Open AccessSOME RESULTS RELATED WITH FUZZY a −NORMED LINEAR SPACE
Author(s) -
M. Arunmaran,
K. Senthamarai Kannan
Publication year - 2019
Publication title -
advances in mathematical sciences
Language(s) - English
Resource type - Journals
ISSN - 2664-598X
DOI - 10.37516/adv.math.sci.2019.0052
Subject(s) - mathematics , fuzzy number , normed vector space , fuzzy logic , fuzzy set , fuzzy mathematics , fuzzy subalgebra , norm (philosophy) , fuzzy classification , discrete mathematics , bounded function , mathematical analysis , computer science , artificial intelligence , political science , law
Zadeh established the concept of fuzzy set based on the characteristic function. Foundation of fuzzy set theory was introduced by him. Throughout this paper, () denotes the set of all fuzzy matrices of order over the fuzzy unit interval [0,1]. Inaddi tion ( (), ) dis called as fuzzy −normed linear space. The objective of this paper is to investigate the relationships between convergent sequences and fuzzy −normed linear space. The set of all fuzzy points in () is denoted by ∗(()). For a fuzzy −normed linear space ( (), ), we have |() −() | ≤ (,). Besides is a continuous function on (). That is, if → as → ∞ then ( ) → () as → ∞, where is a sequence in ( (), ). Hence, is always bounded on (). Next we introduce the following result: Let , ∈ ∗ (()) with and converge to and respectively as → ∞. Then + converge to + as → ∞. Furthermore, we are able to compare two different fuzzy −norms with convergent sequence. The result states that for a fuzzy −normed linear space ((), ), we have ()1 ≥ ( )2 , for some > 0 and ∈ ∗ (()). If converges to under fuzzy 1 −norm then converges to under fuzzy 2 −norm. Moreover, if ( (), ) has finite dimension then it should be complete. Through these results, we are able to get clear understanding about the concept fuzzy −normed linear space and its properties.