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Inner Product Fuzzy Quasilinear Spaces and Some Fuzzy Sequence Spaces

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  • Yılmaz Yılmaz
  • Hacer Bozkurt
  • Halise Levent
  • Ãœmit Çetinkaya
  • Francisco J. Garcia Pacheco

Abstract

It has been shown that the class of fuzzy sets has a quasilinear space structure. In addition, various norms are defined on this class, and it is given that the class of fuzzy sets is a normed quasilinear space with these norms. In this study, we first developed the algebraic structure of the class of fuzzy sets F℠n and gave definitions such as quasilinear independence, dimension, and the algebraic basis in these spaces. Then, with special norms, namely, uq=∫01supx∈uαxqdα1/q where 1≤q≤∞, we stated that F℠n,uq is a complete normed space. Furthermore, we introduced an inner product in this space for the case q=2. The inner product must be in the formu,v=∫01uα,vα K℠ndα=∫01a,b℠ndα  :a∈uα,b∈vα. For u,v∈F℠n. We also proved that the parallelogram law can only be provided in the regular subspace, not in the entire of F℠n. Finally, we showed that a special class of fuzzy number sequences is a Hilbert quasilinear space.

Suggested Citation

  • Yılmaz Yılmaz & Hacer Bozkurt & Halise Levent & Ãœmit Çetinkaya & Francisco J. Garcia Pacheco, 2022. "Inner Product Fuzzy Quasilinear Spaces and Some Fuzzy Sequence Spaces," Journal of Mathematics, Hindawi, vol. 2022, pages 1-15, July.
    • Handle: RePEc:hin:jjmath:2466817
    DOI: 10.1155/2022/2466817
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