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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.

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