A note on smooth rotund norms which are not midpoint locally uniformly rotund
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913778864488448 |
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| author | De Bernardi, Carlo Alberto Preti, Alessandro Somaglia, Jacopo |
| author_facet | De Bernardi, Carlo Alberto Preti, Alessandro Somaglia, Jacopo |
| contents | We prove that every separable infinite-dimensional Banach space admits a Gâteaux smooth and rotund norm which is not midpoint locally uniformly rotund. Moreover, by using a similar technique, we provide in every infinite-dimensional Banach space with separable dual a Fréchet smooth and weakly uniformly rotund norm which is not midpoint locally uniformly rotund. These two results provide a positive answer to some open problems by A. J. Guirao, V. Montesinos, and V. Zizler. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13869 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on smooth rotund norms which are not midpoint locally uniformly rotund De Bernardi, Carlo Alberto Preti, Alessandro Somaglia, Jacopo Functional Analysis We prove that every separable infinite-dimensional Banach space admits a Gâteaux smooth and rotund norm which is not midpoint locally uniformly rotund. Moreover, by using a similar technique, we provide in every infinite-dimensional Banach space with separable dual a Fréchet smooth and weakly uniformly rotund norm which is not midpoint locally uniformly rotund. These two results provide a positive answer to some open problems by A. J. Guirao, V. Montesinos, and V. Zizler. |
| title | A note on smooth rotund norms which are not midpoint locally uniformly rotund |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2402.13869 |