Some aspects of norm and inner product on fuzzy linear spaces over fuzzy field and interretionship between them

Abstract

The fundamental concept of fuzzy set was introduced by L. A. Zadeh [146] in 1965 and fuzzy topology was introduced by C.L. Chang [24] in 1968. There after many researchers introduced the notions of fuzzy norm and fuzzy inner product from different point of view. In 1984 Katsaras [82] defined a fuzzy norm on a linear space and there after Wu and Fang [136] introduced a fuzzy normed space. R. Biswas [6] in 1991 defined fuzzy norm and fuzzy inner product of elements on a linear space. In 1992, Felbin [51] introduced fuzzy norm on a linear space by assigning a fuzzy real number to each element of the linear space. Another important approach of fuzzy norm on a linear space was introduced in 1994, by Cheng and Mordeson [25], on a parallel line as the corresponding fuzzy metric is of Kramosil and Michelek [87] type. There after Krishna and Sarma [86], Xiao and zhu [144] discussed fuzzy norms on linear spaces at different points of aspects. All these researchers have done their work in the area of crisp linear space. Gu Wenxiang and Lu Tu [138] were the first to introduce the concept of fuzzy fields and fuzzy linear spaces over fuzzy fields. In 2011, C.P. Santhosh and T.V. Ramakrishnan [122] introduced norm on fuzzy linear space over fuzzy field. In 2018, Noori F. AL-Mayahi and Suadad M. Abbas [4] defined fuzzy normed algebra over fuzzy field. A satisfactory theory of 2-inner product space and n-inner product space has been effectively constructed by C.R. Diminnie, S. Gähler and A. White [36]. In 2005, Bag and Samanta [10], introduced an idea of fuzzy norm of a linear operator from a fuzzy normed linear space to another fuzzy normed linear space and defined various notions of continuities and boundedness of linear operators over fuzzy normed linear spaces such as fuzzy continuity, sequential fuzzy continuity, weakly fuzzy continuity, strongly fuzzy continuity, weakly and strongly fuzzy boundedness. In 2012, Srinivas, Narasimha Swamy, and Nagaiah [124] introduced anti-fuzzy near-algebras over anti-fuzzy fields. In 2022, B