The Radon-Nikod$\acute{Y}$m property of $\mathbb{L}$-Banach spaces and the dual representation theorem of $\mathbb{L}$-Bochner function spaces
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929494123610112 |
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| author | Zhang, Xia Yan, Xiangle Liu, Ming |
| author_facet | Zhang, Xia Yan, Xiangle Liu, Ming |
| contents | In this paper, we first introduce $\mathbb{L}$-$μ$-measurable functions and $\mathbb{L}$-Bochner integrable functions on a finite measure space $(S,\mathcal{F},μ),$ and give an $\mathbb{L}$-valued analogue of the canonical $L^{p}(Ω,\mathcal{F},μ).$ Then we investigate the completeness of such an $\mathbb{L}$-valued analogue and propose the Radon-Nikod$\acute{y}$m property of $\mathbb{L}$-Banach spaces. Meanwhile, an example constructed in this paper shows that there do exist an $\mathbb{L}$-Banach space which fails to possess the Radon-Nikod$\acute{y}$m property. Finally, based on above work, we establish the dual representation theorem of $\mathbb{L}$-Bochner integrable function spaces, which extends and improves the corresponding classical result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06279 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Radon-Nikod$\acute{Y}$m property of $\mathbb{L}$-Banach spaces and the dual representation theorem of $\mathbb{L}$-Bochner function spaces Zhang, Xia Yan, Xiangle Liu, Ming Functional Analysis Primary 46B22, 46B10, Secondary 46E30 In this paper, we first introduce $\mathbb{L}$-$μ$-measurable functions and $\mathbb{L}$-Bochner integrable functions on a finite measure space $(S,\mathcal{F},μ),$ and give an $\mathbb{L}$-valued analogue of the canonical $L^{p}(Ω,\mathcal{F},μ).$ Then we investigate the completeness of such an $\mathbb{L}$-valued analogue and propose the Radon-Nikod$\acute{y}$m property of $\mathbb{L}$-Banach spaces. Meanwhile, an example constructed in this paper shows that there do exist an $\mathbb{L}$-Banach space which fails to possess the Radon-Nikod$\acute{y}$m property. Finally, based on above work, we establish the dual representation theorem of $\mathbb{L}$-Bochner integrable function spaces, which extends and improves the corresponding classical result. |
| title | The Radon-Nikod$\acute{Y}$m property of $\mathbb{L}$-Banach spaces and the dual representation theorem of $\mathbb{L}$-Bochner function spaces |
| topic | Functional Analysis Primary 46B22, 46B10, Secondary 46E30 |
| url | https://arxiv.org/abs/2409.06279 |