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 IJFSA.2015040104.m01 and IJFSA.2015040104.m02 means fuzzy topological spaces (fts). The notions Cl (A) will stand for the fuzzy closure of a fuzzy set A in a fts IJFSA.2015040104.m03. Support of a fuzzy set A in IJFSA.2015040104.m04 will be denoted by S (A). The fuzzy sets in IJFSA.2015040104.m05 taking on respectively the constant value 0 and 1 are denoted by IJFSA.2015040104.m06 and IJFSA.2015040104.m07 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 IJFSA.2015040104.m08and IJFSA.2015040104.m09of IJFSA.2015040104.m10, the fuzzy subset IJFSA.2015040104.m11is defined by

    IJFSA.2015040104.m12

And for a scalar IJFSA.2015040104.m13 of IJFSA.2015040104.m14 and a fuzzy subset IJFSA.2015040104.m15 of IJFSA.2015040104.m16, the fuzzy subset IJFSA.2015040104.m17 is defined by

  • Definition 1.2: [2] IJFSA.2015040104.m19 is said to be

    • 1.

      Convex if IJFSA.2015040104.m20

    • 2.

      Balanced if IJFSA.2015040104.m21

    • 3.

      Absorbing if IJFSA.2015040104.m22

  • Definition 1.3: [2] Let IJFSA.2015040104.m23 be a topological space and

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