On the Definition of Fuzzy Hilbert Spaces and Its Application

Abstract

Abstract. In this paper we introduce the notion of fuzzy Hilbert spaces and deduce the fuzzy version of Riesz representation theorem. Also we prove some results in fixed point theory and utilize the results to study the existence and uniqueness of solution of Uryson's integral equation.

Fuzzy inner product spaces and fuzzy orthogonality

“…One should note that the condition (FIP-4) in Definition 2.4 is not correct for α < 0. So, we were unable to validate the results of [10] by using this definition. In other words, if α < 0, then the condition (FIP-4) is true for x, y ∈ X.…”

“…In this paper, we first bring some definitions and results from the theory of fuzzy normed spaces and FIP-spaces and then correct the definition of fuzzy inner product space which is introduced by Goudarzi and Vaezpour in [10]. Furthermore, we define an inner product on FIP-spaces and study some basic properties of these spaces.…”

Absence of Non-Trivial Fuzzy Inner Product Spaces and the Cauchy–Schwartz Inequality

“…The concept of a fuzzy inner product was first introduced by Goudarzi et al [6]. Since their works, a study for fuzzy inner product has been progressed actively [6][7][8]. In 2010, Hasankhani et al defined another fuzzy inner product that arises from Felbin's fuzzy norm.…”

Some Expansion of Fuzzy Paranormal Operators

“…Riesz theorem was given by Youngfsu [9] using fuzzy concept in 2007. In 2009 Goudarzi and Vaezpour [6] have been introduced the definition of fuzzy Hilbert space. They introduced triplet ( ℍ, ,*), where ℍ is the Fuzzy Hilbert space, is the Fuzzy set on ℍ 2 × ℝ, and * is continuous tnorm.…”