Pointwise semi-Lipschitz functions and Banach-Stone theorems
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arXiv
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| Format: | Preprint |
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2024
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| author | Durand-Cartagena, Estíbalitz Jaramillo, Jesús Á. M, Francisco Venegas |
| author_facet | Durand-Cartagena, Estíbalitz Jaramillo, Jesús Á. M, Francisco Venegas |
| contents | We study the fundamental properties of pointwise semi-Lipschitz functions between asymmetric spaces, which are the natural asymmetric counterpart of pointwise Lipschitz functions. We also study the influence that partial symmetries of a given space may have on the behavior of pointwise semi-Lipschitz functions defined on it. Furthermore, we are interested in characterizing the pointwise semi-Lipschitz structure of an asymmetric space in terms of real-valued pointwise semi-Lipschitz functions defined on it. By using two algebras of functions naturally associated to our spaces of pointwise real-valued semi-Lipschitz functions, we are able to provide two Banach-Stone type results in this context. In fact, these results are obtained as consequences of a general Banach-Stone type theorem of topological nature, stated for abstract functional spaces, which is quite flexible and can be applied to many spaces of continuous functions over metric and asymmetric spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_17574 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pointwise semi-Lipschitz functions and Banach-Stone theorems Durand-Cartagena, Estíbalitz Jaramillo, Jesús Á. M, Francisco Venegas Functional Analysis 46T99, 30L99, 51F30 We study the fundamental properties of pointwise semi-Lipschitz functions between asymmetric spaces, which are the natural asymmetric counterpart of pointwise Lipschitz functions. We also study the influence that partial symmetries of a given space may have on the behavior of pointwise semi-Lipschitz functions defined on it. Furthermore, we are interested in characterizing the pointwise semi-Lipschitz structure of an asymmetric space in terms of real-valued pointwise semi-Lipschitz functions defined on it. By using two algebras of functions naturally associated to our spaces of pointwise real-valued semi-Lipschitz functions, we are able to provide two Banach-Stone type results in this context. In fact, these results are obtained as consequences of a general Banach-Stone type theorem of topological nature, stated for abstract functional spaces, which is quite flexible and can be applied to many spaces of continuous functions over metric and asymmetric spaces. |
| title | Pointwise semi-Lipschitz functions and Banach-Stone theorems |
| topic | Functional Analysis 46T99, 30L99, 51F30 |
| url | https://arxiv.org/abs/2407.17574 |