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 $F\left({\mathrm{ℝ}}^n\right)$ and gave definitions such as quasilinear independence, dimension, and the algebraic basis in these spaces. Then, with special norms, namely, ${‖u‖}_q=$ ${\left({\displaystyle {\int }_0^1{\left({\text{sup}}_{x\in {\left[u\right]}^{\alpha }}‖x‖\right)}^q}d\alpha \right)}^{1/q}$ where $1\le q\le \infty$ , we stated that $\left(F\left({\mathrm{ℝ}}^n\right),{‖u‖}_q\right)$ 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 $〈u,v〉={\displaystyle {\int }_0^1〈{\left[u\right]}^{\alpha },{\left[v\right]}^{\alpha }〉{}_{K\left({\mathrm{ℝ}}^n\right)}}d\alpha =\left\{{\displaystyle {\int }_0^1{〈a,b〉}_{{\mathrm{ℝ}}^n}}d\alpha :a\in {\left[u\right]}^{\alpha },b\in {\left[v\right]}^{\alpha }\right\}$ . For $u,v\in F\left({\mathrm{ℝ}}^n\right)$ . We also proved that the parallelogram law can only be provided in the regular subspace, not in the entire of $F\left({\mathrm{ℝ}}^n\right).$ Finally, we showed that a special class of fuzzy number sequences is a Hilbert quasilinear space.