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A Tingley’s type problem in n-normed spaces

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Abstract

This paper solves a Tingley’s type problem in n-normed spaces and states that for \(n\ge 2\), every n-isometry on the unit sphere of an n-normed space is an n-isometry on the whole space except the origin 0. Also, using analytical approach we give a short proof to show that for \(n\ge 3\), any mapping which preserves n-norms of values one and zero is, up to pointwise multiplication by \(\pm 1\)-, a linear n-isometry. This gives a Wigner-type theorem in n-normed spaces which was proven in a recent paper.

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Acknowledgements

The authors wish to express their appreciation to Guanggui Ding for many very helpful comments regarding isometric theory in Banach spaces.

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  1. Department of Mathematics, Tianjin University of Technology, 300384, Tianjin, China

    Xujian Huang & Dongni Tan

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  1. Xujian Huang
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  2. Dongni Tan
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Correspondence to Dongni Tan.

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The authors are supported by the Natural Science Foundation of China (Grant Nos. 11371201, 11201337, 11201338, 11301384).

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Huang, X., Tan, D. A Tingley’s type problem in n-normed spaces. Aequat. Math. 93, 905–918 (2019). https://doi.org/10.1007/s00010-019-00637-w

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