On the Scottish Book Problem 155 by Mazur and Sternbach

Michiya Mori

doi:10.5802/crmath.572

Research article - Functional analysis

On the Scottish Book Problem 155 by Mazur and Sternbach

Michiya Mori ^1,²

¹ Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
² Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 813-816.

Abstract
Résumé

Problem 155 of the Scottish Book asks whether every bijection $U : X \to Y$ between two Banach spaces $X, Y$ with the property that, each point of $X$ has a neighborhood on which $U$ is isometric, is globally isometric on $X$ . We prove that this is true under the additional assumption that $X$ is separable and the weaker assumption of surjectivity instead of bijectivity.

Le problème 155 du Scottish Book demande si toute bijection $U : X \to Y$ entre deux espaces de Banach $X, Y$ ayant la propriété que chaque point de $X$ a un voisinage sur lequel U est isométrique, est globalement isométrique sur $X$ . Nous prouvons que ceci est vrai sous l’hypothèse supplémentaire que $X$ est séparable et l’hypothèse plus faible de surjectivité au lieu de bijectivité.

Received: 2023-08-14
Revised: 2023-09-17
Accepted: 2023-09-18
Published online: 2024-10-07

DOI: 10.5802/crmath.572

Classification: 46B04

Author's affiliations:

Michiya Mori ^{1, 2}

¹ Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
² Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Michiya Mori},
     title = {On the {Scottish} {Book} {Problem} 155 by {Mazur} and {Sternbach}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {813--816},
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     year = {2024},
     doi = {10.5802/crmath.572},
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Michiya Mori. On the Scottish Book Problem 155 by Mazur and Sternbach. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 813-816. doi : 10.5802/crmath.572. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.572/

References
Cited by

[1] P. Mankiewicz On extension of isometries in normed linear spaces, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astron. Phys., Volume 20 (1972), pp. 367-371 | MR | Zbl

[2] R. D. Mauldin The Scottish Book. Mathematics from the Scottish Café. With selected problems from the New Scottish Book, Birkhäuser/Springer, 2015 | DOI | Zbl

[3] S. Mazur; S. Ulam Sur les transformationes isométriques d’espaces vectoriels normés, C. R. Acad. Sci. Paris, Volume 194 (1932), pp. 946-948 | Zbl

[4] S. Willard General Topology, Dover Publications, Mineola, 2004 | MR | Zbl

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