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Isometries and additive mapping on the unit spheres of normed spaces

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Abstract

In this paper, we investigate isometric extension problem in general normed space. We prove that an isometry between spheres can be extended to a linear isometry between the spaces if and only if the natural positive homogeneous extension is additive on spheres. Moreover, this conclusion still holds provided that the additivity holds on a restricted domain of spheres.

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Acknowledgements

The authors express their deep gratitude to Professor Ding Guanggui for his encouragement and advice.

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Authors and Affiliations

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

    Rui Dong Wang & Xu Jian Huang

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  1. Rui Dong Wang
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  2. Xu Jian Huang
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Correspondence to Xu Jian Huang.

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Supported by National Natural Science Foundation of China (Grant Nos. 11301384, 11371201, 11201337 and 11201338)

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Wang, R.D., Huang, X.J. Isometries and additive mapping on the unit spheres of normed spaces. Acta. Math. Sin.-English Ser. 33, 1431–1442 (2017). https://doi.org/10.1007/s10114-017-6589-1

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  • DOI: https://doi.org/10.1007/s10114-017-6589-1

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