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Isometry and phase-isometry of non-Archimedean normed spaces

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Abstract

In this paper, we study isometries and phase-isometries of non-Archimedean normed spaces. We show that every isometry f : Sr (X) → Sr (X), where X is a finite-dimensional non-Archimedean normed space and Sr(X) is a sphere with radius r ∈ ∥X∥, is surjective if and only if \(\mathbb{K}\) is spherically complete and k is finite. Moreover, we prove that if X and Y are non-Archimedean normed spaces over non-trivially non-Archimedean valued fields with |2| = 1, any phase-isometry f: X → Y is phase equivalent to an isometric operator.

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Authors and Affiliations

  1. College of Science, Tianjin University of Technology, Tianjin, 300384, China

    Ruidong Wang & Wenting Yao

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  1. Ruidong Wang
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  2. Wenting Yao
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Correspondence to Ruidong Wang.

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The authors declare no conflict of interest.

Additional information

Wang’s research was supported by the Natural Science Foundation of China (12271402) and the Natural Science Foundation of Tianjin City (22JCYBJC00420).

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Wang, R., Yao, W. Isometry and phase-isometry of non-Archimedean normed spaces. Acta Math Sci 43, 2377–2386 (2023). https://doi.org/10.1007/s10473-023-0603-8

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  • DOI: https://doi.org/10.1007/s10473-023-0603-8

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