Fuzzy Real Pre-Hilbert Space and Some of Their Properties

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Mayada N. Mohammedali


In this work, two different structures are proposed which is fuzzy real normed space (FRNS) and fuzzy real Pre-Hilbert space (FRPHS). The basic concept of fuzzy norm on a real linear space is first presented to construct  space, which is a FRNS with some modification of the definition introduced by G. Rano and T. Bag. The structure of fuzzy real Pre-Hilbert space (FRPHS) is then presented which is based on the structure of FRNS. Then, some of the properties and related concepts for the suggested space FRN such as -neighborhood, closure of the set  named , the necessary condition for separable, fuzzy linear manifold (FLM) are discussed. The definition for a fuzzy seminorm on  is also introduced with the prove that a fuzzy seminorm on  is FRNS. The relationship between the -convergent sequence, -Cauchy sequence and -completeness is then investigated in this work. The structure of FRPHS with some important properties concerning on this space are introduced and proved. In addition, the property of orthogonality with some important properties for these spaces is included, for example the annihilator of the set . The relation between FRPHS and FRNS is investigated in the present work. Finally, after introducing the structure of FRPHS, it leads naturally to the definition of the most important class of FRPHS, namely the fuzzy real Hilbert space (FRHS).


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Mohammedali MN. Fuzzy Real Pre-Hilbert Space and Some of Their Properties. Baghdad Sci.J [Internet]. 2022Apr.1 [cited 2022Jun.26];19(2):0313. Available from: https://bsj.uobaghdad.edu.iq/index.php/BSJ/article/view/4776


Rano G, Bag T. Fuzzy Normed Linear Spaces. Int J Compu Math. 2012; 2(2):16-19.

Goguen JA. L-Fuzzy Sets. J Math Anal Appl. 1967; 18:145-174.

Miin-Shen Y, Zahid H. Fuzzy Entropy for Pythagorean Fuzzy Sets with Application to Multicriterion Decision Making. Hindawi. 2018 Oct; 1-14.

Humberto B, Javier F, Esteban I. Introduction to Special Issue. New Trends in Fuzzy Set Theory and Related Items. 2018 Jun; 7(37):1-2.

Abbas M, Mehrbakhsh N, Jurgita A, Madjid T, Romualdas B, Othman I. Recent Fuzzy Generalisations of Rough Sets Theory. A Systematic Review and Methodological Critique of the Literature. Complexity, 2017 Oct; 1-33.

Chao Z, Deyu L, Yimin M, Dong S. An interval-valued hesitant fuzzy multigranulation rough set over two universes model for steam turbine fault diagnosis. Appl. Math. Model. 2017 Feb; 42:693–704.

Soni M, Mene S, Singh MM. Fuzzy Set Theory in Sociology. IJRASET. 2017 Sep; 5(IX):1148-1151.

Antonovich MV, Druzhina OS, Serebryakova VO, Butusov DN, Kopets EE. The Analysis of Oscillations in Chaotic Circuit with Sensitive Inductive Coil. In2019 IEEE Conference of Russian Young Researchers in Electrical and Electronic Engineering (EIConRus) 2019 Jan 28 (pp. 61-65). IEEE. 2019 Jan; 61–65.

Zhao J, Lin CM, Chao F. Wavelet Fuzzy Brain Emotional Learning Control System Design for MIMO Uncertain Nonlinear Systems. Front. Neurosci. 2018; 12, 918.

Felbin C. The completion of a fuzzy normed linear space. J Math Anal Appl. 1993; 174(2):428–440.

Cheng SC, Mordeson JN. Fuzzy linear operators and fuzzy normed linear spaces. News Bull. Calcutta Math. Soc.1994; 86(5):429–436.

Sorin N. Fuzzy pseudo-norms and fuzzy F-spaces. Fuzzy Set Syst. 2016 Jan; 282:99–114.

Bag T, Samanta SK. A comparative study of fuzzy norms on a linear space. Fuzzy Set Syst. 2008; 159(6): 670–684.

Jebril IH, Samanta TK. Fuzzy anti-normed linear space. Journal of Mathematics and Technology. 2010; 66–77.

Schweizer B, Sklar A. Statistical metric spaces. Pac J Math. 1960 Sep; 10(1):313–334.

Biswas R. Fuzzy inner product spaces and fuzzy norm function. Inf. Sci. 1991; 53:185–190.

El-Abyad A M, El-Hamouly H M. Fuzzy inner product spaces. Fuzzy Set Syst. 1991; 44(2):309–326.

Pinaki M, Samanta SK. On fuzzy inner product spaces. J. Fuzzy Math. 2008; 16(2): 377–392.

Hasankhani A, Nazari A, Saheli M. Some properties of fuzzy Hilbert spaces and norm of operators. Iran J Fuzzy Syst. 2010; 7(3):129–157.

Mukherjee S, Bag T. Fuzzy real inner product space and its properties. Ann. Fuzzy Math. Inform. 2013; 6(2): 377–389.

Alizadeh HN. A Characterization for Fuzzy Normed Linear Spaces and its Applications. J. Math. Ext. 2017 Feb; 11(3): 19-30

Vijayabalaji S, Thillaigovindan N, Bae J Y. Intuitionistic fuzzy n-normed linear space. Bull. Korean Math. Soc. 2007; 44:291-308.

Nazarkandi H A. A Characterization for Fuzzy Normed Linear Spaces and its Applications. J. Math. Ext. 2017 Feb; 11(3):19-30.

Daraby B, Khosravi N, Rahimi A. Weak fuzzy topology on fuzzy topological vector spaces. Cornell University. 2019 Oct; 1-18.

Gani AN, Kannan K, Manikandan AR. Inner Product over Fuzzy Matrices. Journal of Mathematics. Hindawi. 2016 Feb; 1-10.

Hacer B, Yilmaz Y. Some new results on inner product quasilinear spaces. Congent Mathematics. 2016 Jun; 3(1194801):1-10.