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

and Applied Analysis 3 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 x, t ⎧ ⎪⎨ ⎪⎩ t t ‖x‖ if t > 0, 0 if t ≤ 0; ν x, t ⎧ ⎪⎨ ⎪⎩ ‖x‖ t |x| if t > 0, 1 if t ≤ 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 < 1 implies x 0 ∀t ∈ . 1.2 Let ‖x‖α inf{t > 0 : μ x, t ≥ α and ν 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 xk is said to be i weakly intuitionistic fuzzy convergent to x ∈ X if and only if, for every > 0 and α ∈ 0, 1 , there exists some k0 k0 α, such that μ xk − x, ≥ 1 − α and ν xk − x, ≤ α for all k ≥ k0. In this case we write xk wif −−→ x, 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 xk − x, t ≥ 1 − α and ν xk − x, t ≤ α for all t > 0. In this case we write xk sif −→ x. 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 -limx→ x0 f x L if and only if, for each α ∈ 0, 1 lim ‖x−x0‖ 1 α → 0 ∥ ∥f x − L∥ 2 α 0; 2.1 ii sif -limx→ x0 f x L if and only if lim ‖x−x0‖ 1 α → 0 ∥ ∥f x − L∥ 2 α 0 uniformly in α, 2.2 where ‖ · ‖ 1 α and ‖ · ‖ 2 α are α-norms of the intuitionistic fuzzy norms μ1, ν1 and μ2, ν2 , respectively. 4 Abstract and Applied Analysis Proof. Here we prove the case ii . Suppose that sif -limx→x0f x L. For a given > 0, there exists some δ δ > 0 such that μ2 ( f x − L, ) ≥ μ1 x − x0, δ , ν2 ( f x − L, ) ≥ ν1 x − x0, δ , 2.3 for all x ∈ X. For each α ∈ 0, 1 , if ‖x − x0‖ 1 α inf { t > 0 : μ1 x − x0, t ≥ α, ν1 x − 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, ≥ α and ν2 f x −L, ≤ 1−α and so that ‖f x − L‖ 2 α < . Since δ does not depend on α, we get lim ‖x−x0‖ 1 α → 0 ∥ ∥f x − L∥ 2 α 0 uniformly in α. 2.5 Conversely, let lim‖x−x0‖ 1 α → 0‖f x − L‖ 2 α 0 uniformly in α. Given > 0, there exists some δ δ > 0 such that ‖x − x0‖ 1 α < δ implies ∥ ∥f x − L∥ 2 α < 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 μ1 x − x0, δ sup { α ∈ 0, 1 : ‖x − x0‖ 1 α < δ } , 2.7 there exists some α0 ∈ 0, 1 such that λ < α0 and ‖x − x0‖ 1 α0 < δ. Hence ‖f x − L‖ 2 α0 < by the hypothesis, that is, μ2 ( f x − L, ) ≥ α0 > λ, ν2 ( f x − L, ) ≤ 1 − α0 < 1 − λ. 2.8 So, we get μ2 f x − L, ≥ μ1 x − x0, δ and ν2 f x − L, ≤ μ1 x − x0, δ . Proposition 2.3. Let xk be a sequence in the IFNS X, μ1, ν1, ∗, satisfying 1.2 . Then i xk wif −−→ x if and only if, for each α ∈ 0, 1 lim k→∞ ‖xk − x‖α 0. 2.9 ii xk sif −→ x if and only if lim k→∞ ‖xk − x‖α 0 uniformly in α, 2.10 where ‖ · ‖α are α-norms of the intuitionistic fuzzy norms μ, ν . Abstract and Applied Analysis 5 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 xn ∈ B such that xn sif wif −−−−−→ x. In this case, we write B sif B wif . B is said to be sif wif -closedwhenever B sif B wif B. It is easy to see that B sif ⊆ B . The following example shows that inclusion is strict. Example 2.5. Let X andand Applied Analysis 5 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 xn ∈ B such that xn sif wif −−−−−→ x. In this case, we write B sif B wif . B is said to be sif wif -closedwhenever B sif B wif B. It is easy to see that B sif ⊆ B . The following example shows that inclusion is strict. Example 2.5. Let X and μ x, t ⎧ ⎪⎨ ⎪⎩ t − |x| t |x| if t > |x|, 0 if t ≤ |x|; ν x, t ⎧ ⎪⎨ ⎪⎩ 2|x| t |x| if t > |x|, 1 if t ≤ |x|, 2.11 on X. Let UX {x ∈ X : |x| < 1} and we show that U wif X BX {x ∈ X : |x| ≤ 1}. For every x ∈ BX , there exists a sequence xn ⊂ UX such that ‖xn − x‖α → 0 as n → ∞, for each α ∈ 0, 1 . This is accomplished by taking xn 1 − 1/ n 1 x since each xn ∈ UX and ‖xn − x‖α 1 α 1 − α |xn − x| < ( 1 α 1 − α ) 1 n 1 −→ 0 as n −→ ∞, 2.12 for each α ∈ 0, 1 . However, U X UX . Indeed for x ∈ U sif X , there exists xn ⊂ UX such that ‖xn − x‖α → 0 as n → ∞, uniformly in α. This means that, given > 0, there exists an integer n◦ > 0 such that for every α ∈ 0, 1 and n ≥ n◦, ‖xn − x‖α < . 2.13