Copies of $ℓ_{∞}$ in the space of Pettis integrable functions with integrals of finite variation

Juan Carlos Ferrando

Displaying similar documents to “Copies of $ℓ_{\infty}$ in the space of Pettis integrable functions with integrals of finite variation”

Correction to the paper ’Copies of $ℓ_{\infty}$ in the space of Pettis integrable functions with integrals of finite variation’ (Studia Math. 210 (2012), 93-98)

Juan Carlos Ferrando (2016)

Studia Mathematica

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Order-bounded operators from vector-valued function spaces to Banach spaces

Marian Nowak (2005)

Banach Center Publications

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Let E be an ideal of L⁰ over a σ-finite measure space (Ω,Σ,μ). For a real Banach space $(X, | | \cdot | |_{X})$ let E(X) be a subspace of the space L⁰(X) of μ-equivalence classes of strongly Σ-measurable functions f: Ω → X and consisting of all those f ∈ L⁰(X) for which the scalar function ${| | f (\cdot) | |}_{X}$ belongs to E. Let E(X)˜ stand for the order dual of E(X). For u ∈ E⁺ let $D_{u} (= {f \in E (X) : | | f (\cdot) | |}_{X} \leq u)$ stand for the order interval in E(X). For a real Banach space $(Y, | | \cdot | |_{Y})$ a linear operator T: E(X) → Y is said to be order-bounded whenever for each u ∈...

Henstock-Kurzweil integral on $BV$ sets

Jan Malý, Washek Frank Pfeffer (2016)

Mathematica Bohemica

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The generalized Riemann integral of Pfeffer (1991) is defined on all bounded $BV$ subsets of $ℝ^{n}$ , but it is additive only with respect to pairs of disjoint sets whose closures intersect in a set of $σ$ -finite Hausdorff measure of codimension one. Imposing a stronger regularity condition on partitions of $BV$ sets, we define a Riemann-type integral which satisfies the usual additivity condition and extends the integral of Pfeffer. The new integral is lipeomorphism-invariant and closed with respect...

Variational Henstock integrability of Banach space valued functions

Luisa Di Piazza, Valeria Marraffa, Kazimierz Musiał (2016)

Mathematica Bohemica

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We study the integrability of Banach space valued strongly measurable functions defined on $[0, 1]$ . In the case of functions $f$ given by $\sum_{n = 1}^{\infty} x_{n} χ_{E_{n}}$ , where $x_{n}$ are points of a Banach space and the sets $E_{n}$ are Lebesgue measurable and pairwise disjoint subsets of $[0, 1]$ , there are well known characterizations for Bochner and Pettis integrability of $f$ . The function $f$ is Bochner integrable if and only if the series $\sum_{n = 1}^{\infty} x_{n} | E_{n} |$ is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability...

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 $f : [0, 1] \to X$ 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...

Continuity in the Alexiewicz norm

Erik Talvila (2006)

Mathematica Bohemica

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If $f$ is a Henstock-Kurzweil integrable function on the real line, the Alexiewicz norm of $f$ is $∥ f ∥ = {sup}_{I} | \int_{I} f |$ where the supremum is taken over all intervals $I \subset ℝ$ . Define the translation $τ_{x}$ by $τ_{x} f (y) = f (y - x)$ . Then $∥ τ_{x} f - f ∥$ tends to $0$ as $x$ tends to $0$ , i.e., $f$ is continuous in the Alexiewicz norm. For particular functions, $∥ τ_{x} f - f ∥$ can tend to 0 arbitrarily slowly. In general, $∥ τ_{x} f - f ∥ \geq osc f | x |$ as $x \to 0$ , where $osc f$ is the oscillation of $f$ . It is shown that if $F$ is a primitive of $f$ then $∥ τ_{x} F - F ∥ \leq ∥ f ∥ | x |$ . An example shows that the function $y \mapsto τ_{x} F (y) - F (y)$ need not be in $L^{1}$ . However, if...

Algebraic genericity of strict-order integrability

Luis Bernal-González (2010)

Studia Mathematica

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We provide sharp conditions on a measure μ defined on a measurable space X guaranteeing that the family of functions in the Lebesgue space $L^{p} (μ, X)$ (p ≥ 1) which are not q-integrable for any q > p (or any q < p) contains large subspaces of $L^{p} (μ, X)$ (without zero). This improves recent results due to Aron, García, Muñoz, Palmberg, Pérez, Puglisi and Seoane. It is also shown that many non-q-integrable functions can even be obtained on any nonempty open subset of X, assuming that X is a topological...

Application of $(L)$ sets to some classes of operators

Kamal El Fahri, Nabil Machrafi, Jawad H&#039;michane, Aziz Elbour (2016)

Mathematica Bohemica

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The paper contains some applications of the notion of $Ł$ sets to several classes of operators on Banach lattices. In particular, we introduce and study the class of order $(L)$ -Dunford-Pettis operators, that is, operators from a Banach space into a Banach lattice whose adjoint maps order bounded subsets to an $(L)$ sets. As a sequence characterization of such operators, we see that an operator $T : X \to E$ from a Banach space into a Banach lattice is order $Ł$ -Dunford-Pettis, if and only if $| T (x_{n}) | \to 0$ for $σ (E, E^{'})$ for every...

The $L^{r}$ Henstock-Kurzweil integral

Paul M. Musial, Yoram Sagher (2004)

Studia Mathematica

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We present a method of integration along the lines of the Henstock-Kurzweil integral. All $L^{r}$ -derivatives are integrable in this method.

Dunford-Pettis operators on the space of Bochner integrable functions

Marian Nowak (2011)

Banach Center Publications

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Let (Ω,Σ,μ) be a finite measure space and let X be a real Banach space. Let $L^{Φ} (X)$ be the Orlicz-Bochner space defined by a Young function Φ. We study the relationships between Dunford-Pettis operators T from L¹(X) to a Banach space Y and the compactness properties of the operators T restricted to $L^{Φ} (X)$ . In particular, it is shown that if X is a reflexive Banach space, then a bounded linear operator T:L¹(X) → Y is Dunford-Pettis if and only if T restricted to $L^{\infty} (X)$ is $(τ (L^{\infty} (X), L ¹ (X *)), | | \cdot | |_{Y})$ -compact.

Large structures made of nowhere $L^{q}$ functions

Szymon Głąb, Pedro L. Kaufmann, Leonardo Pellegrini (2014)

Studia Mathematica

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We say that a real-valued function f defined on a positive Borel measure space (X,μ) is nowhere q-integrable if, for each nonvoid open subset U of X, the restriction ${f |}_{U}$ is not in $L^{q} (U)$ . When (X,μ) has some natural properties, we show that certain sets of functions defined in X which are p-integrable for some p’s but nowhere q-integrable for some other q’s (0 < p,q < ∞) admit a variety of large linear and algebraic structures within them. The presented results answer a question of Bernal-González,...

A necessary condition for HK-integrability of the Fourier sine transform function

Juan H. Arredondo, Manuel Bernal, Maria G. Morales (2025)

Czechoslovak Mathematical Journal

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The paper is concerned with integrability of the Fourier sine transform function when $f \in {BV}_{0} (ℝ)$ , where ${BV}_{0} (ℝ)$ is the space of bounded variation functions vanishing at infinity. It is shown that for the Fourier sine transform function of $f$ to be integrable in the Henstock-Kurzweil sense, it is necessary that $f / x \in L^{1} (ℝ)$ . We prove that this condition is optimal through the theoretical scope of the Henstock-Kurzweil integration theory.

Displaying similar documents to “Copies of ℓ ∞ in the space of Pettis integrable functions with integrals of finite variation”

Displaying similar documents to “Copies of $ℓ_{\infty}$ in the space of Pettis integrable functions with integrals of finite variation”