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 ℝ n and gave definitions such as quasilinear independence, dimension, and the algebraic basis in these spaces. Then, with special norms, namely, u q = ∫ 0 1 sup x ∈ u α x q d α 1 / q where 1 ≤ q ≤ ∞ , 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 . The inner product must be in the form u , v = ∫ 0 1 u α , v α K ℝ n d α = ∫ 0 1 a , b ℝ n d α : a ∈ u α , b ∈ v α . For u , v ∈ F ℝ n . We also proved that the parallelogram law can only be provided in the regular subspace, not in the entire of F ℝ n . Finally, we showed that a special class of fuzzy number sequences is a Hilbert quasilinear space.

On the Algebra of Interval Vectors

“…For example, in [17,18] we gave examples of how quasilinear spaces can be used in signal processing. In addition, normed and Hilbert quasilinear space examples of some fuzzy number sets are given in [19] and their properties are examined. A recent study on qasilinear spaces is the concept of quasi-algebra its details can be found in [20,21].…”

On Soft Normed Quasilinear Spaces

“…Then, quasilinear inner product spaces, generalizations of inner product spaces, were defined to develop quasilinear functional analysis in [7][8][9][10]. In addition, Yılmaz et al [11] demonstrated that Hilbert quasilinear spaces are a special class of fuzzy number sequences. In [12,13], Levent and Yılmaz included some quasilinear applications, such as signal processing.…”