Inner Product Fuzzy Quasilinear Spaces and Some Fuzzy Sequence Spaces


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YILMAZ Y., Bozkurt H., Levent H., Cetinkaya U.

JOURNAL OF MATHEMATICS, vol.2022, 2022 (Peer-Reviewed Journal) identifier

  • Publication Type: Article / Article
  • Volume: 2022
  • Publication Date: 2022
  • Doi Number: 10.1155/2022/2466817
  • Journal Name: JOURNAL OF MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded, Scopus

Abstract

t 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 thealgebraic structure of the class of fuzzy setsF(Rn)and gave definitions such as quasilinear independence, dimension, and thealgebraic basis in these spaces. .en, with special norms, namely,?u?q = ( integral(1)(0)(sup(x is an element of[u]alpha)?x?)(q)d alpha)(1/q )where 1 <= q <=infinity, we stated that (F(R-n),?u?(q))is a complete normed space. Furthermore, we introduced an inner product in this space for the case q=2. .e innerproduct must be in the form = integral(1)(0)<[u](alpha),[v](alpha)>(K(Rn))d alpha=integral(1)(0){(Rn)d alpha:a is an element of[u](alpha),b is an element of[v](alpha)}. Foru,v is an element of F(Rn). We alsoproved that the parallelogram law can only be provided in the regular subspace, not in the entire ofF(Rn).Finally, we showed thata special class of fuzzy number sequences is a Hilbert quasilinear space.