Fuzzy soft hilbert spaces

In this work, we define the fuzzy soft Hilbert space ˜H based on the definition of the fuzzy soft inner product space (Ũ, ˜<; ·, · >), introduced by Faried et al. [N. Faried, M. S. S. Ali, H. H. Sakr, Appl. Math. Inf. Sci., 14 (2020), 709–720], in terms of the fuzzy soft vector ṽ_fG(e) . Moreover, we show that Cⁿ (A), Rⁿ (A) and ℓ₂ (A) are suitable examples of fuzzy soft Hilbert spaces. In addition, it is proved that the fuzzy soft orthogonal complement of any non-empty fuzzy soft subset of ˜H is a fuzzy soft closed fuzzy soft subspace of ˜H and we study some of the fuzzy soft Hilbert spaces properties and some of the fuzzy soft inner product spaces properties. Furthermore, we introduce the definition of the fuzzy soft orthogonal family and the fuzzy soft orthonormal family and introduce examples satisfying them. Moreover, we present the fuzzy soft Bessel’s inequality and the fuzzy soft Parseval’s formula in this generalized setting.