞ / c ₀ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that ℓ ∞ / c ₀ does not have an orthogonal ℓ ∞ -decomposition, that is, it is not of the form ℓ ∞ ( X ) for any Banach space X. The main local result is that it is consistent that ℓ ∞ ( c ₀ ( ) ) does not embed isomorphically into ℓ ∞ / c ₀ , where is the cardinality of the continuum,...

Selivanovski hard sets are hard

Janusz Pawlikowski (2015)

Fundamenta Mathematicae

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Let H ⊆ Z ⊆ 2 ω . For n ≥ 2, we prove that if Selivanovski measurable functions from 2 ω to Z give as preimages of H all Σₙ¹ subsets of 2 ω , then so do continuous injections.