On the Relation Between Distances and Seminorms on Fréchet Spaces, with Application to Isometries
A study is made of linear isometries on Fréchet spaces for which the metric is given in terms of a sequence of seminorms. This establishes sufficient conditions on the growth of the function that defines the metric in terms of the seminorms to ensure that a linear operator preserving the metric also...
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| Language: | English |
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MDPI AG
2025-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/13/13/2053 |
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| author | Isabelle Chalendar Lucas Oger Jonathan R. Partington |
| author_facet | Isabelle Chalendar Lucas Oger Jonathan R. Partington |
| author_sort | Isabelle Chalendar |
| collection | DOAJ |
| description | A study is made of linear isometries on Fréchet spaces for which the metric is given in terms of a sequence of seminorms. This establishes sufficient conditions on the growth of the function that defines the metric in terms of the seminorms to ensure that a linear operator preserving the metric also preserves each of these seminorms. As an application, characterizations are given of the isometries on various spaces including those of holomorphic functions on complex domains and continuous functions on open sets, extending the Banach–Stone theorem to surjective and nonsurjective cases. |
| format | Article |
| id | doaj-art-eb8046dcc69845a9a28a505f12d40aae |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-eb8046dcc69845a9a28a505f12d40aae2025-08-20T03:16:47ZengMDPI AGMathematics2227-73902025-06-011313205310.3390/math13132053On the Relation Between Distances and Seminorms on Fréchet Spaces, with Application to IsometriesIsabelle Chalendar0Lucas Oger1Jonathan R. Partington2Université Gustave Eiffel, LAMA, (UMR 8050), UPEM, UPEC, CNRS, F-77454 Marne-la-Vallée, FranceUniversité Gustave Eiffel, LAMA, (UMR 8050), UPEM, UPEC, CNRS, F-77454 Marne-la-Vallée, FranceSchool of Mathematics, University of Leeds, Leeds LS2 9JT, UKA study is made of linear isometries on Fréchet spaces for which the metric is given in terms of a sequence of seminorms. This establishes sufficient conditions on the growth of the function that defines the metric in terms of the seminorms to ensure that a linear operator preserving the metric also preserves each of these seminorms. As an application, characterizations are given of the isometries on various spaces including those of holomorphic functions on complex domains and continuous functions on open sets, extending the Banach–Stone theorem to surjective and nonsurjective cases.https://www.mdpi.com/2227-7390/13/13/2053fréchet spaceisometrydistanceoperator theoryBanach–Stone theorem |
| spellingShingle | Isabelle Chalendar Lucas Oger Jonathan R. Partington On the Relation Between Distances and Seminorms on Fréchet Spaces, with Application to Isometries Mathematics fréchet space isometry distance operator theory Banach–Stone theorem |
| title | On the Relation Between Distances and Seminorms on Fréchet Spaces, with Application to Isometries |
| title_full | On the Relation Between Distances and Seminorms on Fréchet Spaces, with Application to Isometries |
| title_fullStr | On the Relation Between Distances and Seminorms on Fréchet Spaces, with Application to Isometries |
| title_full_unstemmed | On the Relation Between Distances and Seminorms on Fréchet Spaces, with Application to Isometries |
| title_short | On the Relation Between Distances and Seminorms on Fréchet Spaces, with Application to Isometries |
| title_sort | on the relation between distances and seminorms on frechet spaces with application to isometries |
| topic | fréchet space isometry distance operator theory Banach–Stone theorem |
| url | https://www.mdpi.com/2227-7390/13/13/2053 |
| work_keys_str_mv | AT isabellechalendar ontherelationbetweendistancesandseminormsonfrechetspaceswithapplicationtoisometries AT lucasoger ontherelationbetweendistancesandseminormsonfrechetspaceswithapplicationtoisometries AT jonathanrpartington ontherelationbetweendistancesandseminormsonfrechetspaceswithapplicationtoisometries |