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Embedding Cross-Dimensional Vector Space into ℓ2

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Abstract

The cross-dimensional dynamical systems have received increasing research attention in recent years. This paper characterizes the structure features of the cross-dimensional vector space. Specifically, it is proved that the completion of cross-dimensional vector space is an infinite-dimensional separable Hilbert space. Hence, it means that one can isometrically and linearly embed the cross-dimensional vector space into the ℓ2, which is known as the space of square summable sequences. This result will be helpful in the modeling and analyzing the dynamics of cross-dimensional dynamical systems.

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Authors and Affiliations

  1. School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an, 710000, China

    Shengli Xue & Lijun Zhang

  2. School of Mathematics and Statistics, Yulin University, Yulin, 719000, China

    Shengli Xue, Zhiqi Xie & Weijun Yan

  3. Department of Computer Science, University of Surrey, GU27XH, Guildford, UK

    Kuize Zhang

Authors
  1. Shengli Xue
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  2. Lijun Zhang
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  3. Zhiqi Xie
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  4. Weijun Yan
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  5. Kuize Zhang
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Corresponding author

Correspondence to Lijun Zhang.

Ethics declarations

ZHANG Kuize is a youth editorial board member for Journal of Systems Science & Complexity and was not involved in the editorial review or the decision to publish this article. All authors declare that there are no competing interests.

Additional information

This work was supported by the National Natural Science Foundation of China under Grant No. 61673129, and the Key Programs in Shaanxi Province of China under Grant No. 2021JZ-12, and Science and the Technology Bureau Project of Yulin under Grant Nos. 2019-89-2 and 2019-89-4.

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Xue, S., Zhang, L., Xie, Z. et al. Embedding Cross-Dimensional Vector Space into ℓ2. J Syst Sci Complex 36, 2309–2324 (2023). https://doi.org/10.1007/s11424-023-2303-9

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  • DOI: https://doi.org/10.1007/s11424-023-2303-9

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