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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

On the Relation Between Distances and Seminorms on Fréchet Spaces, with Application to Isometries

by
Isabelle Chalendar
1,*,†,
Lucas Oger
1,† and
Jonathan R. Partington
2,†
1
Université Gustave Eiffel, LAMA, (UMR 8050), UPEM, UPEC, CNRS, F-77454 Marne-la-Vallée, France
2
School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
*
Author to whom correspondence should be addressed.
†
These authors contributed equally to this work.
Mathematics 2025, 13(13), 2053; https://doi.org/10.3390/math13132053
Submission received: 26 May 2025 / Revised: 16 June 2025 / Accepted: 18 June 2025 / Published: 20 June 2025
(This article belongs to the Section C4: Complex Analysis)
Versions Notes

Abstract

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.
Keywords: fréchet space; isometry; distance; operator theory; Banach–Stone theorem fréchet space; isometry; distance; operator theory; Banach–Stone theorem

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MDPI and ACS Style

Chalendar, I.; Oger, L.; Partington, J.R. On the Relation Between Distances and Seminorms on Fréchet Spaces, with Application to Isometries. Mathematics 2025, 13, 2053. https://doi.org/10.3390/math13132053

AMA Style

Chalendar I, Oger L, Partington JR. On the Relation Between Distances and Seminorms on Fréchet Spaces, with Application to Isometries. Mathematics. 2025; 13(13):2053. https://doi.org/10.3390/math13132053

Chicago/Turabian Style

Chalendar, Isabelle, Lucas Oger, and Jonathan R. Partington. 2025. "On the Relation Between Distances and Seminorms on Fréchet Spaces, with Application to Isometries" Mathematics 13, no. 13: 2053. https://doi.org/10.3390/math13132053

APA Style

Chalendar, I., Oger, L., & Partington, J. R. (2025). On the Relation Between Distances and Seminorms on Fréchet Spaces, with Application to Isometries. Mathematics, 13(13), 2053. https://doi.org/10.3390/math13132053

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