Bounded and Semi Bounded Inverse Theorems in Fuzzy Normed Spaces

Bounded and Semi Bounded Inverse Theorems in Fuzzy Normed Spaces

Hamid Reza Moradi (Islamic Azad University, Mashhad, Iran)
Copyright: © 2015 |Pages: 9
DOI: 10.4018/IJFSA.2015040104

Abstract

In this paper, the author introduces the notion of the complete fuzzy norm on a linear space. And the author considers some relations between the fuzzy completeness and ordinary completeness on a linear space, moreover a new form of fuzzy compact spaces, namely b-compact spaces, b-closed space is introduced. Some characterization of their properties is obtained. Also some basic properties for linear operators between fuzzy normed spaces are further studied. The notions of fuzzy vector spaces and fuzzy topological vector spaces were introduced in Katsaras and Liu (1977). These ideas were modified by Katsaras (1981), and in (1984) Katsaras defined the fuzzy norm on a vector space. In (1991) Krishna and Sarma discussed the generation of a fuzzy vector topology from an ordinary vector topology on vector spaces. Also Krishna and Sarma (1992) observed the convergence of sequence of fuzzy points. Rhie et al. (1997) Introduced the notion of fuzzy a-Cauchy sequence of fuzzy points and fuzzy completeness.
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1. Introduction And Preliminaries

The notions of fuzzy vector spaces and fuzzy topological vector spaces were introduced in Katsaras and Liu (1977). These ideas were modified by Katsaras (1981), and in (1984) Katsaras defined the fuzzy norm on a vector space. In (1991) Krishna and Sarma discussed the generation of a fuzzy vector topology from an ordinary vector topology on vector spaces. Also Krishna and Sarma (1992) observed the convergence of sequence of fuzzy points. Rhie et al. (1997) Introduced the notion of fuzzy Cauchy sequence of fuzzy points and fuzzy completeness.

Throughout this paper and means fuzzy topological spaces (fts). The notions Cl (A) will stand for the fuzzy closure of a fuzzy set A in a fts . Support of a fuzzy set A in will be denoted by S (A). The fuzzy sets in taking on respectively the constant value 0 and 1 are denoted by and respectively.

In this paper, we first observe a type of the convergence of sequences as an analogy of Bag and Samanta (2003) in a fuzzy normed linear space. Secondly, we introduce the notion of a complete fuzzy norm, using the convergence of a sequence of a linear space. And we consider some relations between the fuzzy completeness and the ordinary completeness on a linear space.

  • Definition 1.1: [4] For two fuzzy subsets and of , the fuzzy subset is defined by

And for a scalar of and a fuzzy subset of , the fuzzy subset is defined by

  • Definition 1.2: [2] is said to be

    • 1.

      Convex if

    • 2.

      Balanced if

    • 3.

      Absorbing if

  • Definition 1.3: [2] Let be a topological space and

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