no. G8
Article de recherche - Analyse fonctionnelle
On the Scottish Book Problem 155 by Mazur and Sternbach
[Sur le problème 155 du Scottish Book par Mazur et Sternbach]
Mori, Michiya1, 2
1Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
2Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan
Comptes Rendus. Mathématique, Tome 362 (2024) no. G8, pp. 813-816

Problem 155 of the Scottish Book asks whether every bijection U:X→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→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.

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DOI : 10.5802/crmath.572
Classification : 46B04

Mori, Michiya  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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {On the {Scottish} {Book} {Problem} 155 by {Mazur} and {Sternbach}},
     journal = {Comptes Rendus. Math\'ematique},
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Mori, Michiya. On the Scottish Book Problem 155 by Mazur and Sternbach. Comptes Rendus. Mathématique, Tome 362 (2024) no. G8, pp. 813-816. doi: 10.5802/crmath.572

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

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

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

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

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