Finite Dimensional Convex Fuzzy Normed Space and Its Basic Properties

Abstract

Here the notion of convex fuzzy absolute value space is represented with an example which shows that the existence of such space. The reason behind introducing the definition of convex fuzzy absolute value and does not using the ordinary absolute value is the main definition in this paper will be not correct with ordinary absolute value. After that the main definition of convex fuzzy normed space is recalled with example which shows that the existence of such space. Then other definitions and theorems is recalled that will be used later in the section of main results such as the convex fuzzy norm is convex fuzzy continuous function. So our goal in the section of results and discussion is to prove some properties of finite dimensional convex fuzzy normed space which not true ingeneral for convex fuzzy normed space. Thus the last section contains the following results with proofs if  is a subspace of the convex fuzzy normed space  with dim  then  is convex fuzzy complete subspace of . Moreover when  is a convex fuzzy bounded as well as convex fuzzy closed subspace of the c-FNS,  also dim   then  is convex fuzzy compact. As well as if dim  for a linear space   then there is a unique convex fuzzy norm on . Finally if ={: the convex fuzzy norm of  belongs to I } a convex fuzzy closed subset of  as well as convex fuzzy compact this implies that dim where   is convex fuzzy normed space.

Received 27/12/2023

Revised 11/05/2024

Accepted 13/05/2024

Published Online First 20/11/2024