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Keywords = Köthe space

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16 pages, 318 KiB  
Article
Barrelled Weakly Köthe–Orlicz Summable Sequence Spaces
by Issam Aboutaib, Janusz Brzdęk and Lahbib Oubbi
Mathematics 2024, 12(1), 88; https://doi.org/10.3390/math12010088 - 26 Dec 2023
Viewed by 1038
Abstract
Let E be a Hausdorff locally convex space. We investigate the space Λφ[E] of weakly Köthe–Orlicz summable sequences in E with respect to an Orlicz function φ and a perfect sequence space Λ. We endow [...] Read more.
Let E be a Hausdorff locally convex space. We investigate the space Λφ[E] of weakly Köthe–Orlicz summable sequences in E with respect to an Orlicz function φ and a perfect sequence space Λ. We endow Λφ[E] with a Hausdorff locally convex topology and determine the continuous dual of the so-obtained space in terms of strongly Köthe–Orlicz summable sequences from the dual space E of E. Next, we give necessary and sufficient conditions for Λφ[E] to be barrelled or quasi-barrelled. This contributes to the understanding of different spaces of vector-valued sequences and their topological properties. Full article
(This article belongs to the Special Issue Topological Space and Its Applications)
13 pages, 274 KiB  
Article
Upper Local Uniform Monotonicity in F-Normed Musielak–Orlicz Spaces
by Yanli Liu, Yangyang Xue and Yunan Cui
Axioms 2023, 12(6), 539; https://doi.org/10.3390/axioms12060539 - 30 May 2023
Viewed by 1166
Abstract
In this paper, the necessary and sufficient conditions for the upper strict monotonicity point and the upper local uniform monotonicity point are given in the case of Musielak–Orlicz spaces equipped with the Mazur–Orlicz F-norm. Moreover, strict monotonicity and upper local uniform monotonicity are [...] Read more.
In this paper, the necessary and sufficient conditions for the upper strict monotonicity point and the upper local uniform monotonicity point are given in the case of Musielak–Orlicz spaces equipped with the Mazur–Orlicz F-norm. Moreover, strict monotonicity and upper local uniform monotonicity are easily deduced in the case of Musielak–Orlicz spaces endowed with the Mazur–Orlicz F-norm, and the work by Kaczmarek presented in the references is encompassed by the corollaries presented in this paper. Full article
(This article belongs to the Special Issue Modern Functional Analysis and Related Applications)
12 pages, 293 KiB  
Article
On Bilinear Narrow Operators
by Marat Pliev, Nonna Dzhusoeva and Ruslan Kulaev
Mathematics 2021, 9(22), 2892; https://doi.org/10.3390/math9222892 - 13 Nov 2021
Cited by 2 | Viewed by 1697
Abstract
In this article, we introduce a new class of operators on the Cartesian product of vector lattices. We say that a bilinear operator T:E×FW defined on the Cartesian product of vector lattices E and F and taking [...] Read more.
In this article, we introduce a new class of operators on the Cartesian product of vector lattices. We say that a bilinear operator T:E×FW defined on the Cartesian product of vector lattices E and F and taking values in a vector lattice W is narrow if the partial operators Tx and Ty are narrow for all xE,yF. We prove that, for order-continuous Köthe–Banach spaces E and F and a Banach space X, the classes of narrow and weakly function narrow bilinear operators from E×F to X are coincident. Then, we prove that every order-to-norm continuous C-compact bilinear regular operator T is narrow. Finally, we show that a regular bilinear operator T from the Cartesian product E×F of vector lattices E and F with the principal projection property to an order continuous Banach lattice G is narrow if and only if |T| is. Full article
14 pages, 329 KiB  
Article
Kuelbs–Steadman Spaces for Banach Space-Valued Measures
by Antonio Boccuto, Bipan Hazarika and Hemanta Kalita
Mathematics 2020, 8(6), 1005; https://doi.org/10.3390/math8061005 - 19 Jun 2020
Cited by 6 | Viewed by 2258
Abstract
We introduce Kuelbs–Steadman-type spaces ( K S p spaces) for real-valued functions, with respect to countably additive measures, taking values in Banach spaces. We investigate the main properties and embeddings of L q -type spaces into K S p spaces, considering both the [...] Read more.
We introduce Kuelbs–Steadman-type spaces ( K S p spaces) for real-valued functions, with respect to countably additive measures, taking values in Banach spaces. We investigate the main properties and embeddings of L q -type spaces into K S p spaces, considering both the norm associated with the norm convergence of the involved integrals and that related to the weak convergence of the integrals. Full article
(This article belongs to the Special Issue Set-Valued Analysis)
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