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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/2401.03211 |
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| _version_ | 1866916440097947648 |
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| author | Ghorbanalizadeh, Arash Seraji, Reza Roohi |
| author_facet | Ghorbanalizadeh, Arash Seraji, Reza Roohi |
| contents | In this paper, we investigate the geometric properties of the variable mixed Lebesgue-sequence space $\ell^{q(\cdot)} (L^{p(\cdot)})$ as a Banach space. We show that, if $ 1<q_-,p_-,q_+,p_+<\infty $, then $\ell^{q(\cdot)} (L^{p(\cdot)})$ is strictly and uniformly convex. We also prove that when $ 1\le q_-,p_-,q_+,p_+<\infty, $ the convergence in norm implies the convergence in measure, and under some conditions on exponents, the approximation identity holds in the space $ \ell^1(L^{\frac{p(\cdot)}{q(\cdot)}}) $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03211 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some Remarks on the Vector-Valued Variable Exponent Lebesgue Spaces $\ell^{q(\cdot)} (L^{p(\cdot)})$ Ghorbanalizadeh, Arash Seraji, Reza Roohi Functional Analysis 46E35 In this paper, we investigate the geometric properties of the variable mixed Lebesgue-sequence space $\ell^{q(\cdot)} (L^{p(\cdot)})$ as a Banach space. We show that, if $ 1<q_-,p_-,q_+,p_+<\infty $, then $\ell^{q(\cdot)} (L^{p(\cdot)})$ is strictly and uniformly convex. We also prove that when $ 1\le q_-,p_-,q_+,p_+<\infty, $ the convergence in norm implies the convergence in measure, and under some conditions on exponents, the approximation identity holds in the space $ \ell^1(L^{\frac{p(\cdot)}{q(\cdot)}}) $. |
| title | Some Remarks on the Vector-Valued Variable Exponent Lebesgue Spaces $\ell^{q(\cdot)} (L^{p(\cdot)})$ |
| topic | Functional Analysis 46E35 |
| url | https://arxiv.org/abs/2401.03211 |