Variational Henstock integrability of Banach space valued functions

Luisa Di Piazza; Valeria Marraffa; Kazimierz Musiał

The search session has expired. Please query the service again.

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

Similarity:

We present the full descriptive characterizations of the strong McShane integral (or the variational McShane integral) of a Banach space valued function $f : W \to X$ 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 $F : ℐ_{W} \to X$ 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

Similarity:

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...

The topology of the space of $ℋ𝒦$ integrable functions in $ℝ^{n}$

Varayu Boonpogkrong (2025)

Czechoslovak Mathematical Journal

Similarity:

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 ${∥ \cdot ∥}^{X}$ is defined. A ${∥ \cdot ∥}^{X}$ -convergent sequence is equivalent to a control-convergent sequence. Furthermore, an $F$ -norm is also defined for a ${∥ \cdot ∥}^{X}$ -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

Similarity:

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...

Elementary construction of Hölder functions such that the Kurzweil-Stieltjes integral does not exist

Martin Rmoutil (2025)

Czechoslovak Mathematical Journal

Similarity:

For any $α, β > 0$ with $α + β < 1$ we provide a simple construction of an $α$ -Hölde function $f : [0, 1] \to ℝ$ and a $β$ -Hölder function $g : [0, 1] \to ℝ$ such that the integral $\int_{0}^{1} f d g$ fails to exist even in the Kurzweil-Stieltjes sense.

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

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

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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 $p$ , $1 \leq p < \infty$ .

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

Ioana Ghenciu (2018)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Equivalent formulations of the Dunford-Pettis property of order $p$ ( ${D P P}_{p}$ ), $1 < p < \infty$ , 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

Similarity:

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 $X \otimes_{π} Y$ 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

Similarity:

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 $X \otimes_{ϵ} Y$ have the $w G P$ property, when $X$ and $Y$ have the $w G P$ property.