Copmleteness of Fuzzy Gamma-M-Normed Linear Space

Abstract

This paper introduces the conception of fuzzy gamma-m-normed linear space in accordance with the theory of fuzzy n-normed linear space and it is proved that in a finite dimensional fuzzy gamma m-normed linear space, fuzzy m-normed linear spaces are the same, up to the fuzzy norm equivalence. Also the paper introduces fuzzy gamma-2-normed linear spaces, fuzzy right gamma-m-normed linear spaces and its properties and fuzzy gamma-m-normed linear space which can be analyzed by using the fuzzy n-normed linear space. The paper uses an applications with examples for algebraic operations of fuzzy set theory. The most important concepts fuzzy gamma ring, fuzzy gamma divison ring, fuzzy gamma vector space, fuzzy gamma ring have already been introduced. Using these concepts.  Using these concepts it initiated the fuzzy Gamma-m-normed linear space and suggested a theorem for the gamma norm function which is continuous. So far the earlier research has been done in the general t-norm in a fuzzy n-normed linear space and proven that if t-norm is chosen other than “minimum” then the decomposition theorem of a fuzzy norm into a family of crisp norms may not hold.  The paper identified completeness of fuzzy gamma m-normed linear space and constructed a norm function and satisfies the axioms of it fuzzy gamma m-normed linear space and additionally provided an example with proof in which a sequence is a Cauchy sequence and converges sequence in fuzzy gamma m-normed linear space if and only if it is Cauchy sequence and Convergence sequence in completeness of fuzzy gamma m-normed linear space. This paper originates the notion of completeness, and produces some results on it in fuzzy Gamma-m-normed linear space. Also a necessary condition and theorem for completeness of a sequence in fuzzy gamma m-normed linear space is suggested.