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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.05117 |
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| _version_ | 1866914829613137920 |
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| author | Tuǧ, Orhan Malkowsky, Eberhard Rakočević, Vladimir Yaying, Taja |
| author_facet | Tuǧ, Orhan Malkowsky, Eberhard Rakočević, Vladimir Yaying, Taja |
| contents | Double sequence spaces have become a significant area of research within functional analysis due to their applications in various branches of mathematics and mathematical physics. In this study, we investigate Hahn double sequence space denoted as $\mathcal{H}_{\vartheta}$, where $\vartheta\in\{p,bp,r\}$, as an extension of the Hahn sequence space $h$. Our investigation begins with an analysis of several topological properties of $\mathcal{H}_{\vartheta}$, apart from a comprehensive analysis of the relationship between Hahn double sequences and some other classical double sequence spaces. The $α-$dual, algebraic dual and $β(bp)-$dual, and $γ-$dual of the space $\mathcal{H}_{\vartheta}$ are detrmined. Furthermore, we define the determining set of $\mathcal{H}_{\vartheta}$ and we state the conditions concerning the characterization of four-dimensional (4D) matrix classes $(\mathcal{H}_{\vartheta},λ)$, where $λ=\{\mathcal{H}_{\vartheta},\mathcal{BV}, \mathcal{BV}_{\vartheta 0}, \mathcal{CS}_{\vartheta},\mathcal{CS}_{\vartheta 0},\mathcal{BS}\}$ and $(μ,\mathcal{H}_{\vartheta})$, where $μ=\{\mathcal{L}_u, \mathcal{C}_{\vartheta 0}, \mathcal{C}_{\vartheta},\mathcal{M}_{u}\}$. In conclusion, this research contributes non-standard investigation and various significant results into the space $\mathcal{H}_{\vartheta}$. The conducted results are deepen the understanding of the space $\mathcal{H}_{\vartheta}$ and open up new avenues for further research and applications in sequence space theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05117 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Double Sequence Space $\mathcal{H}_{\vartheta}$ as an Extension of Hahn Space $h$ Tuǧ, Orhan Malkowsky, Eberhard Rakočević, Vladimir Yaying, Taja Functional Analysis 46A45, 40C05 Double sequence spaces have become a significant area of research within functional analysis due to their applications in various branches of mathematics and mathematical physics. In this study, we investigate Hahn double sequence space denoted as $\mathcal{H}_{\vartheta}$, where $\vartheta\in\{p,bp,r\}$, as an extension of the Hahn sequence space $h$. Our investigation begins with an analysis of several topological properties of $\mathcal{H}_{\vartheta}$, apart from a comprehensive analysis of the relationship between Hahn double sequences and some other classical double sequence spaces. The $α-$dual, algebraic dual and $β(bp)-$dual, and $γ-$dual of the space $\mathcal{H}_{\vartheta}$ are detrmined. Furthermore, we define the determining set of $\mathcal{H}_{\vartheta}$ and we state the conditions concerning the characterization of four-dimensional (4D) matrix classes $(\mathcal{H}_{\vartheta},λ)$, where $λ=\{\mathcal{H}_{\vartheta},\mathcal{BV}, \mathcal{BV}_{\vartheta 0}, \mathcal{CS}_{\vartheta},\mathcal{CS}_{\vartheta 0},\mathcal{BS}\}$ and $(μ,\mathcal{H}_{\vartheta})$, where $μ=\{\mathcal{L}_u, \mathcal{C}_{\vartheta 0}, \mathcal{C}_{\vartheta},\mathcal{M}_{u}\}$. In conclusion, this research contributes non-standard investigation and various significant results into the space $\mathcal{H}_{\vartheta}$. The conducted results are deepen the understanding of the space $\mathcal{H}_{\vartheta}$ and open up new avenues for further research and applications in sequence space theory. |
| title | On the Double Sequence Space $\mathcal{H}_{\vartheta}$ as an Extension of Hahn Space $h$ |
| topic | Functional Analysis 46A45, 40C05 |
| url | https://arxiv.org/abs/2406.05117 |