Home Decomposition in direct sum of seminormed vector spaces and Mazur–Ulam theorem
Article
Licensed
Unlicensed Requires Authentication

Decomposition in direct sum of seminormed vector spaces and Mazur–Ulam theorem

  • Oleksiy Dovgoshey EMAIL logo , Jürgen Prestin and Igor Shevchuk
Published/Copyright: May 13, 2024
Become an author with De Gruyter Brill

Abstract

It was proved by S. Mazur and S. Ulam in 1932 that every isometric surjection between normed real vector spaces is affine. We generalize the Mazur–Ulam theorem and find necessary and sufficient conditions under which distance-preserving mappings between seminormed real vector spaces are linear.


Oleksiy Dovgoshey was supported by Volkswagen Stiftung Project “From Modeling and Analysis to Approximation”


Acknowledgement

We would like to thank the anonymous referee for the very careful reading of the paper and many helpful suggestions and improvements.

  1. (Communicated by Marcus Waurick)

References

[1] Baker, J. A.: Isometries in normed spaces, Amer. Math. Monthly 78 (1971), 655–658.Search in Google Scholar

[2] Bilet, V.—Dovgoshey, O.: Pseudometric spaces. From minimality to maximality in the groups of combinatorial self-similarities, https://arxiv.org/abs/2304.03822.Search in Google Scholar

[3] Bilet, V.—Dovgoshey, O.: When all permutations are combinatorial similarities, Bull. Korean Math. Soc. 60 (2023), 733–746.Search in Google Scholar

[4] Dovgoshey, O.: Combinatorial properties of ultrametrics and generalized ultrametrics, Bull. Belg. Math. Soc. Simon Stevin 27 (2020), 379–417.Search in Google Scholar

[5] Dovgoshey, O.—Luukkainen, J.: Combinatorial characterization of pseudometrics, Acta Math. Hungar. 161 (2020), 257–291.Search in Google Scholar

[6] Gudder, S.—Strawther, D.: Strictly convex normed linear spaces, Proc. Amer. Math. Soc. 59 (1976), 263–267.Search in Google Scholar

[7] Kelley, J. L.: General Topology . Grad. Texts in Math., vol. 27, Springer-Verlag, 1975.Search in Google Scholar

[8] Kurepa, Đ.: Tableaux ramifiés d’ensemples, espaces pseudodistacies, C. R. Acad. Sci. Paris 198 (1934), 1563–1565.Search in Google Scholar

[9] Mazur, S.—Ulam, S. M.: Sur les transformations isométriques d’espaces vectoriels normés, C. R. Acad. Sci. Paris 194 (1932), 946–948.Search in Google Scholar

[10] Nica, B.: The Mazur-Ulam theorem, Expo. Math. 30 (2012), 397–398.Search in Google Scholar

[11] Przeworska-Rolewicz, D.—Rolewicz, S.: Equations in Linear Spaces . Monografie Matematyczne, vol. 47, Państwowe Wydawnictwo Naukowe, Warszawa, 1968.Search in Google Scholar

[12] Rudin, V.: Functional Analysis, McGraw-Hill, Inc., New York, 1991.Search in Google Scholar

[13] Väisälä, J.: A proof of the Mazur–Ulam theorem, Amer. Math. Monthly 110 (2003), 633–635.Search in Google Scholar

Received: 2022-10-22
Accepted: 2023-08-15
Published Online: 2024-05-13
Published in Print: 2024-02-26

© 2024 Mathematical Institute Slovak Academy of Sciences

Downloaded on 26.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2024-0010/html
Scroll to top button