Fuzzy Linear space using triangular fuzzy numbers

In this paper, we discussed the elementary structure of linear space for triangular fuzzy system, linear subspace of the triangular fuzzy numbers and linear transformations between two triangular fuzzy spaces. We introduce a study of new concept of a Hamel base for any linear space and the uniqueness of its cardinality of fuzzy numbers. We begin with some basic definitions and properties of linear space triangular fuzzy system which will be used throughout this paper. Also the axioms of the primary field and the addition and scalar multiplication operations for triangular fuzzy system have been included.


Introduction
A linear space,denoted by F,is a primary field of (real or complex) numbers. The elements of the linear space are called scalars which are denoted by lower case Greek letters,for example, λ and μ. Re λ denotes real part and Im λ denotes the imaginary part of a complex number. However, once in a while the symbols m or M may be used to denote a scalar which is used to represent a type of boundedness.
The upper case letter Tdenotes the linear transformation between linear spaces. When the range and domain of the transformation are linear topological spaces, it can be referred to as a linear map.
In 1965, the concept of fuzzy set theory was introduced first by Zadeh [6] and thereafter, several authors have developed the concept through the contribution of the different articles and applied the same on several branches of pure and applied mathematics. Katsaras [4] introduced the concept of fuzzy norm in 1984 and in 1992. The idea of fuzzy norm linear space was introduced byFelbin [2]. A different idea of fuzzy norm on a linear space was introduced byCheng -Moderson [1] whose related metric is same as defined byKatsaras [4]. Later on Bag and Samanta [3] customized the definition of fuzzy norm of Cheng -Moderson and there after they have studied finite dimensional fuzzy normed linear spaces and established the concept of continuity and boundedness of a function with respect to their fuzzy norm. Moreover, the definition of intuitionistic fuzzy n-normed linear space was introduced in the paper [5] and a sufficient condition for an intuitionistic fuzzy n-normed linear space to be complete was established.
In this paper, we use the above generalized notion of fuzzy linear space and triangular fuzzy numbers in order to introduce a new generalized triangular fuzzy linear space, triangular fuzzy Hamel space and also the triangular fuzzy linear transformation and the work is extended to obtain the important theorem results.

Preliminaries
In this section, some basic definitions of fuzzy numbers have been given. Fuzzy numbers are of incredible significance in fuzzy frameworks. The fuzzy numbers that generally applied as a part of utilizations are the triangular (shaped) and the trapezoidal (shaped) fuzzy numbers [20].

Definition
A fuzzy number is defined as iii) There are real numbers a, b in such a way that c a b d ≤ ≤ ≤ and E denotes the set of all fuzzy numbers. An identical parametric is additionally given in [16]. Another definition or parametric form of a fuzzy number which gives the same ( ) ℜ F is given by Kaleva [30].
Arithmetic operations between two triangular fuzzy numbers defined on universal set of real numbers ℜ are investigated in [14].

Triangular Fuzzy Number
It is a fuzzy number represented with three points as given: This is interpreted as membership functions and holds the accompanying conditions i) 1  A fuzzy number   1 2 3 ( , , ) U u u u = % % % % is said to be positive triangular if all j u % 's > 0 for j=1, 2, 3. A fuzzy number   1 2 3 ( , , ) U u u u = % % % % is said to be negative triangular ifall j u % 's < 0 for j=1, 2, 3.

Note.
A negative triangular fuzzy number can be composed as the negative multiplication of a positive triangular fuzzy number.

3.Elementary Properties
Some elementary properties of linear space are discussed here.

Definition of linearly independent
Let Y % be a linear space. A subset A % of it is called linearly independent given that for every finite subset

Definition of linear span
, then the y % -translate of U % is given by

Linear transformation and theorem on fuzzy linear space
In the following theorem, Hamel base of linear system and the uniqueness of its cardinality of triangular fuzzy numbers have been proved.

Definition of linear transformation
A linear transformation T % from a linear space 1 Y % to a linear space 2 Y % (on the same scalar field F ) is a function which satisfies i)

Theorem
Let T % bethe linear transformation from the linear space 1 Y % to the linear space 2 Y % .