The Intuitionistic Fuzzy Normed Space of Coefficients

Abstract

Intuitionistic fuzzy normed space is defined using concepts of 𝑡-norm and 𝑡-conorm. The concepts of fuzzy completeness, fuzzy minimality, fuzzy biorthogonality, fuzzy basicity, and fuzzy space of coefficients are introduced. Strong completeness of fuzzy space of coefficients with regard to fuzzy norm and strong basicity of canonical system in this space are proved. Strong basicity criterion in fuzzy Banach space is presented in terms of coefficient operator.

1. Introduction

The fuzzy theory, dating back to Zadeh [1], has emerged as the most active area of research in many branches of mathematics and engineering. Fuzzy set theory is a powerful handset for modeling uncertainty and vagueness in various problems arising in the field of science and engineering. The concept of fuzzy topology may have very important applications in quantum particle physics, particularly in connection with both string and 𝜀^(∞) theories introduced and studied by El Naschie [2–4] and further developed in [5]. So, further development in E-Infinity may lead to a set transitional resolution of quantum entanglement [6]. A large number of research works are appearing these days which deal with the concept of fuzzy set numbers, and the fuzzification of many classical theories has also been made. The concept of Schauder basis in intuitionistic fuzzy normed space and some results related to this concept have recently been studied in [7–9]. These works introduced the concepts of strongly and weakly intuitionistic fuzzy (Schauder) basis in intuitionistic fuzzy Banach spaces (IFBS in short). Some of their properties are revealed. The concepts of strongly and weakly intuitionistic fuzzy approximation properties (sif-AP and wif-AP in short, resp.) are also introduced in these works. It is proved that if the intuitionistic fuzzy space has a sif-basis, then it has a sif-AP. All the results in these works are obtained on condition that IFBS admits equivalent topology using the family of norms generated by 𝑡-norm and 𝑡-conorm (we will define them later).

In our work, we define the basic concepts of classical basis theory in intuitionistic fuzzy normed spaces (IFNS in short). Concepts of weakly and strongly fuzzy spaces of coefficients are introduced. Strong completeness of these spaces with regard to fuzzy norm and strong basicity of canonical system in them is proved. Strong basicity criterion in fuzzy Banach space is presented in terms of coefficient operator.

In Section 2, we recall some notations and concepts. In Section 3, we state our main results. We first define the fuzzy space of coefficients and then introduce the corresponding fuzzy norms. We prove that for nondegenerate system the corresponding fuzzy space of coefficients is strongly fuzzy complete. Moreover, we show that the canonical system forms a strong basis for this space.

2. Some Preliminary Notations and Concepts

We will use the usual notations: 𝑁 will denote the set of all positive integers, 𝑅 will be the set of all real numbers, 𝐶 will be the set of complex numbers, and 𝐾 will denote a field of scalars (𝐾≡𝑅, or 𝐾≡𝐶), 𝑅₊≡(0,+∞). We state some concepts and facts from IFNS theory to be used later.

One of the most important problems in fuzzy topology is to obtain an appropriate concept of intuitionistic fuzzy normed space. This problem has been investigated by Park [10]. He has introduced and studied a notion of intuitionistic fuzzy metric space. We recall it.

Definition 2.1. A binary operation ∗∶[0,1]²→[0,1] is a continuous 𝑡-norm if it satisfies the following conditions:(a)∗ is associative and commutative,(b)∗ is continuous,(c)𝑎∗1=𝑎, ∀𝑎∈[0,1],(d)𝑎∗𝑏≤𝑐∗𝑑 whenever 𝑎≤𝑐 and 𝑏≤𝑑, ∀𝑎, 𝑏, 𝑐, 𝑑∈[0,1].

Example 2.2. Two typical examples of continuous 𝑡-norm are 𝑎∗𝑏=𝑎𝑏 and 𝑎∗𝑏=min{𝑎;𝑏}.

Definition 2.3. A binary operation ⋄∶[0,1]²→[0,1] is a continuous 𝑡-conorm if it satisfies the following conditions:(𝛼)⋄ is associative and commutative,(𝛽)⋄ is continuous,(𝛾)𝑎⋄0=𝑎, ∀𝑎∈[0,1], (𝜂)𝑎⋄𝑏≤𝑐⋄𝑑 whenever 𝑎≤𝑐 and 𝑏≤𝑑, ∀𝑎,𝑏,𝑐,𝑑∈[0,1].

Example 2.4. Two typical examples of continuous 𝑡-conorm are 𝑎⋄𝑏=min{𝑎+𝑏;1} and 𝑎⋄𝑏=max{𝑎;𝑏}.

Definition 2.5. Let 𝑋be a linear space over a field 𝐾. Functions 𝜇;𝜈∶𝑋×𝑅→[0,1] are called fuzzy norms on 𝑋if they hold the following conditions:(1)𝜇(𝑥;𝑡)=0, ∀𝑡≤0, ∀𝑥∈𝑋,(2)𝜇(𝑥;𝑡)=1, ∀𝑡>0⇒𝑥=0,(3)𝜇(𝑐𝑥;𝑡)=𝜇(𝑥;𝑡/|𝑐|), ∀𝑐≠0,(4)𝜇(𝑥;⋅)∶𝑅→[0,1] is a nondecreasing function of 𝑡 for ∀𝑥∈𝑋 and lim_𝑡→∞𝜇(𝑥;𝑡)=1, ∀𝑥∈X,(5)𝜇(𝑥;𝑠)∗𝜇(𝑦;𝑡)≤𝜇(𝑥+𝑦;𝑠+𝑡), ∀𝑥,𝑦∈𝑋, ∀𝑠,𝑡∈𝑅,(6)𝜈(𝑥;𝑡)=1, ∀𝑡≤0, ∀𝑥∈𝑋,(7)𝜈(𝑥;𝑡)=0, ∀𝑡<0⇒𝑥=0,(8)𝜈(𝑐𝑥;𝑡)=𝜈(𝑥;𝑡/|𝑐|), ∀𝑐≠0,(9)𝜈(𝑥;⋅)∶𝑅→[0,1] is a nonincreasing function of 𝑡 for ∀𝑥∈𝑋 and lim_𝑡→∞𝜈(𝑥;𝑡)=0, ∀𝑥∈𝑋,(10)𝜈(𝑥;𝑠)⋄𝜈(𝑦;𝑡)≥𝜈(𝑥+𝑦;𝑠+𝑡), ∀𝑥,𝑦∈𝑋, ∀𝑠,𝑡∈𝑅,(11)𝜇(𝑥;𝑡)+𝜈(𝑥;𝑡)≤1, ∀𝑥∈𝑋, ∀𝑡∈𝑅.Then the 5-tuple (𝑋;𝜇;𝜈;∗;⋄) is said to be an intuitionistic fuzzy normed space (shortly IFNS).

Example 2.6. Let (𝑋;‖⋅‖) be a normed space. Denote 𝑎∗𝑏=𝑎𝑏 and 𝑎⋄𝑏=min{𝑎+𝑏;1}, for∀𝑎,𝑏∈[0,1], and define 𝜇 and 𝜈 as follows: 𝜇()=𝑡𝑥;𝑡‖𝑥‖𝜈()=𝑡𝑡+,𝑡>0,0,𝑡≤0,𝑥;𝑡𝑡+‖𝑥‖,𝑡>0,1,𝑡≤0. (2.1) Then (𝑋;𝜇;𝜈;∗;⋄) is an IFNS.
The above concepts allow to introduce the following kinds of convergence (or topology) in IFNS.

Definition 2.7. Let (𝑋;𝜇;𝜈) be a fuzzy normed space, and let {𝑥_𝑛}_𝑛∈𝑁⊂𝑋 be some sequence, then it is said to be strongly intuitionistic fuzzy convergent to 𝑥∈𝑋 (denoted by 𝑥−→_𝑛^𝑠𝑥, 𝑛→∞ or 𝑠-lim_𝑛→∞𝑥_𝑛=𝑥 in short) if and only if for ∀𝜀>0, ∃𝑛₀=𝑛₀(𝜀)∶𝜇(𝑥_𝑛−𝑥;𝑡)≥1−𝜀, 𝜈(𝑥_𝑛−𝑥;𝑡)≤𝜀, ∀𝑛≥𝑛₀, ∀𝑡∈𝑅.

Definition 2.8. Let (𝑋;𝜇;𝜈) be a fuzzy normed space, and let{𝑥_𝑛}_𝑛∈𝑁⊂𝑋 be some sequence, then it is said to be weakly intuitionistic fuzzy convergent to 𝑥∈𝑋 (denoted by 𝑥−_𝑛^𝑤→𝑥, 𝑛→∞, or 𝑤-lim_𝑛→∞𝑥_𝑛=𝑥 in short) if and only if for ∀𝑡∈𝑅₊, ∀𝜀>0, ∃𝑛₀=𝑛₀(𝜀;𝑡)∶𝜇(𝑥_𝑛−𝑥;𝑡)≥1−𝜀, 𝜈(𝑥_𝑛−𝑥;𝑡)≤𝜀, ∀𝑛≥𝑛₀. More details on these concepts can be found in [10–19].

Let (𝑋;𝜇;𝜈) be an IFNS, and let 𝑀⊂𝑋 be some set. By 𝐿[𝑀], we denote the linear span of 𝑀 in 𝑋. The weakly (strongly) intuitionistic fuzzy convergent closure of 𝐿[𝑀] will be denoted by 𝐿_𝑠[𝑀] (𝐿_𝑤[𝑀]). If 𝑋 is complete with respect to the weakly (strongly) intuitionistic fuzzy convergence, then we will call it intuitionistic fuzzy weakly (strongly) Banach space (𝐼𝐹𝐵_𝑤𝑆 or 𝑋_𝑤 (𝐼𝐹𝐵_𝑠𝑆 or 𝑋_𝑠) in short). Let 𝑋 be an 𝐼𝐹𝐵_𝑠𝑆 (𝐼𝐹𝐵_𝑤𝑆). We denote by 𝑋^∗_𝑠 (𝑋^∗_𝑤) the linear space of linear and continuous in 𝐼𝐹𝐵_𝑠𝑆 (𝐼𝐹𝐵_𝑤𝑆) functionals over the same field 𝐾.

Now, we define the corresponding concepts of basis theory for IFNS. Let {𝑥_𝑛}_𝑛∈𝑁⊂𝑋 be some system.

Definition 2.9. System {𝑥_𝑛}_𝑛∈𝑁 is called 𝑠-complete (𝑤-complete) in 𝑋_𝑠 (in 𝑋_𝑤) if 𝐿_𝑠[{𝑥_𝑛}_𝑛∈𝑁]≡𝑋_𝑠(𝐿_𝑤[{𝑥_𝑛}_𝑛∈𝑁]≡𝑋_𝑤).

Definition 2.10. System {𝑥^∗_𝑛}_𝑛∈𝑁⊂𝑋^∗_𝑠({𝑥^∗_𝑛}_𝑛∈𝑁⊂𝑋^∗_𝑤) is called 𝑠-biorthogonal (𝑤-biorthogonal) to the system {𝑥_𝑛}_𝑛∈𝑁 if 𝑥^∗_𝑛(𝑥_𝑘)=𝛿_𝑛𝑘, ∀𝑛,𝑘∈𝑁, where 𝛿_𝑛𝑘 is the Kronecker symbol.

Definition 2.11. System {𝑥_𝑛}_𝑛∈𝑁⊂𝑋_𝑠({𝑥_𝑛}_𝑛∈𝑁⊂𝑋_𝑤) is called 𝑠-linearly (𝑤-linearly) independent in 𝑋 if ∑^∞_𝑛=1𝜆_𝑛𝑥_𝑛=0 in 𝑋_𝑠 (in 𝑋_𝑤) implies 𝜆_𝑛=0, ∀𝑛∈𝑁.

Definition 2.12. System {𝑥_𝑛}_𝑛∈𝑁⊂𝑋_𝑠({𝑥_𝑛}_𝑛∈𝑁⊂𝑋_𝑤) is called 𝑠-basis ( 𝑤-basis) for 𝑋_𝑠 (for 𝑋_𝑤) if ∀𝑥∈𝑋, ∃!{𝜆_𝑛}_𝑛∈𝑁⊂𝐾∶∑^∞_𝑛=1𝜆_𝑛𝑥_𝑛=𝑥 in 𝑋_𝑠 (in 𝑋_𝑤).

We will also need the following concept.

Definition 2.13. System {𝑥_𝑛}_𝑛∈𝑁⊂𝑋 is called nondegenerate if 𝑥_𝑛≠0, ∀𝑛∈𝑁.

3. Main Results

3.1. Space of Coefficients

Let 𝑋 be an IFNS, and let {𝑥_𝑛}_𝑛∈𝑁⊂𝑋 be some system.

Assume that𝒦^𝑤_𝑥≡𝜆_{𝑛𝑛∈𝑁}⊂𝐶∶^∞_𝑛=1𝜆_𝑛𝑥_𝑛convergesin𝑋_𝑤,𝒦^𝑠_𝑥≡𝜆_{𝑛𝑛∈𝑁}⊂𝐶∶^∞_𝑛=1𝜆_𝑛𝑥_𝑛convergesin𝑋_𝑠. (3.1)

It is not difficult to see that 𝒦^𝑤_𝑥 and 𝒦^𝑠_𝑥 are linear spaces with regard to component-specific summation and component-specific multiplication by a scalar. Take ∀𝜆≡{𝜆_𝑛}_𝑛∈𝑁∈𝒦^𝑤_𝑥, and assume that𝜇_𝐾𝜆;𝑡=inf_𝑚𝜇^𝑚_𝑛=1𝜆_𝑛𝑥_𝑛;𝑡;𝜈_𝐾𝜆;𝑡=sup_𝑚𝜈^𝑚_𝑛=1𝜆_𝑛𝑥_𝑛;𝑡. (3.2)

Let us show that 𝜇_𝐾 and 𝜈_𝐾 satisfy the conditions (1)–(11).(1)It is clear that 𝜇_𝐾(𝜆;𝑡)=0, ∀𝑡≤0.(2)Let 𝜇_𝐾(𝜆;𝑡)=1, ∀𝑡>0. Hence, 𝜇(∑^𝑚_𝑛=1𝜆_𝑛𝑥_𝑛;𝑡)=1, ∀𝑚∈𝑁, ∀𝑡>0. Suppose that the system {𝑥_𝑛}_𝑛∈𝑁 is nondegenerate. It follows from the above-stated relations that, for 𝑚=1, we have 𝜇(𝜆₁𝑥₁;𝑡)=1, ∀𝑡>0. Hence, 𝜆₁𝑥₁=0⇒𝜆₁=0. Continuing this way, we get at the end of this process that𝜆_𝑛=0, ∀𝑛∈𝑁, that is, 𝜆=0.(3)The validity of relation 𝜇_𝐾(𝐴𝜆;𝑡)=𝜇_𝐾(𝜆;𝑡/|𝑐|), ∀𝑐≠0 is beyond any doubt.(4) As 𝜇(𝑥;⋅) is a nondecreasing function on 𝑅 it is not difficult to see that 𝜇_𝐾(𝜆;⋅) has the same property. Let us show that lim_𝑡→∞𝜇_𝐾(𝜆;𝑡)=1. Take ∀𝜀>0. Let 𝑆_𝑚=∑^𝑚_𝑛=1𝜆_𝑛𝑥_𝑛 and 𝑤-lim_𝑚→∞𝑆_𝑚=𝑆∈𝑋_𝑤. It is clear that ∃𝑡₀>0∶𝜇(𝑆;𝑡₀)≥1−𝜀. Then it follows from the definition of 𝑤-lim_𝑚 that∃𝑚₀(𝜀;𝑡₀)∶𝜇(𝑆_𝑚−𝑆;𝑡₀)≥1−𝜀,∀𝑚≥𝑚₀(𝜀;𝑡₀). Property (4) implies 𝜇𝑆_𝑚;2𝑡₀=𝜇𝑆_𝑚−𝑆+𝑆;𝑡₀+𝑡₀≥𝜇𝑆_𝑚−𝑆;𝑡₀∗𝜇𝑆;𝑡₀. (3.3) As a result, we get 𝜇𝑆_𝑚;𝑡₀≥1−𝜀,∀𝑚≥𝑚₀𝜀;𝑡₀. (3.4) As𝜇(𝑥;⋅) is a nondecreasing function of𝑡, it follows from (3.4) that 𝜇𝑆_𝑚;𝑡≥1−𝜀,∀𝑚≥𝑚₀𝜀;𝑡₀,∀𝑡≥𝑡₀. (3.5) We have 𝜇_𝐾𝜆;𝑡=inf_𝑚𝜇𝑆_𝑚;𝑡=min𝜇𝑆₁;𝑡;…;𝜇𝑆_𝑚0−1;𝑡;inf_𝑚≥𝑚0𝜇𝑆_𝑚;𝑡, (3.6) where𝑚₀=𝑚₀(𝜀;𝑡₀). Aslim_𝑡→∞𝜇(𝑆_𝑘;𝑡)=1 for∀𝑘∈𝑁, we have∃𝑡_𝑘(𝜀);∀𝑡≥𝑡_𝑘(𝜀)∶𝜇(𝑆_𝑘;𝑡)≥1−𝜀,𝑘=1,𝑚₀−1. Let𝑡⁰_𝜀=max{𝑡_𝑘(𝜀),𝑘=1,𝑚₀−1}, then it is clear that 𝜇𝑆_𝑘;𝑡≥1−𝜀,∀𝑡≥𝑡⁰_𝜀. (3.7) It follows from (3.5) and (3.6) that inf_𝑚≥𝑚0𝜇𝑆_𝑚;𝑡≥1−𝜀,∀𝑡≥𝑡₀. (3.8) Let𝑡_𝜀=max{𝑡₀;𝑡⁰_𝜀}. Hence, we obtain from (3.6) and (3.7) that 𝜇_𝐾𝜆;𝑡≥1−𝜀,∀𝑡≥𝑡_𝜀. (3.9) Thus,lim_𝑡→∞𝜇_𝐾(𝜆;𝑡)=1,∀𝜆∈𝒦^𝑤_𝑥.(5) Let 𝜆,𝜇∈𝒦^𝑤_𝑥(𝜆≡{𝜆_𝑛}_𝑛∈𝑁;𝜇≡{𝜇_𝑛}_𝑛∈𝑁) and 𝑠,𝑡∈𝑅. We have 𝜇_𝐾𝜆+𝜇;𝑠+𝑡=inf_𝑚𝜇^𝑚_𝑛=1𝜆_𝑛+𝜇_𝑛𝑥_𝑛;𝑠+𝑡=inf_𝑚𝜇^𝑚_𝑛=1𝜆_𝑛𝑥_𝑛+^𝑚_𝑛=1𝜇_𝑛𝑥_𝑛;𝑠+𝑡≥inf_𝑚𝜇^𝑚_𝑛=1𝜆_𝑛𝑥_𝑛;𝑠∗𝜇^𝑚_𝑛=1𝜇_𝑛𝑥_𝑛=;𝑡inf_𝑚𝜇^𝑚_𝑛=1𝜆_𝑛𝑥_𝑛;𝑠∗inf_𝑚𝜇^𝑚_𝑛=1𝜇_𝑛𝑥_𝑛;𝑡=𝜇𝜆;𝑠∗𝜇𝜇;𝑡. (3.10)(6) As 𝜈(𝑥;𝑡)=1, ∀𝑡≤0, it is clear that 𝜈_𝐾(𝜆;𝑡)=1, ∀𝑡≤0, ∀𝜆∈𝒦^𝑤_𝑥.(7) Let the system {𝑥_𝑛}_𝑛∈𝑁 be nondegenerate. Assume that 𝜈_𝐾(𝜆;𝑡)=0, ∀𝑡>0, then𝜈(∑^𝑚_𝑛=1𝜆_𝑛𝑥_𝑛;𝑡)=0,∀𝑡>0,∀𝑚∈𝑁. For𝑚=1, we have𝜈(𝜆₁𝑥₁;𝑡)=0,∀𝑡>0⇒𝜆₁𝑥₁=0⇒𝜆₁=0. Continuing this process, we get𝜆_𝑛=0, ∀𝑛∈𝑁⇒𝜆=0.(8) Clearly, 𝜈_𝐾(𝑐𝜆;𝑡)=𝜈_𝐾(𝜆;𝑡/|𝑐|), ∀𝑐≠0.(9) It follows from the property (9) that 𝜈(𝑥;⋅) is a nonincreasing function on 𝑅. Therefore, 𝜈_𝐾(𝜆;⋅) is a nonincreasing function on 𝑅. Let us show that lim_𝑡→∞𝜈_𝐾(𝜆;𝑡)=0. Let 𝑆_𝑚=∑^𝑚_𝑛=1𝜆_𝑛𝑥_𝑛 and 𝑤-lim_𝑚→∞𝑆_𝑚=𝑆∈𝑋. Take ∀𝜀>0. It is clear that ∃𝑡₀>0∶𝜈(𝑆;𝑡₀)≤𝜀. Then it follows from the definition of 𝑤-lim_𝑚 that ∃𝑚₀=𝑚₀(𝜀;𝑡₀)∶𝜈(𝑆_𝑚−𝑆;𝑡₀)≤𝜀, ∀𝑚≥𝑚₀. We have 𝜈𝑆_𝑚;𝑡₀=𝜈𝑆_𝑚−𝑆+𝑆;𝑡₀+𝑡₀≤𝜈𝑆_𝑚−𝑆;𝑡₀⋄𝜈𝑆;𝑡₀≤𝜀,∀𝑚≥𝑚₀ (3.11) As 𝜈(𝑥;⋅) is a nonincreasing function, it is clear that 𝜈𝑆_𝑚;𝑡≤𝜀,∀𝑚≥𝑚₀,∀𝑡≥𝑡₀. (3.12) We have 𝜈_𝐾𝜆;𝑡=sup_𝑚𝜈𝑆_𝑚;𝑡=max𝜈𝑆₁;𝑡;…;𝜈𝑆_𝑚0−1;𝑡;sup_𝑚≥𝑚0𝜈𝑆_𝑚;𝑡. (3.13) As lim_𝑡→∞𝜈(𝑆_𝑘;𝑡)=0 for ∀𝑘∈𝑁, we have ∃𝑡_𝑘(𝜀);∀𝑡≥𝑡_𝑘(𝜀)∶𝜈(𝑆_𝑘;𝑡)≤𝜀, 𝑘=1,𝑚₀−1. Let 𝑡⁰_𝜀=max{𝑡_𝑘(𝜀),𝑘=1,𝑚₀−1}. It is clear that 𝜈(𝑆_𝑘;𝑡)≤𝜀, ∀𝑡≥𝑡⁰_𝜀. It follows from (3.12) that sup_𝑚≥𝑚0𝜈(𝑆_𝑚;𝑡)≤𝜀, ∀𝑡≥𝑡₀. Let 𝑡_𝜀=max{𝑡₀;𝑡⁰_𝜀}, then it is clear that 𝜈_𝐾(𝜆;𝑡)≤𝜀, ∀𝑡≥𝑡_𝜀⇒lim_𝑡→∞𝜈_𝐾(𝜆;𝑡)=0.(10) Let 𝜆,𝜇∈𝒦^𝑤_𝑥(𝜆≡{𝜆_𝑛}_𝑛∈𝑁;𝜇≡{𝜇_𝑛}_𝑛∈𝑁) and 𝑠,𝑡∈𝑅. We have 𝜈_𝐾𝜆+𝜇;𝑠+𝑡=sup_𝑚𝜈^𝑚_𝑛=1𝜆_𝑛+𝜇_𝑛𝑥_𝑛;𝑠+𝑡≤sup_𝑚𝜈^𝑚_𝑛=1𝜆_𝑛𝑥_𝑛;𝑠⋄𝜈^𝑚_𝑛=1𝜇_𝑛𝑥_𝑛=;𝑡sup_𝑚𝜈^𝑚_𝑛=1𝜆_𝑛𝑥_𝑛;𝑠⋄sup_𝑚𝜈^𝑚_𝑛=1𝜇_𝑛𝑥_𝑛;𝑡=𝜈_𝐾𝜆;𝑠⋄𝜈_𝐾𝜇;𝑡. (3.14)(11)Consider the following: 𝜇_𝐾𝜆;𝑡+𝜈_𝐾𝜆;𝑡=inf_𝑚𝜇^𝑚_𝑛=1𝜆_𝑛𝑥_𝑛;𝑡+sup_𝑚𝜈^𝑚_𝑛=1𝜆_𝑛𝑥_𝑛;𝑡≤sup_𝑚𝜇^𝑚_𝑛=1𝜆_𝑛𝑥_𝑛;𝑡+𝜈^𝑚_𝑛=1𝜆_𝑛𝑥_𝑛;𝑡≤1,∀𝜆∈𝒦^𝑤_𝑥,∀𝜆∈𝑅. (3.15) Thus, we have proved the validity of the following.

Theorem 3.1. Let (𝑋;𝜇;𝜈) be a fuzzy normed space, and let {𝑥_𝑛}_𝑛∈𝑁⊂𝑋 be a nondegenerate system, then the space of coefficients (𝒦^𝑠_𝑥;𝜇_𝐾;𝜈_𝐾) is also strongly fuzzy normed space.

The following theorem is proved in absolutely the same way.

Theorem 3.2. Let (𝑋;𝜇;𝜈) be a fuzzy normed space, and let {𝑥_𝑛}_𝑛∈𝑁⊂𝑋 be a nondegenerate system, then the space of coefficients (𝒦^𝑤_𝑥;𝜇_𝐾;𝜈_𝐾) is also weakly fuzzy normed space.

3.2. Completeness of the Space of Coefficients

Subsequently, we assume that (𝑋;𝜇;𝜈) is IFBS. Let us show that (𝒦^𝑠_𝑥;𝜇_𝐾;𝜈_𝐾) is a strongly fuzzy complete normed space. First, we prove the following.

Lemma 3.3. Let 𝑥₀≠0, 𝑥₀∈𝑋, and let {𝜆_𝑛}_𝑛∈𝑁⊂𝑅 be some sequence. If 𝑠-lim_𝑛→∞(𝜆_𝑛𝑥₀)=0, that is, ∀𝜀>0, ∃𝑛₀=𝑛₀(𝜀)∶𝜇(𝜆_𝑛𝑥₀;𝑡)>1−𝜀, 𝜈(𝜆_𝑛𝑥₀;𝑡)<𝜀, ∀𝑡∈𝑅₊, and ∀𝑛≥𝑛₀, then 𝜆_𝑛→0, 𝑛→∞.

Proof. As 𝑥₀≠0, it is clear that ∃𝑡₀>0∶𝜇(𝑥₀;𝑡₀)<1. We have 𝜇(𝜆_𝑛𝑥₀;𝑡)=𝜇(𝑥₀;(𝑡)/|𝜆_𝑛|) for 𝜆_𝑛≠0. Assume that the relation lim_𝑛→∞𝜆_𝑛=0 is not true, then ∃{𝜆_𝑛𝑘}_𝑘∈𝑁 and ∃𝛿>0∶|𝜆_𝑛𝑘|≥𝛿, ∀𝑘∈𝑁. It is clear that lim_𝑘→∞𝜇(𝜆_𝑛𝑘𝑥₀;𝑡)=1 uniformly in 𝑡. On the other hand, for 𝑡_𝑘=|𝜆_𝑛𝑘|𝑡₀, we have 𝜇(𝜆_𝑛𝑘𝑥₀;𝑡_𝑘)=𝜇(𝑥₀;𝑡₀)<1. So we came upon a contradiction which proves the lemma.

Further, we assume that the following condition is also fulfilled.(12) The functions 𝜇(𝑥;⋅), 𝜈(𝑥;⋅)∶𝑅→[0,1] are continuous for ∀𝑥∈𝑋.

Take𝑠-fundamental sequence{𝜆_𝑛}_𝑛∈𝑁⊂𝒦^𝑠_𝑥,𝜆_𝑛≡{𝜆_𝑘^(𝑛)}_𝑘∈𝑁. Thenlim_{𝑛,𝑚→∞}𝜇_𝐾(𝜆_𝑛−𝜆_𝑚;𝑡)=1 uniformly in𝑡∈𝑅, that is, lim_{𝑛,𝑚→∞}inf_𝑟𝜇^𝑟_𝑘=1𝜆_𝑘^(𝑛)−𝜆_𝑘^(𝑚)𝑥_𝑘=1, (3.16)

uniformly in 𝑡∈𝑅. Take ∀𝑘₀∈𝑁 and fix it. We have𝜆_𝑘^(𝑛)₀−𝜆_𝑘^(𝑚)₀𝑥_𝑘0=^𝑘0_𝑘=1𝜆_𝑘^(𝑛)−𝜆_𝑘^(𝑚)𝑥_𝑘−^𝑘0−1_𝑘=1𝜆_𝑘^(𝑛)−𝜆_𝑘^(𝑚)𝑥_𝑘. (3.17)

Then from property (5), we get𝜇𝜆_𝑘^(𝑛)₀−𝜆_𝑘^(𝑚)₀𝑥_𝑘0;𝑡≥𝜇^𝑘0_𝑘=1𝜆_𝑘^(𝑛)−𝜆_𝑘^(𝑚)𝑥_𝑘;𝑡2∗𝜇^𝑘0−1_𝑘=1𝜆_𝑘^(𝑛)−𝜆_𝑘^(𝑚)𝑥_𝑘;𝑡2. (3.18)

It follows directly from this relation that lim_{𝑛,𝑚→∞}𝜇((𝜆_𝑘^(𝑛)₀−𝜆_𝑘^(𝑚)₀)𝑥_𝑘0;𝑡)=1 uniformly in 𝑡. As 𝑥_𝑘0≠0, Lemma 3.3 implies lim_{𝑛,𝑚→∞}|𝜆_𝑘^(𝑛)₀−𝜆_𝑘^(𝑚)₀|=0, that is, the sequence {𝜆_𝑘^(𝑛)₀}_𝑛∈𝑁 is fundamental in 𝑅. Let 𝜆_𝑘^(𝑛)₀→𝜆_𝑘0, as 𝑛→∞. Denote 𝜆≡{𝜆_𝑛}_𝑛∈𝑁. Let us show that lim_𝑛→∞𝜇_𝐾(𝜆_𝑛−𝜆;𝑡)=1 uniformly in 𝑡. Take ∀𝜀>0. It is clear that ∃𝑛₀, ∀𝑛≥𝑛₀, ∀𝑝∈𝑁∶𝜇_𝐾(𝜆_𝑛−𝜆_𝑛+𝑝;𝑡)>1−𝜀, ∀𝑡∈𝑅. Consequently,inf_𝑟𝜇^𝑟_𝑘=1𝜆_𝑘^(𝑛)−𝜆_𝑘^(𝑛+𝑝)𝑥_𝑘;𝑡>1−𝜀,∀𝑛≥𝑛₀,∀𝑝∈𝑁,∀𝑡∈𝑅₊. (3.19)

Hence, 𝜇^𝑟_𝑘=1𝜆_𝑘^(𝑛)−𝜆_𝑘^(𝑛+𝑝)𝑥_𝑘;𝑡>1−𝜀,∀𝑛≥𝑛₀,∀𝑟,𝑝∈𝑁,∀𝑡∈𝑅₊. (3.20) As shown above, lim_{𝑛,𝑚→∞}𝜇((𝜆_𝑘^(𝑛)−𝜆_𝑘^(𝑚))𝑥_𝑘;𝑡)=1 uniformly in 𝑡∈𝑅₊. Now let us take into account the fact that lim_𝑚→∞𝜇(𝜆_𝑘^(𝑚)𝑥_𝑘;𝑡)=𝜇(𝜆_𝑘𝑥_𝑘;𝑡), ∀𝑡∈𝑅₊. Indeed, if 𝜆_𝑘=0, then 𝜇(0;𝑡)=1, ∀𝑡∈𝑅₊, and clearly, lim_𝑚→∞𝜇(𝜆_𝑘^(𝑚)𝑥_𝑘;𝑡)=1 for ∀𝑡∈𝑅₊. If 𝜆_𝑘≠0, then for sufficiently large values of 𝑚 we have 𝜆_𝑘^(𝑚)≠0, and as a result,𝜇𝜆_𝑘^(𝑚)𝑥_𝑘;𝑡=𝜇𝑥_𝑘;𝑡𝜆_𝑘^{(𝑚)𝑚→∞}⟶𝜇𝑥_𝑘;𝑡𝜆_𝑘=𝜇𝜆_𝑘𝑥_𝑘;𝑡,∀𝑡∈𝑅₊. (3.21)

Passage to the limit in the inequality (3.20) as 𝑝→∞ yields𝜇^𝑟_𝑘=1𝜆_𝑘^(𝑛)−𝜆_𝑘𝑥_𝑘;𝑡≥1−𝜀,∀𝑛≥𝑛₀,∀𝑟∈𝑁,∀𝑡∈𝑅₊. (3.22) We have𝜇^𝑟+𝑝_𝑘=𝑟𝜆_𝑘^(𝑛)−𝜆_𝑘𝑥_𝑘;𝑡=𝜇^𝑟+𝑝_𝑘=1𝜆_𝑘^(𝑛)−𝜆_𝑘𝑥_𝑘−^𝑟−1_𝑘=1𝜆_𝑘^(𝑛)−𝜆_𝑘𝑥_𝑘;𝑡≥𝜇^𝑟+𝑝_𝑘=1𝜆_𝑘^(𝑛)−𝜆_𝑘𝑥_𝑘;𝑡2∗𝜇^𝑟−1_𝑘=1𝜆_𝑘^(𝑛)−𝜆_𝑘𝑥_𝑘;𝑡2≥1−𝜀,∀𝑛≥𝑛₀,∀𝑟,𝑝∈𝑁,∀𝑡∈𝑅₊. (3.23)

As 𝜆_𝑛∈𝒦^𝑠_𝑥, it is clear that ∃𝑚₀^(𝑛)∶∀𝑚≥𝑚₀^(𝑛), ∀𝑝∈𝑁:𝜇^𝑚+𝑝_𝑘=𝑚𝜆_𝑘^(𝑛)𝑥_𝑘;𝑡>1−𝜀,∀𝑡∈𝑅₊. (3.24) We have 𝜇^𝑚+𝑝_𝑘=𝑚𝜆_𝑘𝑥_𝑘;𝑡=𝜇^𝑚+𝑝_𝑘=𝑚𝜆_𝑘−𝜆_𝑘^(𝑛)𝑥_𝑘+^𝑚+𝑝_𝑘=𝑚𝜆_𝑘^(𝑛)𝑥_𝑘;𝑡≥𝜇^𝑚+𝑝_𝑘=𝑚𝜆_𝑘−𝜆_𝑘^(𝑛)𝑥_𝑘;𝑡2∗𝜇^𝑚+𝑝_𝑘=𝑚𝜆_𝑘^(𝑛)𝑥_𝑘;𝑡2≥1−𝜀,∀𝑚≥𝑚₀^(𝑛),∀𝑝∈𝑁,∀𝑡∈𝑅₊. (3.25)

It follows that the series ∑^∞_𝑘=1𝜆_𝑘𝑥_𝑘 is strongly fuzzy convergent, that is, ∃𝑠-lim_𝑚→∞∑^m_k=1𝜆_𝑘x_𝑘. Consequently,𝜆∈𝒦^𝑠_𝑥, and the relation (3.22) implies that lim_𝑛→∞𝜇_𝐾(𝜆_𝑛−𝜆;𝑡)=1 uniformly in, ∀𝑡∈𝑅₊. It can be proved in a similar way that lim_𝑛→∞𝜈_𝐾(𝜆_𝑛−𝜆;t)=0 uniformly in ∀𝑡∈𝑅₊. As a result, we obtain that the space (𝒦^𝑠_𝑥;𝜇_𝐾;𝜈_𝐾) is strongly fuzzy complete. Thus, we have proved the following.

Theorem 3.4. Let (𝑋;𝜇;𝜈) be a fuzzy Banach space with condition (12), and let {𝑥_𝑛}_𝑛∈𝑁⊂𝑋 be a nondegenerate system, then the space of coefficients (𝒦^𝑠_𝑥;𝜇_𝐾;𝜈_𝐾) is a strongly fuzzy complete normed space.

Consider operator 𝑇∶𝒦^𝑠_𝑥→𝑋 defined by𝑇𝜆=^∞_𝑛=1𝜆_𝑛𝑥_𝑛,𝜆≡𝜆_{𝑛𝑛∈𝑁}∈𝒦^𝑠_𝑥. (3.26)

Let 𝑠-lim_𝑛→∞𝜆_𝑛=𝜆 in 𝒦^𝑠_𝑥, where 𝜆_𝑛≡{𝜆_𝑘^(𝑛)}_𝑘∈𝑁∈𝒦^𝑠_𝑥. We have𝜇𝑇𝜆_𝑛−𝑇𝜆;𝑡=𝜇^∞_𝑘=1𝜆_𝑘^(𝑛)−𝜆_𝑘𝑥_𝑘;𝑡≥inf_𝑚𝜇^𝑚_𝑘=1𝜆_𝑘^(𝑛)−𝜆_𝑘𝑥_𝑘;𝑡=𝜇𝜆_𝑛−𝜆;𝑡. (3.27)

It follows directly that 𝑠-lim_𝑛→∞𝑇𝜆_𝑛=𝑇𝜆, that is, the operator 𝑇 is strongly fuzzy continuous. Let 𝜆∈Ker𝑇, that is, 𝑇𝜆=0⇒∑^∞_𝑛=1𝜆_𝑛𝑥_𝑛=0, where 𝜆≡{𝜆_𝑛}_𝑛∈𝑁∈𝒦^𝑠_𝑥. It is clear that if the system{𝑥_𝑛}_𝑛∈𝑁 is 𝑠-linearly independent, then 𝜆_𝑛=0, ∀𝑛∈𝑁, and as a result, Ker𝑇={0}. In this case,∃𝑇⁻¹∶ ImT→𝒦^𝑠_𝑥. If, in addition, ImT is 𝑠-closed in 𝑋, then𝑇⁻¹ is also continuous.

Denote by{𝑒_𝑛}_𝑛∈𝑁⊂𝒦^𝑠_𝑥 a canonical system in𝒦^𝑠_𝑥, where 𝑒_𝑛={𝛿_𝑛𝑘}_𝑘∈𝑁∈𝒦^𝑠_𝑥. Obviously, 𝑇𝑒_𝑛=𝑥_𝑛, ∀𝑛∈𝑁. Let us prove that {𝑒_𝑛}_𝑛∈𝑁 forms an 𝑠-basis for 𝒦^𝑠_𝑥. Take ∀𝜆≡{𝜆_𝑛}_𝑛∈𝑁∈𝒦^𝑠_𝑥 and show that the series ∑^∞_𝑛=1𝜆_𝑛𝑒_𝑛 is strongly fuzzy convergent in 𝒦^𝑠_𝑥. In fact, the existence of 𝑠-lim_𝑚→∞∑^m_n=1𝜆_𝑛𝑥_𝑛 in 𝑋_𝑠 implies that ∀𝜀>0, ∃𝑚₀∈𝑁,𝜇^𝑚+𝑝_𝑛=𝑚𝜆_𝑛𝑥_𝑛;𝑡>1−𝜀,∀𝑚≥𝑚₀,∀𝑝∈𝑁,∀𝑡∈𝑅₊. (3.28)

We have𝜇_𝐾^𝑚+𝑝_𝑛=𝑚𝜆_𝑛𝑒_𝑛;𝑡=inf_𝑟^𝑟_𝑛=𝑚𝜆_𝑛𝑥_𝑛;𝑡≥1−𝜀,∀𝑚≥𝑚₀,∀𝑝∈𝑁,∀𝑡∈𝑅₊. (3.29)

It follows that the series ∑^∞_𝑛=1𝜆_𝑛𝑒_𝑛 is strongly fuzzy convergent in 𝒦^𝑠_𝑥. Moreover,𝜇_𝐾𝜆−^𝑚_𝑛=1𝜆_𝑛𝑒_𝑛;𝑡=𝜇_𝐾…;0;𝜆_𝑚+1;…;𝑡=inf_𝑟𝜇^𝑟_𝑛=𝑚+1𝜆_𝑛𝑥_𝑛;𝑡≥1−𝜀,∀𝑚≥𝑚₀,∀𝑡∈𝑅₊. (3.30)

Consequently, 𝑠-lim_𝑚→∞∑^𝑚_𝑛=1𝜆_𝑛𝑒_𝑛=𝜆, that is, 𝜆=^𝑠∑^𝑚_𝑛=1𝜆_𝑛𝑒_𝑛. Consider the functionals 𝑒^∗_𝑛(𝜆)=𝜆_𝑛, ∀𝑛∈𝑁. Let us show that they are 𝑠-continuous. Let 𝑠-lim_𝑛→∞𝜆_𝑛=𝜆, where 𝜆_𝑛≡{𝜆_𝑘^(𝑛)}_𝑘∈𝑁∈𝒦^𝑠_𝑥. As established in the proof of Theorem 3.4, we have 𝜆_𝑘^(𝑛)→𝜆_𝑘 as 𝑛→∞, that is, 𝑒^∗_𝑘(𝜆_𝑛)→𝑒^∗_𝑘(𝜆) as 𝑛→∞ for ∀𝑘∈𝑁. Thus, 𝑒^∗_𝑘 is 𝑠-continuous in 𝒦^𝑠_𝑥 for ∀𝑘∈𝑁. On the other hand, it is easy to see that 𝑒^∗_𝑛(𝑒_𝑘)=𝛿_𝑛𝑘, ∀𝑛,𝑘∈𝑁, that is, {𝑒^∗_𝑛}_𝑛∈𝑁 is 𝑠-biorthogonal to {𝑒_𝑛}_𝑛∈𝑁. As a result, we obtain that the system {𝑒_𝑛}_𝑛∈𝑁 forms an 𝑠-basis for 𝒦^𝑠_𝑥. So we get the validity of the following.

Theorem 3.5. Let (𝑋;𝜇;𝜈) be a fuzzy Banach space with condition (12), and let {𝑥_𝑛}_𝑛∈𝑁⊂𝑋 be a nondegenerate system. Then the corresponding space of coefficients (𝒦^𝑠_𝑥;𝜇_𝐾;𝜈_𝐾) is strongly fuzzy complete with canonical 𝑠-basis {𝑒_𝑛}_𝑛∈𝑁.

Suppose that the system {𝑥_𝑛}_𝑛∈𝑁 is𝑠-linearly independent and ImT is closed, then it is easily seen that {𝑥_𝑛}_𝑛∈𝑁 forms an 𝑠-basis for ImT, and in case of its 𝑠-completeness in 𝑋_𝑠, it forms an 𝑠-basis for 𝑋_𝑠. In this case, 𝒦^𝑠_𝑥 and 𝑋_𝑠 are isomorphic, and 𝑇 is an isomorphism between them. The opposite of it is also true, that is, if the above-defined operator 𝑇 is an isomorphism between 𝒦^𝑠_𝑥 and 𝑋_𝑠, then the system {𝑥_𝑛}_𝑛∈𝑁 forms an 𝑠-basis for 𝑋_𝑠. We will call 𝑇 a coefficient operator. Thus, the following theorem holds.

Theorem 3.6. Let (𝑋;𝜇;𝜈) be a fuzzy Banach space with condition (12), let {𝑥_𝑛}_𝑛∈𝑁⊂𝑋 be a nondegenerate system, let (𝒦^𝑠_𝑥;𝜇_𝐾;𝜈_𝐾) be a corresponding strongly fuzzy complete normed space, and let 𝑇∶𝒦^𝑠_𝑥→𝑋_s be a coefficient operator. System {𝑥_𝑛}_𝑛∈𝑁 forms an 𝑠-basis for 𝑋_𝑠 if and only if the operator 𝑇 is an isomorphism between 𝒦^𝑠_𝑥 and 𝑋_s.