Inner Product Fuzzy Quasilinear Spaces and Some Fuzzy Sequence Spaces

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(ℝn) and gave definitions such as quasilinear independence, dimension, and the algebraic basis in these spaces. Then, with special norms, namely, ‖u‖q = where 1 ≤ q ≤ ∞, we stated that (F(ℝn), ‖u‖q) 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 form. For u, v ∈ F(ℝn). We also proved that the parallelogram law can only be provided in the regular subspace, not in the entire of F(ℝn). Finally, we showed that a special class of fuzzy number sequences is a Hilbert quasilinear space.