Schauder Basis, Separability, and Approximation Property in Intuitionistic Fuzzy Normed Space

Abstract

We define and study the concepts of Schauder basis, separability, and approximation property in intuitionistic fuzzy normed spaces and establish some results related to these concepts. We also display here some interesting examples by using classical sequence spaces ℓ_𝑝(1≤𝑝≤∞).

1. Introduction and Background

We will write 𝑤 for the set of all complex sequences 𝑥=(𝑥_𝑘)^∞_𝑘=0. Let ℓ_∞, 𝑐, and 𝑐₀ denote the sets of all bounded, convergent, and null sequences, respectively. We write ℓ_𝑝∶={𝑥∈𝑤∶∑^∞_𝑘=0|𝑥_𝑘|^𝑝<∞} for 1≤𝑝<∞. By 𝑒 and 𝑒^(𝑛)(𝑛∈ℕ), we denote the sequences such that 𝑒_𝑘=1 for 𝑘=0,1,…,𝑒_𝑛^(𝑛)=1, and 𝑒_𝑘^(𝑛)=0(𝑘≠𝑛). Note that 𝑐₀, 𝑐, and ℓ_∞ are Banach spaces with the sup-norm ‖𝑥‖_∞=sup_𝑘|𝑥_𝑘|, and ℓ_𝑝(1≤𝑝<∞) are Banach spaces with the norm ‖𝑥‖_𝑝=(∑|𝑥_𝑘|^𝑝)^1/𝑝.

A sequence (𝑏^(𝑛))^∞_𝑛=0 in a normed linear space 𝑋 is called a Schauder basis [1] if for every 𝑥∈𝑋, there is a unique sequence (𝛽_𝑛)^∞_𝑛=0 of scalars such that 𝑥=∑^∞_𝑛=0𝛽_𝑛𝑏^(𝑛), that is, lim_𝑚→∞‖𝑥−∑^𝑚_𝑛=0𝛽_𝑛𝑏^(𝑛)‖_𝑋=0.

Recently, the concept of intuitionistic fuzzy normed space has been introduced and studied by Saadati and Park [2] and further studied by Mursaleen and Mohiuddine [3–5]. The concept of Schauder basis and its applications have recently been studied by Palomares et al. [6] and by Yılmaz [7]. In this paper, we define and study the concept of Schauder basis, separability, and approximation property in intuitionistic fuzzy normed spaces and establish some results related to these concepts analogous to those of Yılmaz [7]. We also display here some interesting examples by using classical sequence spaces ℓ_𝑝(1≤𝑝≤∞).

In this section, we recall some notations and basic definitions used in this paper.

Definition 1.1. A binary operation ∗:[0,1]×[0,1]→[0,1] is said to be a continuous 𝑡-norm if it satisfies the following conditions:(a)∗ is associative and commutative, (b)∗ is continuous, (c)𝑎∗1=𝑎 for all 𝑎∈[0,1],(d)𝑎∗𝑏≤𝑐∗𝑑 whenever 𝑎≤𝑐 and 𝑏≤𝑑 for each 𝑎,𝑏,𝑐,𝑑∈[0,1].

Definition 1.2. A binary operation ⋄:[0,1]×[0,1]→[0,1] is said to be a continuous 𝑡-conorm if it satisfies the following conditions:(𝑎^′)⋄ is associative and commutative, (𝑏^′)⋄ is continuous, (𝑐^′)𝑎⋄0=𝑎 for all 𝑎∈[0,1],(𝑑^′)𝑎⋄𝑏≤𝑐⋄𝑑 whenever 𝑎≤𝑐 and 𝑏≤𝑑 for each 𝑎,𝑏,𝑐,𝑑∈[0,1].

Using the notions of continuous 𝑡-norm and 𝑡-conorm, Saadati and Park [2] have recently introduced the concept of intuitionistic fuzzy normed space as follows.

Definition 1.3. The five-tuple (𝑋,𝜇,𝜈,∗,⋄) is said to be an intuitionistic fuzzy normed spaces (for short, IFNS) if 𝑋 is a vector space, ∗ is a continuous 𝑡-norm, ⋄ is a continuous 𝑡-conorm, and 𝜇,𝜈 are fuzzy sets on 𝑋×(0,∞) satisfying the following conditions. For every 𝑥,𝑦∈𝑋 and 𝑠,𝑡>0,(i)𝜇(𝑥,𝑡)+𝜈(𝑥,𝑡)≤1, (ii)𝜇(𝑥,𝑡)>0, (iii)𝜇(𝑥,𝑡)=1 if and only if 𝑥=0, (iv)𝜇(𝛼𝑥,𝑡)=𝜇(𝑥,𝑡/|𝛼|) for each 𝛼≠0, (v)𝜇(𝑥,𝑡)∗𝜇(𝑦,𝑠)≤𝜇(𝑥+𝑦,𝑡+𝑠), (vi)𝜇(𝑥,⋅)∶(0,∞)→[0,1] is continuous,(vii)lim_𝑡→∞𝜇(𝑥,𝑡)=1 and lim_𝑡→0𝜇(𝑥,𝑡)=0, (viii)𝜈(𝑥,𝑡)<1, (ix)𝜈(𝑥,𝑡)=0 if and only if 𝑥=0, (x)𝜈(𝛼𝑥,𝑡)=𝜈(𝑥,𝑡/|𝛼|) for each 𝛼≠0, (xi)𝜈(𝑥,𝑡)⋄𝜈(𝑦,𝑠)≥𝜈(𝑥+𝑦,𝑡+𝑠),(xii)𝜈(𝑥,⋅)∶(0,∞)→[0,1] is continuous, and(xiii)lim_𝑡→∞𝜈(𝑥,𝑡)=0 andlim_𝑡→0𝜈(𝑥,𝑡)=1.
In this case (𝜇,𝜈) is called an intuitionistic fuzzy norm.

Example 1.4. Let (𝑋,‖⋅‖) be a normed space, 𝑎∗𝑏=𝑎𝑏, and 𝑎⋄𝑏=min{𝑎+𝑏,1} for all 𝑎,𝑏∈[0,1]. For all 𝑥∈𝑋 and every 𝑡>0, consider 𝜇()=𝑡𝑥,𝑡‖𝑥‖𝜈()=‖𝑥‖𝑡+if𝑡>0,0if𝑡≤0;𝑥,𝑡𝑡+|𝑥|if𝑡>0,1if𝑡≤0. (1.1) Then (𝑋,𝜇,𝜈,∗,⋄) is an IFNS.

Remark 1.5 (see [3]). Let (𝑋,𝜇,𝜈,∗,⋄) be an IFNS with the condition 𝜇(𝑥,𝑡)>0,𝜈(𝑥,𝑡)<1implies𝑥=0∀𝑡∈ℝ. (1.2) Let ‖𝑥‖_𝛼=inf{𝑡>0∶𝜇(𝑥,𝑡)≥𝛼and𝜈(𝑥,𝑡)≤1−𝛼}, for all 𝛼∈(0,1). Then {‖⋅‖_𝛼∶𝛼∈(0,1)} is an ascending family of norms on 𝑋. These norms are called α-norms on 𝑋 corresponding to intuitionistic fuzzy norm (𝜇,𝜈).

2. Some Topological Concepts in IFNS

Recently, the strong and weak intuitionistic fuzzy convergence as well as strong and weak intuitionistic fuzzy limit were discussed by Mursaleen and Mohiuddine [3].

Definition 2.1. Let (𝑋,𝜇,𝜈,∗,⋄) be an IFNS. Then, a sequence (𝑥_𝑘) is said to be(i)weakly intuitionistic fuzzy convergent to 𝑥∈𝑋 if and only if, for every𝜖>0 and 𝛼∈(0,1), there exists some 𝑘₀=𝑘₀(𝛼,𝜖) such that 𝜇(𝑥_𝑘−𝑥,𝜖)≥1−𝛼 and 𝜈(𝑥_𝑘−𝑥,𝜖)≤𝛼for all 𝑘≥𝑘₀. In this case we write 𝑥−−→_𝑘^wif𝑥,(ii)strongly intuitionistic fuzzy convergent to 𝑥∈𝑋 if and only if, for every 𝛼∈(0,1), there exists some 𝑘₀=𝑘₀(𝛼) such that 𝜇(𝑥_𝑘−𝑥,𝑡)≥1−𝛼 and 𝜈(𝑥_𝑘−𝑥,𝑡)≤𝛼 for all 𝑡>0. In this case we write 𝑥−−→_𝑘^sif𝑥.

The following result characterizes the (wif)- and (sif)-limit through α-norms.

Proposition 2.2. Let (𝑋,𝜇₁,𝜈₁,∗,⋄) and (𝑌,𝜇₂,𝜈₂,∗,⋄) be two IFNS satisfying (1.2) and 𝑓∶𝑋→𝑌 be a mapping. Then(i)(wif)-lim_𝑥→𝑥0𝑓(𝑥)=𝐿 if and only if, for each𝛼∈(0,1)lim_{𝑥−𝑥0𝛼(1)→0}𝑓(𝑥)−𝐿_𝛼⁽²⁾=0; (2.1)(ii)(sif)-lim_𝑥→𝑥0𝑓(𝑥)=𝐿 if and only if lim_{𝑥−𝑥0𝛼(1)→0}𝑓(𝑥)−𝐿_𝛼⁽²⁾=0uniformlyin𝛼, (2.2)where ‖⋅‖_𝛼⁽¹⁾ and ‖⋅‖_𝛼⁽²⁾ are α-norms of the intuitionistic fuzzy norms (𝜇₁,𝜈₁) and (𝜇₂,𝜈₂), respectively.

Proof. Here we prove the case (ii). Suppose that (sif)-lim_𝑥→𝑥0𝑓(𝑥)=𝐿. For a given 𝜖>0, there exists some 𝛿=𝛿(𝜖)>0 such that 𝜇₂𝑓(𝑥)−𝐿,𝜖≥𝜇₁𝑥−𝑥₀,𝛿,𝜈₂𝑓(𝑥)−𝐿,𝜖≥𝜈₁𝑥−𝑥₀,𝛿, (2.3) for all 𝑥∈𝑋. For each 𝛼∈(0,1), if 𝑥−𝑥_0𝛼⁽¹⁾=inf𝑡>0∶𝜇₁𝑥−𝑥₀,𝑡≥𝛼,𝜈₁𝑥−𝑥₀,𝑡≤1−𝛼<𝛿, (2.4) then 𝜇₁(𝑥−𝑥₀,𝛿)≥𝛼 and 𝜈₁(𝑥−𝑥₀,𝛿)≤1−𝛼. Hence 𝜇₂(𝑓(𝑥)−𝐿,𝜖)≥𝛼 and 𝜈₂(𝑓(𝑥)−𝐿,𝜖)≤1−𝛼 and so that ‖𝑓(𝑥)−𝐿‖_𝛼⁽²⁾<𝜖. Since 𝛿 does not depend on α, we get lim_{‖𝑥−𝑥0‖𝛼(1)→0}𝑓(𝑥)−𝐿_𝛼⁽²⁾=0uniformlyin𝛼. (2.5)
Conversely, let lim_{‖𝑥−𝑥0‖𝛼}⁽¹⁾_→0‖𝑓(𝑥)−𝐿‖_𝛼⁽²⁾=0 uniformly in α. Given 𝜖>0, there exists some 𝛿=𝛿(𝜖)>0 such that𝑥−𝑥_0𝛼⁽¹⁾<𝛿implies𝑓(𝑥)−𝐿_𝛼⁽²⁾<𝜖 (2.6) for all 𝑥∈𝑋 and 𝛼∈(0,1). Choose some 𝜆<𝜇₁(𝑥−𝑥₀,𝛿) and 𝜆>𝜈₁(𝑥−𝑥₀,𝛿) or 𝜈₁(𝑥−𝑥₀,𝛿)<𝜆<𝜇₁(𝑥−𝑥₀,𝛿). Since 𝜇₁𝑥−𝑥₀,𝛿=sup𝛼∈(0,1)∶𝑥−𝑥_0𝛼⁽¹⁾<𝛿, (2.7) there exists some 𝛼₀∈(0,1) such that 𝜆<𝛼₀ and ‖𝑥−𝑥₀‖_𝛼⁽¹⁾₀<𝛿. Hence ‖𝑓(𝑥)−𝐿‖_𝛼⁽²⁾₀<𝜖 by the hypothesis, that is, 𝜇₂𝑓(𝑥)−𝐿,𝜖≥𝛼₀>𝜆,𝜈₂𝑓(𝑥)−𝐿,𝜖≤1−𝛼₀<1−𝜆. (2.8) So, we get 𝜇₂(𝑓(𝑥)−𝐿,𝜖)≥𝜇₁(𝑥−𝑥₀,𝛿) and 𝜈₂(𝑓(𝑥)−𝐿,𝜖)≤𝜇₁(𝑥−𝑥₀,𝛿).

Proposition 2.3. Let (𝑥_𝑘) be a sequence in the IFNS (𝑋,𝜇₁,𝜈₁,∗,⋄) satisfying (1.2). Then(i)𝑥−−→_𝑘^wif𝑥 if and only if, for each 𝛼∈(0,1)lim_𝑘→∞𝑥_𝑘−𝑥_𝛼=0. (2.9)(ii)𝑥−−→_𝑘^sif𝑥 if and only if lim_𝑘→∞𝑥_𝑘−𝑥_𝛼=0uniformlyin𝛼, (2.10)where ‖⋅‖_𝛼 are α-norms of the intuitionistic fuzzy norms (𝜇,𝜈).

The proof of the above theorem directly follows from Propositions 2.2.

We define the following concepts analogous to that of Yılmaz [7].

Definition 2.4. The sif(wif)-closure of a subset 𝐵 in IFNS (𝑋,𝜇,𝜈,∗,⋄) is the set of all 𝑥∈𝑋 such that there exists a sequence (𝑥_𝑛)∈𝐵 such that 𝑥−−−−−→_𝑛^sif(wif)𝑥. In this case, we write 𝐵^sif(𝐵^wif). 𝐵 is said to be sif(wif)-closed whenever 𝐵^sif(𝐵^wif)=𝐵.

It is easy to see that 𝐵^sif⊆(𝐵^wif). The following example shows that inclusion is strict.

Example 2.5. Let 𝑋=ℝ and 𝜇()=|𝑥|𝑥,𝑡𝑡−|𝑥||𝑥|,|𝑥|;𝜈()=2|𝑥|𝑡+if𝑡>0if𝑡≤𝑥,𝑡|𝑥||𝑥|,|𝑥|,𝑡+if𝑡>1if𝑡≤ (2.11) on 𝑋. Let 𝑈_𝑋={𝑥∈𝑋∶|𝑥|<1} and we show that 𝑈^wif_𝑋=𝐵_𝑋={𝑥∈𝑋∶|𝑥|≤1}. For every 𝑥∈𝐵_𝑋, there exists a sequence (𝑥_𝑛)⊂𝑈_𝑋 such that ‖𝑥_𝑛−𝑥‖_𝛼→0 as 𝑛→∞, for each 𝛼∈(0,1). This is accomplished by taking 𝑥_𝑛=(1−1/(𝑛+1))𝑥 since each 𝑥_𝑛∈𝑈_𝑋 and 𝑥_𝑛−𝑥_𝛼=1+𝛼𝑥1−𝛼^𝑛−𝑥<1+𝛼11−𝛼𝑛+1⟶0as𝑛⟶∞, (2.12) for each 𝛼∈(0,1). However, 𝑈^sif_𝑋=𝑈_𝑋. Indeed for 𝑥∈𝑈^sif_𝑋, there exists (𝑥_𝑛)⊂𝑈_𝑋 such that ‖𝑥_𝑛−𝑥‖_𝛼→0 as 𝑛→∞, uniformly in α. This means that, given 𝜖>0, there exists an integer 𝑛_∘(𝜖)>0 such that for every 𝛼∈(0,1) and 𝑛≥𝑛_∘, 𝑥_𝑛−𝑥_𝛼<𝜖. (2.13) On the other hand, |𝑥|≤𝑥_𝑛−𝑥+𝑥_𝑛<𝑥_𝑛−𝑥+1=1−𝛼𝑥1+𝛼^𝑛−𝑥_𝛼+1<1−𝛼1+𝛼𝜖+1, (2.14) for all 𝛼∈(0,1) and 𝑛≥𝑛_∘. Letting 𝜖→0 (and hence 𝛼→1), we get |𝑥|<1. Therefore, 𝑈^sif_𝑋⊆𝑈_𝑋.

Definition 2.6. A subset 𝑆 of an IFNS (𝑋,𝜇,𝜈,∗,⋄) is said to be dense in (𝑋,𝜇,𝜈,∗,⋄) if and only if 𝑆^(sif)(𝑆^(wif))=𝑋.

Definition 2.7. An IFNS (𝑋,𝜇,𝜈,∗,⋄) is said to separable if it contains a countable dense subset, that is, there is a countable set {𝑥_𝑘} with the following property: for each 𝜖>0 and each 𝑥∈𝑋, there is at least one 𝑥_𝑛 with 𝜇𝑥_𝑛−𝑥,𝜖≥1−𝛼,𝜈𝑥_𝑛−𝑥,𝜖≤𝛼 (2.15) for 𝛼∈(0,1).

Theorem 2.8. Every finite dimension IFNS is separable.

Proof. Let (𝑋,𝜇,𝜈,∗,⋄) be a finite dimension normed linear space and {𝑢₁,𝑢₂,…,𝑢_𝑛} a basis of 𝑋. Since 𝑄_𝕂 is a countable subset of 𝕂, it follows that 𝐷_𝑛=^𝑛_𝑗=1𝛼_𝑗𝑢_𝑗∶𝛼_𝑗∈𝑄_𝕂 (2.16) is countable subset of span{𝑢₁,𝑢₂,…,𝑢_𝑛}=𝑋. Also, 𝐷 is dense in (𝑋,𝜇,𝜈,∗,⋄). To see this, let 𝑥∈𝑋 and 𝜖>0. Let 𝛽₁,…,𝛽_𝑛 be scalars such that 𝑥=∑^𝑛_𝑗=1𝛽_𝑗𝑢_𝑗. By the denseness of 𝑄_𝕂 in 𝕂, there exist 𝛼₁,𝛼₂,…,𝛼_𝑛 in 𝑄_𝕂 such that 𝜇𝛽_𝑗−𝛼_𝑗,𝜖∑^𝑛_𝑗=1𝑢_𝑗≥1−𝛼,𝜈𝛽_𝑗−𝛼_𝑗,𝜖∑^𝑛_𝑘=1𝑢_𝑗≤𝛼, (2.17) for all 𝑗∈{1,2,…,𝑛}. Then it follows that 𝜇𝑥−^𝑛_𝑗=1𝛼_𝑗𝑢_𝑗,𝜖≥𝜇𝛽_𝑗−𝛼_𝑗^𝑛_𝑗=1𝑢_𝑗,𝜖=𝜇𝛽_𝑗−𝛼_𝑗,𝜖∑^𝑛_𝑗=1𝑢_𝑗≥1−𝛼. (2.18) Similarly, 𝜈𝑥−^𝑛_𝑗=1𝛼_𝑗𝑢_𝑗,𝜖≤𝛼. (2.19) This implies that 𝐷 is dense in (𝑋,𝜇,𝜈,∗,⋄).

Theorem 2.9. Every IFNS having wif-basis is separable.

Proof. Let 𝑌 be IFNS with wif-basis {𝑢₁,𝑢₂,…}. Since 𝑌=⋃^∞_𝑛=1𝑌_𝑛 with 𝑌_𝑛=span{𝑢₁,𝑢₂,…,𝑢_𝑛} for all 𝑛∈ℕ is dense in 𝑌, it is enough to show that 𝑌 has a countable dense subset. Let 𝐷_𝑛=^𝑛_𝑗=1𝛼_𝑗𝑢_𝑗∶𝛼_𝑗∈𝑄_𝕂, (2.20) for all 𝑛∈ℕ. Then 𝐷_𝑛 will be a countable dense subset of 𝑌_𝑛 (see Theorem 2.8). Thus ⋃^∞_𝑛=1𝐷_𝑛 is a countable dense subset of 𝑌.

3. Intuitionistic Fuzzy Schauder Bases

In this section, we define strong and weak intuitionistic fuzzy Schauder bases.

Definition 3.1. Let (𝑋,𝜇,𝜈,∗,⋄) be an IFNS. Then, a sequence (𝑥_𝑛) is said to be(i)strongly intuitionistic fuzzy (Schauder) basis (for short, sif-basis) of 𝑋 if and only if for every 𝑥∈𝑋 there exists a unique sequence (𝑎_𝑛) of scalars such that ^𝑛_𝑘=1𝑎_𝑘𝑥−−→_𝑘^sif𝑥, (3.1) this means that for each 𝛼∈(0,1) there exists 𝑛_∘=𝑛_∘(𝛼) such that 𝑛≥𝑛_∘ implies 𝜇𝑥−^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘,𝑡≥1−𝛼,𝜈𝑥−^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘,𝑡≤𝛼, (3.2) for all 𝑡>0,(ii)weak intuitionistic fuzzy (Schauder) basis (for short, wif-basis) of 𝑋 if and only if for every 𝑥∈𝑋 there exists a unique sequence (𝑎_𝑛) of scalars such that ^𝑛_𝑘=1𝑎_𝑘𝑥−−→_𝑘^wif𝑥, (3.3) this means, for each 𝛼∈(0,1) and 𝜖>0, there exists 𝑛_∘=𝑛_∘(𝛼,𝜖) such that 𝑛≥𝑛_∘ implies 𝜇𝑥−^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘,𝜖≥1−𝛼,𝜈𝑥−^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘,𝜖≤𝛼. (3.4)

Proposition 3.2. Let (𝑥_𝑘) be a sequence in the IFNS (𝑋,𝜇,𝜈,∗,⋄) satisfying (1.2). Then(i)(𝑥_𝑘) is a wif-basis of 𝑋 if and only if for every 𝑥∈𝑋 there exists a unique sequence (𝑎_𝑛) of scalars such that for each 𝛼∈(0,1), lim_𝑛→∞𝑥−^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘𝛼=0; (3.5)(ii)(𝑥_𝑘) is a sif-basis of 𝑋 if and only if for every 𝑥∈𝑋 there exists a unique sequence (𝑎_𝑛) of scalars such that lim_𝑛→∞𝑥−^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘𝛼=0uniformlyin𝛼, (3.6)where ‖⋅‖_𝛼 are α-norms of the intuitionistic fuzzy norms (𝜇,𝜈).

The proof of the above theorem is similar to Propositions 2.2.

Definition 3.3. By the notations in Definition 3.1., the mappings 𝑔_𝑛∶𝑋⟶ℝsuchthat𝑔_𝑛(𝑥)=𝑔_𝑛^∞_𝑘=1𝑎_𝑘𝑥_𝑘=𝑎_𝑛,ℎ_𝑛∶𝑋⟶𝑋suchthatℎ_𝑛(𝑥)=ℎ_𝑛^∞_𝑘=1𝑎_𝑘𝑥_𝑘=^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘,𝑛=1,2,… (3.7) are called coordinate functionals and natural projections, respectively, associated to the sif(wif)-basis (𝑥_𝑛) in 𝑋.

Proposition 3.4. Let (𝑥_𝑛) be a basis in wif-complete IFNS (𝑋,𝜇,𝜈,∗,⋄) satisfying (1.2). Then each 𝑔_𝑛 and ℎ_𝑛 is wif-continuous.

Proof. By Proposition 3.2, (𝑥_𝑛) is also a Schauder basis in the Banach space (𝑋,‖⋅‖_𝛼) for each 𝛼∈(0,1). Thus 𝑔_𝑛∶𝑋,‖⋅‖_𝛼⟶ℝsuchthat𝑔_𝑛(𝑥)=𝑔_𝑛^∞_𝑘=1𝑎_𝑘𝑥_𝑘=𝑎_𝑛,ℎ_𝑛∶𝑋,‖⋅‖_𝛼⟶𝑋,‖⋅‖_𝛼suchthatℎ_𝑛(𝑥)=ℎ_𝑛^∞_𝑘=1𝑎_𝑘𝑥_𝑘=^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘 (3.8) are continuous. Therefore, the mappings are wif-continuous for each 𝑛.

Remark 3.5. It is obvious that, if (𝑥_𝑛) is an sif-basis of 𝑋 then it is wif-basis of 𝑋, but not conversely. For the converse part, let us consider the following example.

Example 3.6. Let 𝑋=ℓ_𝑝(1≤𝑝<∞), the Banach space of all absolutely 𝑝-summable sequences with the norm ‖𝑥‖_𝑝=(∑_𝑘|𝑥_𝑘|^𝑝)^1/𝑝, and consider the intuitionistic fuzzy norm 𝜇()=‖𝑥‖𝑥,𝑡𝑡−_𝑝𝑡+‖𝑥‖_𝑝if𝑡>‖𝑥‖_𝑝,0if𝑡≤‖𝑥‖_𝑝;𝜈()=2‖𝑥‖𝑥,𝑡𝑡+‖𝑥^𝑝‖_𝑝if𝑡>‖𝑥‖_𝑝,1if𝑡≤‖𝑥‖_𝑝, (3.9) on 𝑋. We can find α-norms of intuitionistic fuzzy norm (𝜇,𝜈) since it satisfies condition (1.2). Thus 𝜇()‖𝑥‖𝑥,𝑡≥𝛼⟺𝑡−_𝑝𝑡+‖𝑥‖_𝑝≥𝛼⟺1+𝛼‖𝑥‖1−𝛼^𝑝𝜈()2‖𝑥‖≤𝑡,𝑥,𝑡≤1−𝛼⟺^𝑝𝑡+‖𝑥‖_𝑝≤1−𝛼⟺1+𝛼‖𝑥‖1−𝛼^𝑝≤𝑡. (3.10) This shows that ‖𝑥‖_𝛼=inf𝑡>0∶𝜇(𝑥,𝑡)≥𝛼,𝜈(𝑥,𝑡)≤1−𝛼=1+𝛼‖𝑥‖1−𝛼^𝑝. (3.11) Now, we show that the sequence (𝑒_𝑘)^∞_𝑘=1 is wif-basis but not sif-basis. Take any 𝑥=(𝑥_𝑘)∈ℓ_𝑝. Put 𝑦_𝑛=𝑥−𝑥₁,𝑥₂,…,𝑥_𝑛,0,0,…, (3.12) then 𝑦_𝑛=0,0,…,𝑥_𝑛+1,𝑥_𝑛+2,…. (3.13) Hence lim_𝑛𝑦_𝑛𝛼=1+𝛼1−𝛼lim_𝑛𝑦_𝑛𝑝=1+𝛼1−𝛼lim_𝑛^∞_𝑘=𝑛+1𝑥_𝑘^𝑝1/𝑝=0 (3.14) and by Proposition 3.2(𝑒_𝑘) is wif-basis for ℓ_𝑝. However, this convergence is not uniform in α since (1+𝛼)/(1−𝛼)𝜖→∞ as 𝛼→1.
However, if we put𝜇₁()=‖𝑥‖𝑥,𝑡1if𝑡>_𝑝,0if𝑡≤‖𝑥‖_𝑝;𝜈₁()=‖𝑥‖𝑥,𝑡0if𝑡>_𝑝,1if𝑡≤‖𝑥‖_𝑝, (3.15) on 𝑋, then (ℓ_𝑝,𝜇₁,𝜈₁,∗,⋄) is an IFNS satisfying (1.2), and (𝑒_𝑘) is a sif-basis for ℓ_𝑝 since ‖𝑥‖_𝛼=‖𝑥‖_𝑝 for each 𝛼∈(0,1).

Remark 3.7. In finite-dimensional spaces, the definition of basis is independent of the intuitionistic fuzzy norm and hence coincides with the definition of a classical vector space basis (Hamel basis).

We know that every intuitionistic fuzzy normed space induces a topology 𝜏 such that for some 𝐴⊂𝑋,𝐴∈𝜏 if and only if for each 𝑥∈𝐴 there exists some 𝑡>0 and 𝛼∈(0,1) such that 𝐵(𝑥,𝛼,𝑡)⊂𝐴, where 𝐵(𝑥,𝛼,𝑡)∶=𝑦∶𝜇𝑥−𝑦,𝑡≥1−𝛼,𝜈𝑥−𝑦,𝑡≤𝛼. (3.16)

Proposition 3.8. 𝜏 is a vector topology for 𝑋; that is, the vector space operations are continuous in this topology.

Proof. Since the family 𝐵(𝑥,1/𝑛,1/𝑛)∶𝑛=1,2,… is a countable local basis at 𝑥, 𝜏 is the first countable topology of 𝑋. Hence it is sufficient to show that the vector space operations are sequentially continuous in 𝜏. Suppose 𝑥_𝑛→𝑥 and 𝑦_𝑛→𝑦 in the topological space (𝑋,𝜏). This means 𝜇(𝑥_𝑛−𝑥,𝑡/2),𝜇(𝑦_𝑛−𝑦,𝑡/2),𝜈(𝑥_𝑛−𝑥,𝑡/2), and 𝜈(𝑦_𝑛−𝑦,𝑡/2)→1 as 𝑛→∞, for all 𝑡>0. Now 𝜇𝑥_𝑛+𝑦_𝑛−𝑥+𝑦,𝑡≥𝜇𝑥_𝑛𝑡−𝑥,2∗𝜇𝑦_𝑛𝑡−𝑦,2𝜈𝑥⟶1as𝑛⟶∞,_𝑛+𝑦_𝑛−𝑥+𝑦,𝑡≤𝜈𝑥_𝑛𝑡−𝑥,2⋄𝜈𝑦_𝑛𝑡−𝑦,2⟶1as𝑛⟶∞, (3.17) for all 𝑡>0. Further, if 𝜆_𝑛→𝜆 in ℝ or ℂ, the scalar field of 𝑋, then 𝜇𝜆_𝑛𝑥_𝑛−𝜆𝑥,𝑡=𝜇𝜆_𝑛𝑥_𝑛−𝜆𝑥_𝑛+𝜆𝑥_𝑛−𝜆𝑥,𝑡=𝜇𝜆_𝑛−𝜆𝑥_𝑛+𝜆𝑥_𝑛𝑥−𝑥,𝑡≥𝜇_𝑛,𝑡2𝜆_𝑛𝑥−𝜆∗𝜇_𝑛𝑡−𝑥,2|𝜆|𝜈𝜆⟶1as𝑛⟶∞,_𝑛𝑥_𝑛−𝜆𝑥,𝑡=𝜈𝜆_𝑛𝑥_𝑛−𝜆𝑥_𝑛+𝜆𝑥_𝑛−𝜆𝑥,𝑡=𝜈𝜆_𝑛−𝜆𝑥_𝑛+𝜆𝑥_𝑛𝑥−𝑥,𝑡≤𝜈_𝑛,𝑡2𝜆_𝑛𝑥−𝜆⋄𝜈_𝑛𝑡−x,2|𝜆|⟶1as𝑛⟶∞. (3.18)
Analogous to the classical results, we prove here that a normed linear space having a Schauder basis is separable.

Theorem 3.9. Let (𝑋,𝜇,𝜈,∗,⋄) be an IFNS having wif-basis (𝑥_𝑛). Then the topological space (𝑋,𝜏) is separable.

Proof. Let 𝐸 denotes the set of all finite linear combinations ∑^𝑛_𝑘=1𝑏_𝑘𝑥_𝑘, where each 𝑏_𝑘 is a (real or complex) rational number. Obviously, 𝐸 is countable and let us show that it is dense in 𝜏. Suppose 𝑥∈𝑋 is arbitrary. There exists a unique sequence (𝑎_𝑛) of scalars such that for each 𝜖>0 and 𝛼∈(0,1), we can find some integer 𝑛_∘=𝑛_∘(𝛼,𝜖) such that 𝜇𝑥−^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘,𝜖≥1−𝛼,𝜈𝑥−^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘,𝜖≤𝛼. (3.19) That is, for all 𝑛≥𝑛_∘, ^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘∈𝐵(𝑥,𝛼,𝜖). (3.20) On the other hand, one can constitute a sequence (𝑏^𝑖_𝑘)^∞_𝑖=1 of scalars converging to 𝑎_𝑘, for each 𝑘. Hence the sequence (∑^𝑛_𝑘=1𝑏^𝑖_𝑘𝑥_𝑘)^∞_𝑖=1 converges to ∑^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘 in 𝜏 by the continuity of vector space operations. This implies that every 𝑥-centered 𝜏-open sphere 𝐵(𝑥,𝛼,𝜖) includes an element ∑^𝑛_𝑘=1𝑏^𝑖_𝑘𝑥_𝑘 of 𝐸.

Theorem 3.10. Let (𝑋,‖⋅‖) be a normed space and (𝑥_𝑛) a basis in 𝑋. Then (𝑥_𝑛) is a wif-basis for IFNS (𝑋,𝜇,𝜈,∗,⋄), where 𝜇()=𝑡𝑥,𝑡‖𝑥‖𝜈()=‖𝑥‖𝑡+if𝑡>0,0if𝑡≤0;𝑥,𝑡𝑡+‖𝑥‖if𝑡>0,1if𝑡≤0. (3.21)

Proof. By the hypothesis, for each 𝑥∈𝑋, there exists a unique sequence (𝑎_𝑛) of scalars with ∑^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘→0 in the norm topology as 𝑛→∞. Explicitly, for each 𝛿>0, there exists an integer 𝑛_∘=𝑛_∘(𝛿) such that 𝑛≥𝑛_∘ implies 𝑥−^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘≤𝛿. (3.22) Now, for each 𝜖 and 𝛼∈(0,1), take 𝛿=𝛼𝜖/(1−𝛼). So, there exists an integer 𝑛_∘=𝑛_∘(𝛿)=𝑛_∘(𝛼,𝜖) such that 𝑛≥𝑛_∘ implies 𝑥−^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘≤𝛼𝜖1−𝛼 (3.23) if and only if 𝜇𝑥−^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘=𝜖,𝜖𝜖+𝑥−∑^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘𝜈≥1−𝛼,𝑥−^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘=𝜖,𝜖𝜖+𝑥−∑^𝑛_𝑘=1𝑎_𝑘𝑥_𝑘≤𝛼. (3.24)

4. Intuitionistic Fuzzy Approximation Property

In this section, we define strong and weak intuitionistic fuzzy approximation property and prove some interesting results.

Definition 4.1. We say that sif-complete IFNS (𝑋,𝜇,𝜈,∗,⋄) is said to have strong intuitionistic fuzzy approximation property (for short, sif-AP) if for every sif-compact set 𝐾⊂𝑋 and 𝛼∈(0,1) there exists an operator 𝑇∶𝑋→𝑋 of finite rank such that 𝜇(𝑇(𝑥)−𝑥,𝑡)≥1−𝛼,𝜈(𝑇(𝑥)−𝑥,𝑡)≤𝛼, (4.1) for all 𝑥∈𝐾 and 𝑡>0.

Definition 4.2. A wif-complete IFNS (𝑋,𝜇,𝜈,∗,⋄) is said to have weak intuitionistic fuzzy approximation property (for short, wif-AP) if for every wif-compact set 𝐾⊂𝑋 and for each 𝜖>0 and 𝛼∈(0,1) there exists an operator 𝑇_𝛼∶𝑋→𝑋 of finite rank such that 𝜇𝑇_𝛼(𝑥)−𝑥,𝜖≥1−𝛼,𝜈𝑇_𝛼(𝑥)−𝑥,𝜖≤𝛼, (4.2) for all 𝑥∈𝐾.

Remark 4.3. The operator 𝑇 in wif-AP depends both on 𝜖>0 and 𝛼∈(0,1) whereas it depends only on 𝛼∈(0,1) in sif-AP. 𝑇 depends on the set 𝐾 in both situations.

Proposition 4.4. (i) A wif-complete IFNS (𝑋,𝜇,𝜈,∗,⋄) satisfying (1.2) has wif-AP if and only if for every wif-compact set 𝐾∈𝑋 and for each 𝜖>0 and 𝛼∈(0,1) there exists an operator 𝑇_𝛼∶𝑋→𝑋 of finite rank such that 𝑇_𝛼(𝑥)−𝑥_𝛼<𝜖, (4.3) for all 𝑥∈𝐾.
(ii) A sif-complete IFNS (𝑋,𝜇,𝜈,∗,⋄) satisfying (1.2) has sif-AP if and only if for every sif-compact set 𝐾∈𝑋 and for each 𝜖>0 there exists an operator 𝑇∶𝑋→𝑋 of finite rank such that ‖𝑇(𝑥)−𝑥‖_𝛼<𝜖, (4.4) for all 𝑥∈𝐾.

The proof of the above theorem directly follows from Propositions 2.2.

Theorem 4.5. Let (𝑋,𝜇,𝜈,∗,⋄) be an IFNS possessing a wif-basis (𝑥_𝑛). Then 𝑋 has the wif-AP.

Proof. Let 𝐾⊂𝑋 be a wif-compact subset of 𝑋. Let 𝜖>0 and 𝛼∈(0,1) be arbitrary. By the hypothesis, for some 𝑥∈𝐾, there exists a unique sequence (𝑎_𝑛) of scalars such that 𝑃_𝑛(𝑥)=^𝑛_𝑘=1𝑎_𝑘𝑥−−→_𝑘^wif𝑥as𝑛⟶∞. (4.5) Then, there exists some 𝑛_∘(𝛼,𝜖) such that 𝜇𝑃_𝑛(𝑥)−𝑥,𝜖≥1−𝛼,𝜈𝑃_𝑛(𝑥)−𝑥,𝜖≤𝛼 (4.6) for all 𝑛≥𝑛_∘. Further, each 𝑃_𝑛 has a finite rank in the linear space 𝑋 since dim𝑃_𝑛(𝑥)=𝑛. Hence, each 𝑃_𝑛 such that 𝑛≥𝑛_∘ can be taken as a desired finite rank operator in the definition.

Remark 4.6. Theorem 4.5 can also proved for sif-basis.

Example 4.7. Let 𝑋=ℓ_∞, the Banach space of all bounded sequence with sup-norm ‖𝑥‖_∞=sup_𝑛|𝑥_𝑛|. Also, ‖𝑥‖_∘=sup_𝑛|𝑥_𝑛/𝑛| is another norm on ℓ_∞. Define the function 𝜇()=‖𝑥‖𝑥,𝑡1if𝑡>_∞,12if‖𝑥‖_∘<𝑡≤‖𝑥‖_∞,0if𝑡≤‖𝑥‖_∘;𝜈()=‖𝑥‖𝑥,𝑡0if𝑡>_∞,12if‖𝑥‖_∞≤𝑡<‖𝑥‖_∘,1if𝑡≤‖𝑥‖_∘. (4.7) Then (𝜇,𝜈) is an intuitionistic fuzzy norm on ℓ_∞. We can find α-norms of intuitionistic fuzzy norm (𝜇,𝜈) since it satisfies (1.2) condition. Thus ‖𝑥‖_𝛼=‖𝑥‖_∞1for1>𝛼>2,‖𝑥‖_𝛼=‖𝑥‖_∘1for1<𝛼≤2. (4.8) IFNS (ℓ_∞,𝜇,𝜈,∗,⋄) cannot have a wif and hence a sif-basis since (ℓ_∞,‖⋅‖_𝛼)=(ℓ_∞,‖⋅‖_∞) for 1>𝛼>1/2 and the Banach space (ℓ_∞,‖⋅‖_∞) is not separable. However, (ℓ_∞,𝜇,𝜈,∗,⋄) has sif-AP. Recall that the set 𝐷 of all partitions 𝑝=(𝛽₁,𝛽₂,…,𝛽_𝑛) of natural numbers is a directed set by the relation 𝑝_𝑎≪𝑝_𝑏 which means that each 𝛽_𝑖∈𝑝_𝑎 is included in some 𝛽^′_𝑗∈𝑝_𝑏. Now, for each 𝑝∈𝐷Λ_𝑝(𝑥)=^𝑛_𝑖=1𝑥_ℎ𝑖𝜒_𝛽𝑖, (4.9) where ℎ_𝑖 is the distinguished point in 𝛽_𝑖 and 𝜒_𝛽𝑖 is the characteristic function of 𝛽_𝑖 for 1≤𝑖≤𝑛. Then Λ_𝑝 is a projection on ℓ_∞ of finite rank. It is well known that the set (Λ_𝑝(𝑥),𝐷) converges to 𝑥 in (ℓ_∞,‖⋅‖_∞). Let 𝐾⊆ℓ_∞ be sif-compact. Given 𝜖>0 and 𝑥∈𝐾, then there exists a partition 𝑝_∘(𝜖) such that for 𝑝_∘(𝜖)≪𝑝Λ_𝑝(𝑥)−𝑥_∞<𝜖. (4.10) But Λ_𝑝(𝑥)−𝑥_∘<𝜖, (4.11) for 𝑝_∘(𝜖)≪𝑝 since ‖𝑥‖_∘≤‖𝑥‖_∞ for every 𝑥∈ℓ_∞. That is, for 𝑝_∘(𝜖)≪𝑝, Λ_𝑝(𝑥)−𝑥_𝛼<𝜖, (4.12) for all 𝛼∈(0,1). Hence for some Λ_𝑝(𝑝_∘≪𝑝) meets all requirements for sif-AP in Proposition 4.4.

Acknowledgment

This paper was completed when the first author (M. Mursaleen) visited the Yildiz Technical University, Istanbul, during June 20–July 19, 2010 under the project supported by TUBITAK BIDEB.