A note on fuzzy Hilbert spaces

1 Introduction

The idea of fuzzy norms on linear spaces introduced at the first by Katsaras. Later, many other mathematicians like Felbin [9], Cheng and Mordeson [8], Bag and Samanta [4], and so on introduced the notion of fuzzy normed linear spaces in different approach. A large number of papers have been published in fuzzy normed linear spaces. But studies on fuzzy inner product spaces are relatively new and a few research have been done in fuzzy inner product spaces. R. Biswas [6], A.M. El-Abyed and H.M. Hamouly in [7] gave an interesting definition of fuzzy inner product spaces as associated fuzzy norm function. An another definition introduced by A. Hasankhani et al. [11]. This authors introduced the concept of a Felbin-fuzzy inner product and showed that any Felbin-fuzzy inner product space can be imbedded in a complete Felbin-fuzzy inner product space.

In this paper, it is showed that classical Hilbert space is a Felbin-fuzzy Hilbert space and thus the results obtained in classical Hilbert spaces can be established in Felbin-fuzzy Hilbert spaces in general.

Moreover by an example, it is showed that each Felbin-fuzzy Hilbert spaces is not necessarily classical Hilbert spaces. The authors are able to establish a transformation theorem from an ascending family of crisp inner product into a Felbin-fuzzy inner product. The organization of the paper is as follows:

In Section 2, some preliminaries and essential concepts are stated.

In Section 3, our main results are stated and a transformation theorem is got that is helpful to study Felbin-fuzzy inner product spaces.

2 Preliminaries

Let η be a fuzzy subset on R, i.e. a mapping η : R → [0, 1] associating with each real number t its grade of membership η (t).

In this paper, the concept of fuzzy real numbers (fuzzy intervals) is considered in the sense of Xiao and Zhu which is defined below:

Definition 1. [17] A fuzzy subset η on R is called a fuzzy real number (fuzzy intervals), whose α-level set is denoted by [η]  _α , i.e., [η]  _α  = {t : η (t) ≥ α}, if it satisfies two axioms:

There exists r′ ∈ R such that η (r′) =1.

The set of all fuzzy real numbers (fuzzy intervals) is denoted by F (R). If η ∈ F (R) and η (t) =0 whenever t < 0, then η is called a non-negative fuzzy real number and F^∗ (R) denotes the set of all non-negative fuzzy real numbers. Real number η α - for all η ∈ F^∗ (R) and each α ∈ (0, 1] is positive.

The fuzzy real number r ˜ ∈ F ( R ) defined by r ˜ ( t ) = { 1 , t = r , 0 , t ≠ r , it follows that R can be embedded in F (R), that is if r ∈ (- ∞ , ∞), then r ˜ ∈ F ( R ) satisfies r ˜ ( t ) = 0 ˜ ( t - r ) and α-level of r ˜ is given by [ r ˜ ] α = [ r , r ] , α ∈ ( 0 , 1 ] .

Theorem 1. [3]Let [a _α , b _α ] , 0 < α ≤ 1, be a family of non-empty intervals. If

then the family [a _α , b _α ] represents the α-level sets of a fuzzy real number η ∈ F (R) such that η (t) =sup{α ∈ (0, 1] : t ∈ [a _α , b _α ]} and [ η ] α = [ η α - , η α + ] = [ a α , b α ] .

Lemma 1.[12] Let [a ^α , b ^α ] , 0 < α ≤ 1, be a given family of non-empty intervals. Assume

Then the family [a ^α , b ^α ] represents the α-level sets of a fuzzy real number η ∈ F (R). Conversely if [a ^α , b ^α ] , 0 < α ≤ 1, are the α-level sets of a fuzzy number η ∈ F (R), then the conditions (a) , (b) and (c) are satisfied.

Theorem 2. [11] Let η ∈ F^∗ (R) and [ η ] α = [ η α - , η α + ], for all α ∈ (0, 1]. Furthermore, let [ η α - ,η α + ] , 0 < α ≤ 1, be a family of non-empty intervals. Then the conditions (a) , (b) and (c) of Lemma 2 are satisfied.

Definition 2. For a non-negative fuzzy real number η, it is defined η = γ where [ γ ] α = [ η α - , η α + ] , α ∈ ( 0 , 1 ].

Lemma 2. [14] Let γ, δ be fuzzy real numbers. Then ∀ α ∈ ( 0 , 1 ] , [ γ ] α = [ δ ] α ⇔ ∀ t ∈ R , γ ( t ) = δ ( t ) .

According to Mizumoto and Tanaka [13], fuzzy arithmetic operations ⊕, ⊖ , ⊗ and ø on F (R) × F (R) can be defined as: ( γ ⊕ δ ) ( t ) = ∨ t = x + y ( min ( γ ( x ) , δ ( y ) ) ) = sup s ∈ R { γ ( s ) ∧ δ ( t - s ) } , t ∈ R ( γ ⊖ δ ) ( t ) = ∨ t = x - y ( min ( γ ( x ) , δ ( y ) ) ) = sup s ∈ R { γ ( s ) ∧ δ ( s - t ) } , t ∈ R ( γ ⊗ δ ) ( t ) = ∨ t = xy ( min ( γ ( x ) , δ ( y ) ) ) = sup s ∈ R , s ≠ 0 { γ ( s ) ∧ δ ( t / s ) } , t ∈ R ( γ ø δ ) ( t ) = ∨ t = x y ( min ( γ ( x ) , δ ( y ) ) ) = sup s ∈ R { γ ( st ) ∧ δ ( s ) } , t ∈ R , which are special cases of Zadeh’s extension principle. The additive and multiplicative identities in F (R) are 0 ˜ and 1 ˜, respectively. Let ⊖γ be defined as 0 ˜ ⊖ γ. It is clear that γ ⊖ δ = γ ⊕ (⊖ δ) .

Lemma 3. [12] Let γ, δ ∈ F (R) and [ γ ] α = [ γ α - , γ α + ], [ δ ] α = [ δ α - , δ α + ]. Then for all α ∈ (0, 1], [ γ ⊕ δ ] α = [ γ α - + δ α - , γ α + + δ α + ] [ γ ⊖ δ ] α = [ γ α - - δ α + , γ α + - δ α - ] [ γ ⊗ δ ] α = [ γ α - δ α - , γ α + δ α + ] [ 1 / γ ] α = [ 1 γ α + , 1 γ α - ] , γ α - > 0 [ | γ | ] α = [ max ( 0 , γ α - , - γ α + ) , max ( | γ α - | , | γ α + | ) ]

Proposition 3. (Bag and Samanta [5]). Let {a _α } and {b _α } be two, respectively, non-decreasing and non-increasing families of real numbers such that - ∞ < a _α  ≤ b _α  < + ∞ , 0 < α ≤ 1 and γ be the fuzzy real number (fuzzy interval) generated by the families of closed intervals {[a _α , b _α ] ; α ∈ (0, 1]}. Then

sup β < α γ β - = γ α -,

Proposition 4. [2] If η _i , i = 1, 2, are the fuzzy real numbers (fuzzy intervals) generated by the family of nested bounded closed intervals { [ a α s , b α s ] : 0 < α ≤ 1 }, for s = - , +, and for each α ∈ ( 0 , 1 ] , a α - ≤ a α + , b α - ≤ b α + then η₁ ⪯ η₂.

From [5], it is obtained that if η is a fuzzy real number with [ η ] α = [ η α - , η α + ], and η^∗ is the fuzzy number (fuzzy interval) generated by the family of nested bounded closed intervals [ η α - , η α + ] , α ∈ ( 0 , 1 ] , then η = η^∗.

Definition 3. Let X be a real linear space; L and R (respectively, left norm and right norm) be symmetric and non-decreasing mappings from [0, 1] × [0, 1] into [0, 1] satisfying L (0, 0) =0, R (1, 1) =1 . Then ∥· ∥ is called a fuzzy norm and (X, ∥ · ∥ , L, R) is a fuzzy normed linear space (abbreviated to FNLS) if the mapping ∥· ∥ from X into F^∗ (R) satisfies the following axioms, where [ ∥ x ∥ ] α = [ ∥ x ∥ α - , ∥ x ∥ α + ] for x ∈ X and α ∈ (0, 1] :

x = 0 if and only if ∥ x ∥ = 0 ˜ ,

Definition 4. [17] Let (X, ∥ · ∥) be a Felbin-fuzzy normed linear space.

A sequence {x _n } ⊆ X is said to converge to x ∈ X (lim n → ∞ x n = x), if lim n → ∞ ∥ x n - x ∥ α + = 0, for all α ∈ (0, 1] .

Definition 5. [17] Let (X, ∥ · ∥) be a Felbin-fuzzy normed linear space. A subset A of X is said to be complete, if every Cauchy sequence in A converges in A .

Definition 6. [10] fel Two Felbin-fuzzy normed linear spaces (X, ∥ · ∥) and (X^∗, ∥ · ∥ ^∗) are called congruent if there exists an isometry of (X, ∥ · ∥) onto (X^∗, ∥ · ∥ ^∗) .

Definition 7. [10] A complete Felbin-fuzzy normed linear space (X^∗, ∥ · ∥ ^∗) is a completion of a Felbin-fuzzy normed linear space (X, ∥ · ∥) if

(X, ∥ · ∥) is congruent to a subspace (X₀, ∥ · ∥ ^∗) of (X^∗, ∥ · ∥ ^∗) and

In [11], the authors introduced a definition of Felbin-fuzzy inner product space.

Definition 8. [11] Let X be a vector space over R. A Felbin-fuzzy inner product on X is a mapping 〈 · , · 〉 : X × X → F (R) such that for all vectors x, y, z ∈ X and r ∈ R, have

〈x + y, z〉 = 〈x, z〉 ⊕ 〈y, z〉,

The vector space X equipped with a Felbin-fuzzy inner product is called a Felbin-fuzzy inner product space. A Felbin-fuzzy inner product on X defines a fuzzy number

∥ x ∥ = 〈 x , x 〉 , ∀ x ∈ X .(1) A fuzzy Hilbert space is a complete Felbin-fuzzy inner product space with the Felbin-fuzzy norm defined by (1).

Definition 9. [11] Let (X^∗, ∥ · ∥ ^∗) be a completion of a Felbin-fuzzy normed linear space (X, ∥ · ∥) and x^∗, y^∗ ∈ X^∗ with representatives {x _n } and {y _n } , respectively. Suppose α ∈ (0, 1] and {α _k } is a strictly increasing sequence converging to α . Define [ 〈 x ∗ , y ∗ 〉 ] α = [ lim n , k → ∞ 〈 x n , y n 〉 α k - , lim n , k → ∞ 〈 x n , y n 〉 α k + ] .

Lemma 4. [11] The function 〈 · , · 〉 defined in Definition 2, is a Felbin-fuzzy inner product on X^∗ .

Definition 10. [15] Let X be a linear space on R and for fuzzy subset F: X × X × R → [0, 1] the following conditions hold where for x, y, z ∈ X and t ∈ R:

For all t < 0, we have F (x, x, t)=0,

Theorem 5. [3] tor Let (X, ∥ · ∥) be a Felbin-fuzzy normed linear space, [ ∥ x ∥ ] α = [ ∥ x ∥ α - , ∥ x ∥ α + ] , for α ∈ (0, 1] and set A = { α ∈ ( 0 , 1 ] : ∥ x ∥ α - ≤ t } . Let N be a function in X × R defined by N (x, t) = { sup { α ∈ ( 0 , 1 ] : ∥ x ∥ α - ≤ t } , ( x , t ) ≠ ( θ , 0 ) , 0 , ( x , t ) = ( θ , 0 ) ∨ A = ∅ . Then N is B-S-fuzzy norm. Also if it is defined ∥ x ∥ α ′ = inf { t > 0 : N ( x , t ) ≥ α } and ∥ x ∥ α ″ = inf { t > 0 : N ( x , t ) < α } , α ∈ ( 0 , 1 ] then ∥x ∥ ′ and ∥x ∥ ″ are norms on X and for α ∈ (0, 1] , have ∥ x ∥ α - = ∥ x ∥ ′ , ∥ x ∥ α + = ∥ x ∥ ″ .

Definition 11. [5] Let (X, ∥ · ∥) and (Y, ∥ · ∥ ^∼) be Felbin-fuzzy normed linear space. A linear operator T : X → Y is said to be weakly fuzzy bounded if there exists a fuzzy interval 0 ˜ ≺ η ∈ F ∗ such that ∥ Tx ∥ ∼ ø ∥ x ∥ ⪯ η , ∀ x ( ≠ 0 _ ) ∈ X .

3.1 Extension of classical Hilbert spaces

If in Definition 9, the relation 〈 · , · 〉 α ± is classical inner product, then Lemma 4 holds. So, the classical Hilbert space implies Felbin-fuzzy Hilbert space. Subsequently, each Felbin-fuzzy Hilbert space is also a classical Hilbert space, and all results and theorems in classical Hilbert spaces hold for Felbin-fuzzy Hilbert spaces in general. There are many theorems of classical Hilbert spaces which are valid in Felbin-fuzzy Hilbert spaces. Because of this, from now on there is not necessarily need to check the results which have already been proven in classical analysis. The only thing which remains is to investigate the properties in Felbin-fuzzy Hilbert spaces which do not hold in classical analysis.

Likewise, the recent argument remains true on fuzzy normed space with respect to classical normed space as it was seen in [14]. Hence it is emphasized that for instance, inspection of the problem of stability of the Cauchy equation in fuzzy normed linear spaces that is investigated in [14], is redundant and must be omitted.

In the following theorem, an example of a Felbin-fuzzy Hilbert space is given which is not a normed space in the classical sense. Therefore, the spectrum of the category of Felbin-fuzzy Hilbert spaces is broader than the category of classical Hilbert spaces. This is why the study of Felbin-fuzzy Hilbert space is of great importance.

Theorem 6. The linear space C (Ω) (the vector space of all complex valued continuous functions on Ω) is Felbin-fuzzy Hilbert space, but for open subset Ω ⊆ R ⁿ , it can not be classical Hilbert space.

Proof. The linear space C (Ω) is not normable in classical analysis, thus it is not classical Hilbert space (see [1, 14]). Now we define

〈 f , g 〉 ( t ) = sup { α ∈ ( 0 , 1 ] | t ∈ [ f α - g α - , f α + g α + ] }(2) for f, g ∈ C (Ω) . For α ∈ (0, 1] , there exists an n ∈ N such that 1 n + 1 < α ≤ 1 n . It is well known that Ω is a countable union of sets K _n ≠ ∅ which can be chosen so that K _n lies in the interior of K_n+1  (n = 1, 2, 3, ⋯) . Let for an arbitrary function f ∈ C (Ω) define f α - = sup { | f ( x ) | ; x ∈ K 1 } , f α + = sup { | f ( x ) | ; x ∈ K n + 1 } , so { [ f α - , f α + ] ; α ∈ ( 0 , 1 ] } is a family of nested bounded closed intervals. Similarly for g and so on, the definition holds.

We show that the 〈 · , · 〉 that defined in (2) is a fuzzy real number. Suppose that f, g ∈ C (Ω) , it follows that - ∞ < f α - g α - ≤ f α + g α + < ∞ .

So, [〈f, g〉]  _α  ≠ ∅ , for all α ∈ (0, 1] . It is showed that [〈f, g〉]  _α , α ∈ (0, 1] , satisfies the conditions of Lemma [14]:

Let 0 < β₁ < β₂ . Assume that α k ↗ β 1 , α k ′ ↗ β 2 and α k ′ > β 1 , for all k . Then, α k ≤ α k ′ , for all k, hence f α k - ≤ f α k ′ - and thus f α k - g α k - ≤ f α k ′ - g α k ′ - similarly, f α k ′ + g α k ′ + ≤ f α k + g α k + . Hence [〈f, g〉]  _{β 2}  ⊆ [〈f, g〉]  _{β 1}  .

It is worth mentioning that since any Felbin-fuzzy inner product space can be imbedded in a complete Felbin-fuzzy inner product space, then result obtained.□

Theorem 7. The function ∥· ∥ that defined by ∥ · ∥ = 〈 · , · 〉 , is a Felbin-fuzzy norm.

Proof. Proof is similar to Theorem 3.3 in [11].□

Remark 1. If the range of 〈 · , · 〉 in the above theorem is a non-negative fuzzy real number, then by Theorem 2, the conditions of (a)–(c) from Lemma 1 are satisfied.

Corollary 1. By the above discussions, a classical Hilbert space is a Felbin-fuzzzy Hilbert space, so all classical theorems remain true in the Felbin-fuzzy Hilbert spaces.

Here the fuzzy version of the well known classical theorems such as Riesz representation theorem, Bessel’s inequality on Felbin-fuzzy Hilbert space is cited that can be established directly on Felbin-fuzzy inner products, whereas by using Corollary 1 easily the results obtained. On the other hand, the proof of this theorems can be summarized by usingCorollary 1.

Theorem 8. (Riesz representation) If weakly fuzzy bounded linear functional f on a Felbin-fuzzy Hilbert space H can be represented for α ∈ (0, 1] , 0 ≠ y ∈ H, by

f ( x ) = 〈 x , y 〉 α - , ∀ x ∈ H ,(3) then ∥ f ∥ = ∥ y ∥ .

Proof. By Corrolary 1 result is satisfied and direct proof is avoided.□

Theorem 9. (Bessel’s inequality) Let H be a Felbin-fuzzy inner product space and for any x ∈ H there exists {x _n } ⊂ H, x _n  → x, {e _k } be a fuzzy orthonormal sequence in H, then Σ k = 1 ∞ | 〈 x , e k 〉 | 2 ⪯ ∥ x ∥ 2 .

Proof. The direct proof appeared in [16].□

Here the authors are able to establish a transformation theorem from an ascending family of classical inner product into a Felbin-fuzzy inner product. It is proceeded to show that the relation between Felbin-fuzzy inner product and classical inner product is formulated. The authors think that suitable transformation theorem which are established in this paper will be helpful to study fuzzy inner product spaces to a large extent.

In [15], the authors proved a transformation theorem from a B-S-fuzzy inner product into an ascending family of crisp inner products.

Theorem 10. [15] Let (X, F) be a B-S-fuzzy real inner product space, for α ∈ (0, 1] , 〈 x , y 〉 = ∧ { t ∈ R : F ( x , y , t ) ≥ α } and F (x, x, t) >0, for t > 0 then x = 0 . Then {〈 · , · 〉}  _α is an ascending family of crisp inner products on X . We call these inner products as α-inner products of F .

Now the transformation theorem from an ascending family of crisp inner products into a Felbin-fuzzy inner product is stated.

Proposition 11. Let X be a Felbin-fuzzy inner product space, and for x , y ∈ X , [ 〈 x , y 〉 ] α = [ 〈 x , y 〉 α - , 〈 x , y 〉 α + ] , α ∈ ( 0 , 1 ] , where 〈 x , y 〉 α - be crisp inner product. Define for c ≥ 0, F′ (cx, y, t) = { 0 , t = 0 , xy = 0 , ∨ { α ∈ ( 0 , 1 ] : 〈 cx , y 〉 α - ≤ t } , otherwise , and for c < 0, F′ (cx, y, t) = { 1 , t = 0 , xy = 0 , 1 - ∨ { α ∈ ( 0 , 1 ] : 〈 cx , y 〉 α - ≥ t } , otherwise . Thus F′ is a B-S-fuzzy inner product space on X . Again, if it is defined for α ∈ (0, 1] , 〈 x , y 〉 ′ = ∧ { t > 0 : F ′ ( x , y , t ) ≥ α } , 〈 x , y 〉 ″ = ∧ { t > 0 : F ′ ( x , y , t ) < α } , then {〈x, y〉′}  _α and {〈x, y〉″}  _α are families of crisp-inner product on X and for α ∈ (0, 1] , 〈 x , y 〉 α - = 〈 x , y 〉 ′ , 〈 x , y 〉 α + = 〈 x , y 〉 ″ . That is, [〈x, y〉]  _α  = [〈x, y〉′, 〈x, y〉″] . Therefore, presented the existence of a Felbin-fuzzy inner product on X .

Proof. The result is obtained by using Part (I) Proposition 3, Theorem 4.3 [15] and Theorem 5.□

4 Conclusion

In this paper, it is showed that each classical Hilbert space is a Felbin-fuzzy Hilbert space; so all results in classical Hilbert spaces hold on Felbin-fuzzy Hilbert spaces in general, and so there is no need to check the results which have already been proven in classical analysis. Also, it is showed that each Felbin-fuzzy Hilbert space is not necessarily classical Hilbert space. Furthermore by giving a transformation theorem, the above information is formulated.