SOME RESULTS RELATED WITH FUZZY a −NORMED LINEAR SPACE

Abstract

Zadeh established the concept of fuzzy set based on the characteristic function. Foundation of fuzzy set theory was introduced by him. Throughout this paper, 𝑀𝑛(𝐹) denotes the set of all fuzzy matrices of order 𝑛 over the fuzzy unit interval [0,1]. Inaddi tion (𝑀𝑛 (𝐹), 𝜃) dis called as fuzzy 𝛼 −normed linear space. The objective of this paper is to investigate the relationships between convergent sequences and fuzzy 𝛼 −normed linear space. The set of all fuzzy points in 𝑀𝑛 (𝐹) is denoted by 𝑃∗(𝑀𝑛(𝐹)). For a fuzzy 𝛼 −normed linear space (𝑀𝑛 (𝐹), 𝜃), we have |𝜃(𝑃𝐴)𝛼 −𝜃(𝑃𝐵)𝛼 | ≤ 𝜃(𝑃𝐴,𝑃𝐵)𝛼. Besides 𝜃 is a continuous function on 𝑀𝑛 (𝐹). That is, if 𝑃𝐴𝑛 → 𝑃𝐴 as 𝑛 → ∞ then 𝜃(𝑃𝐴𝑛 )𝛼 → 𝜃(𝑃𝐴)𝛼 as 𝑛 → ∞, where 𝑃𝐴𝑛 is a sequence in (𝑀𝑛 (𝐹), 𝜃). Hence, 𝜃 is always bounded on 𝑀𝑛(𝐹). Next we introduce the following result: Let 𝑃𝐴𝑛 , 𝑃𝐵𝑛 ∈ 𝑃 ∗ (𝑀𝑛(𝐹)) with 𝑃𝐴𝑛 and 𝑃𝐵𝑛 converge to 𝑃𝐴 and 𝑃𝐵 respectively as 𝑛 → ∞. Then 𝑃𝐴𝑛 + 𝑃𝐵𝑛 converge to 𝑃𝐴 + 𝑃𝐵 as 𝑛 → ∞. Furthermore, we are able to compare two different fuzzy 𝛼 −norms with convergent sequence. The result states that for a fuzzy 𝛼 −normed linear space (𝑀𝑛(𝐹), 𝜃), we have 𝜃(𝑃𝐴)𝛼1 ≥ 𝑀𝜃(𝑃𝐴 )𝛼2 , for some 𝑀 > 0 and 𝑃𝐴 ∈ 𝑃 ∗ (𝑀𝑛(𝐹)). If 𝑃𝐴𝑛 converges to 𝑃𝐴 under fuzzy 𝛼1 −norm then 𝑃𝐴𝑛 converges to 𝑃𝐴 under fuzzy 𝛼2 −norm. Moreover, if (𝑀𝑛 (𝐹), 𝜃) has finite dimension then it should be complete. Through these results, we are able to get clear understanding about the concept fuzzy 𝛼 −normed linear space and its properties.