Properties of the Adjoint Operator of a General Fuzzy Bounded Operator

: Our goal in the present paper is to recall the concept of general fuzzy normed space and its basic properties in order to define the adjoint operator of a general fuzzy bounded operator from a general fuzzy normed space V into another general fuzzy normed space U. After that basic properties of the adjoint operator were proved then the definition of fuzzy reflexive general fuzzy normed space was introduced in order to prove that every finite dimensional general fuzzy normed space is fuzzy reflexive.


Introduction:
The fuzzy topological structure of a fuzzy normed space was studied by Sadeqi and Kia in 2009 (1).Kider introduced a fuzzy normed space in 2011 (2).Also he proved this fuzzy normed space has a completion in (3).Again Kider introduced a new fuzzy normed space in 2012 (4).The properties of fuzzy continuous mapping which is defined on a fuzzy normed space was studied by Nadaban in 2015 (5).
Kider and Kadhum in 2017 (6) introduced the fuzzy norm for a fuzzy bounded operator on a fuzzy normed space and proved its basic properties then other properties were proved by Kadhum in 2017 (7).Ali in 2018 (8) proved basic properties of complete fuzzy normed algebra.Kider and Ali in 2018 (9) introduced the notion of fuzzy absolute value and studied the properties of finite dimensional fuzzy normed space.
The notion of general fuzzy normed space was introduced by Kider and Gheeab in 2019 (10,11) also they proved basic properties of this space and the general fuzzy normed space GFB(V, U).Kider and Kadhum in 2019 (12) introduced the notion fuzzy compact linear operator and proved its basic properties.
In this paper first, the definition of a general fuzzy normed space and also its basic properties is recalled so that to define the adjoint operator of a general fuzzy bounded operator from a general fuzzy normed space V into another general fuzzy normed space U. Then the important properties for the adjoint operator were proved.Finally the notion fuzzy reflexive general fuzzy normed space is defined in order to prove that every finite dimensional general fuzzy normed space is fuzzy reflexive.Let V be a vector space over the field ℝ and ⊙,⊗ be a continuous t-norms.A fuzzy set   :V×[0,∞) is called a general fuzzy norm on V if it satisfies the following conditions for all u, v∈V and for all  ∈ ℝ, s, t ∈ [0,∞): (G1) 0≤   (u, s)<1 for all s>0.(G2)   (u, s)=1 ⟺ u= 0 for all s>0.(G3)   (u, st) ≥  ℝ (, s) ⊗   (u, t) for all  ≠0 ∈ ℝ.  Suppose that (V,  ,⊙,⊗) is a general fuzzy normed space.A sequence (  ) in V is said to be general fuzzy approaches to u if every 0 <  < 1 and 0< s there is N ∈ ℕ such that   (  u , s) > (1 −) for every n≥ N. If (   ) is general fuzzy approaches to the fuzzy limit u it is written by lim →∞   = u or   → u.Also lim →∞   (  − u, s) = 1 if and only if (   ) is general fuzzy approaches to u.

Definition 11:(10)
Let (V,  ,⊙,⊗) be a general fuzzy normed space.A sequence (   ) is said to be general fuzzy bounded if there exists 0 < q <1 such that   (   , s) >(1−q) for all s>0 and n∈ ℕ. Definition 12: (10) Let (V,  ,⊙ ,⊗)and (U,  ,⊙ ,⊗ ) be two general fuzzy normed spaces the operator S: V→ U is called general fuzzy continuous at  0 ∈V for every s>0 and every 0 <  < 1 there exist t and there exists  such that for all v∈V with Suppose that (V, G V ,⊙ .⊗)and (U, G U ,⊙,⊗) are general fuzzy normed spaces.Then S: V → U is a general fuzzy continuous at u ∈ V if and only if u n →u in V implies S(u n ) →S(u) in U.
Definition 13: (10) Suppose that (V, G V ,⊙,⊗) is a general fuzzy normed space.Then V is called a general complete if every general Cauchy sequence in V is general fuzzy approaches to a vector in V. Definition 14 : (10) Suppose that (V, G V , ⨀, ⨂ ) and (U, G U , ⨀, ⨂ ) are two general fuzzy normed spaces.The operator S: D(S) →U is called general fuzzy bounded if it can find , 0 < α < 1 ; with G U (Sv , t) ≥ (1 − α) for each v ∈ D(S) and t > 0 . . . .( 1) Notation 1: (11) Suppose that (V, G V , ⨀, ⨂ )and (U, G U , ⨀, ⨂ ) are a general fuzzy normed spaces.Put GFB(V,U) = { S:V→U: G U (Sv , t) ≥ (1 − α)} with 0 < α < 1. Theorem 2: (11) Suppose that (V, G V , ⨀, ⨂) and (U, G U , ⨀, ⨂) are two general fuzzy normed spaces.Put G(T, t) =  ∈() G U (, ) for all T ∈ GFB(V, U), t > 0. Then [GFB(V, U), G, ⨀, ⨂] is general fuzzy normed space.Notation 2:(11) Rewrite 1 by : G U (Sv , t) ≥ G(S, t)………………(2) Theorem 3 :(11) Suppose that (V, G V , ⨀, ⨂) and (U, G U , ⨀, ⨂) are general fuzzy normed spaces with U is a general complete.Assume that T: D(T) →U be a linear operator and a general fuzzy bounded.Then T has an extension ̅̅̅̅̅̅ → U with S is linear and general fuzzy bounded such that G(T, t) = G(S, t) for all t > 0 .Theorem 4 :(11) If (V, G V , ⨀, ⨂)and (U, G U , ⨀, ⨂) are general fuzzy normed spaces and let T:D(T) →U be a linear operator where D(T) ⊆V.Then T is general fuzzy continuous if and only if T is general fuzzy bounded.Definition 15 : (11) Suppose that (V, G V , ⨀, ⨂) is a general fuzzy normed space and (ℝ,  ℝ , ⊙,⊗) is fuzzy absolute value space (ℝ,  ℝ , ⊙,⊗).Then a linear function f:V→ ℝ is called general fuzzy bounded if there exists  ∈ (0, 1) with  ℝ [f(v), t] ≥ (1 − σ) for any v ∈ D(f), t > 0. Furthermore, the fuzzy norm of f is L(f, t) = inf  ℝ (f(v), t) and  ℝ (f(v), t) ≥ L(f, t).Corollary 1 : (11) Suppose that (V, G V , ⨀, ⨂) is a general fuzzy normed space and (ℝ,  ℝ , ⊙,⊗) is fuzzy absolute value space.Then a linear function f:V→ ℝ with D(f) ⊆ V is general fuzzy bounded if and only f is general fuzzy continuous.Definition 16 : (11) Suppose that (V, G V , ⨀, ⨂) is a general fuzzy normed space.Then GFB(V, ℝ) = { f:V → ℝ : f is general fuzzy bounded linear } forms a general fuzzy normed space with general fuzzy norm defined by L(f, t) = inf L ℝ (f(v), t) which is said to be the general fuzzy dual space of V. Definition 17: (11) Suppose that (V, G V , ⨀, ⨂) is general fuzzy normed space.A sequence (v n ) in V is general fuzzy weakly approaches if it can find v ∈ V with every h ∈ GFB(V,ℝ) lim n→∞ h(v n ) = h(v).This is written v n → w v the element v is said to be the weak limit to (v n ) and (v n ) is said to be general fuzzy approaches weakly to v. Theorem 5:(11) Suppose that (V, G V , ⨀, ⨂) is a general fuzzy normed space and Definition 18:(11) Suppose that (V, G V , ⨀, ⨂) is a general fuzzy normed space.A sequence (h n ) with h n ∈ GFB(V, ℝ) is called 1)Strong general fuzzy approaches in the general fuzzy norm on GFB(V, ℝ) that is h ∈ FB(V, ℝ) with G[h n − h, t] → 1 for all t > 0 this written h n → h 2)Weak general fuzzy approaches in the fuzzy absolute value on ℝ that is h ∈ GFB(V, ℝ) with h n (v) → h(v) for every v ∈ V written by lim n→∞ h n (v) = h(v).Theorem 6 :(11) Suppose that (V, G V , ⨀, ⨂) and (U, G U , ⨀, ⨂) are two general fuzzy normed spaces.Then GFB(V, U) is general complete when U is general complete.

Theorem 7:
The general fuzzy normed space (ℝ, ℝ , ⨀, ⨂) is general fuzzy complete Proof: Let ( n ) be a general Cauchy sequence in ℝ then ( n ) has a monotonic subsequence (  j ) but ( n ) is a general fuzzy bounded hence (  j ) is a general fuzzy bounded thus (  j ) general approaches to a∈ ℝ.Then for every 0 < r <1 there exists N ∈ ℕ such that  ℝ ( m −  n
(10)eneral fuzzy normed space where s⊙t = s ∧ t and t⊗ s= t. s for all t ,s ∈ [0,1].Then  ⃦ .⃦ is called the standard general fuzzy norm induced by the norm ⃦ .⃦.The next result is proved in(10)Proposition 1:Suppose that (V, ||.||) is a normed space define   (u, s) = radius n and s>0 and GFB[ u, n ,s] = {m ∈V :   (u−m ,s ) ≥ (1−n)} is called a general fuzzy closed ball with center u∈V radius n and s>0.Definition 5:(10)Suppose that (V,   ,⊙,⊗) is a general fuzzy normed space and M⊆ V. Then M is called a r) for all m , n ≥ N.
(10)in M such that m n → m.Definition 10:(10)Suppose that (V,  ,⊙,⊗) is a general fuzzy normed space A sequence is fuzzy continuous at each