Intuitionistic Type-2 Fuzzy Normed Linear Space and Some of Its Basic Properties

Abstract

An intuitionistic fuzzy set is a more generalised tool than a fuzzy set for handling unpredictability as, in an intuitionistic fuzzy set, there is scope for considering a grade of non-membership, independent of the grade of membership, only satisfying the condition that their sum is less or equal to 1. The motivation of this paper is to introduce the notion of intuitionistic type-2 fuzzy normed linear space (IT2FNLS). Here, to each vector x, we assign two fuzzy real number valued grades, one for its norm and the other for the negation of its norm. A theorem of the decomposition of the intuitionistic type-2 fuzzy norm into a family of pairs of Felbin-type fuzzy norms is established. Also, we deal with Cauchyness and convergence of sequences in the IT2FNLS. Later on, in the finite-dimensional IT2FNLS, the completeness property and compactness property are explored. Finally, we define two types of intuitionistic type-2 fuzzy continuity and examine the relations between them.