Some Properties of Fuzzy Anti-Inner Product Spaces

In this paper, the definition of fuzzy anti-inner product in a linear space is introduced. Some results of fuzzy anti-inner product spaces are given, such as the relation between fuzzy inner product space and fuzzy anti-inner product. The notion of minimizing vector is introduced in fuzzy anti-inner product settings.


I. Introduction
Kohli and Kumar, in 1993 [1], introduced the definition of the fuzzy inner product space and fuzzy co-inner product space. In 1997, Alsina et al [2] introduced the ideal of probabilistic inner product space. After that, in 2010, Hasankhain et al. [3] introduced some properties of fuzzy Hilbert spaces and norm of operators. In 2013, the fuzzy real inner product space and its properties were proved by Mukherjee and Bag [4]. Finally, a note on fuzzy Hilbert spaces was introduced by Daraby et al. in 2016 [5].

II. Preliminaries
This section consists of some definitions and results that will be needed later in this paper.

Definition (2.1) [6]
Assume that is a linear space over the field of complex numbers. A mapping satisfies the following conditions for all and (FIP1) is called a fuzzy inner product function on and ( ) is called a fuzzy inner product space.

Definition (2.2) [7]
Let be a linear space over a field . A fuzzy set such that the following holds for all and :

ISSN: 0067-2904
Ali and Hussein is said to be a fuzzy anti-norm on and ( , ) is called a fuzzy anti-normed linear space. Later on, the following condition of fuzzy norm will be required: ( 6) for all with , ( ) implies .

III. Fuzzy anti-inner product space
The definition of fuzzy anti-inner product space on a complex linear space is introduced and some of its results are investigated.

Definition (3.1) [8]
Assume that is a linear space over the filed of complex numbers. Define : to be a mapping such that the following holds for all in and is called a fuzzy anti-inner product function on and ( ) is called a fuzzy anti-inner product space.

Example (3.2) Assume that (
) is an inner product space over . A function is defined by Then is a fuzzy anti-inner product space on Proof: (Fa-IP1) Consider the following cases: For then the property follows from the fact that This completes the proof.

Note (3.4) [8]
Assume that satisfies the condition: ) be a fuzzy anti-inner product space, satisfying (Fa-IP8).Then ( ) ‖ ‖ ⋀ * ( ) + is a crisp norm on , called the -anti norm on generated from In the sequel we shall consider the following condition: (Fa-IP9) and ,

Theorem (3.5) [8]
Let be a fuzzy anti-inner product on the defined function , as follows: Then is a fuzzy anti-norm on From now on, if (Fa-IP8) and (Fa-IP9) hold for each ( ) then ‖ ‖ ⋀ * ( ) + is an ordinary anti-norm on satisfying parallelogram law. So, by using polarization identity, one can get an ordinary inner product, called the -anti-inner product, as follows:

Definition (3.6)
is said to be anti-level complete (AL-complete). If ( ) is a fuzzy anti-inner product space satisfying (Fa-IP8) for any ( ) , then every Cauchy sequence converges in w.r.t the -antinorm ‖ ‖ generated by the fuzzy anti-norm which is induced by fuzzy anti-inner product Theorem (3.7) (Minimizing vector) Let ( ) be a fuzzy anti-inner product space satisfying (Fa-IP8) and (Fa-IP9), let M ( ) be the convex subset of which is anti-level complete, and let . Then for each ( ), in M such that is the fuzzy anti-norm induced by fuzzy anti-inner Proof: We note that if ( ) is a fuzzy anti-inner product space, then for each ( ), ( ‖ ‖ ) is a crisp anti-normed linear space satisfying the parallelogram law. Again ‖ . Hence the result follows from the corresponding crisp minimization vector theorem in ( ‖ ‖ ).

Theorem (3.8)
is a fuzzy anti-inner product space on if and only if is a fuzzy inner product space on Proof: For Hence ( ) is a fuzzy anti-inner product space Theorem (3.10) Let ( ) be a fuzzy anti-inner product space satisfying (Fa-IP8) and (Fa-IP9) and beanti-inner product ( ).