Inner Product Fuzzy Quasilinear Spaces and Some Fuzzy Sequence Spaces - Yılmaz Yılmaz; Hacer Bozkurt; Halise Levent; Ümit Çetinkaya (Journal of Mathematics)
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 Fℝn 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 Fℝn,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 Kℝndα=∫01a,bℝndα  :a∈uα,b∈vα. For u,v∈Fℝn. We also proved that the parallelogram law can only be provided in the regular subspace, not in the entire of Fℝn. Finally, we showed that a special class of fuzzy number sequences is a Hilbert quasilinear space.