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Examples, properties and applications of fuzzy inner product spaces

  • Fuzzy systems and mathematics
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Abstract

In this paper a new definition for the fuzzy inner product spaces of the type Kramosil-Michálek and the type George-Veeramani is presented. In the new setting a fuzzy inner product space can naturally become a fuzzy normed space, and a classical inner product space can be considered as a special case of fuzzy inner product spaces. Several examples are given to illustrate that, the new definition is a nontrivial generalization for classical inner product spaces, and so it has rich contents in fuzziness. By virtue of this definition, some elementary properties are described in terms of the families of semi-inner products and a fuzzy version of Pythagorean theorem is given. As applications, a fixed point theorem for nonlinear contractions is established and the existence of solution of global optimization problem is obtained.

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Acknowledgements

The authors are grateful to the referees and the area editor for their comments and suggestions, which helped to improve the manuscript.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, People’s Republic of China

    Jian-Zhong Xiao & Ying Lu

  2. School of Science, Wuxi University, Wuxi, 214105, People’s Republic of China

    Feng-Qin Zhu

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  1. Jian-Zhong Xiao
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  2. Ying Lu
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  3. Feng-Qin Zhu
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All the authors contributed equally to this research paper.

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Correspondence to Jian-Zhong Xiao.

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Xiao, JZ., Lu, Y. & Zhu, FQ. Examples, properties and applications of fuzzy inner product spaces. Soft Comput 27, 239–256 (2023). https://doi.org/10.1007/s00500-022-07584-w

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