# Schauder Basis, Separability, and Approximation Property

Authors: M. Mursaleen; V. Karakaya; S. A. Mohiuddine
Journal: Abstract and Applied Analysis
 (2010)

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## 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 spacesℓp(1≤p≤∞).


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## Body

## 1. Introduction and Background

We will writew for the set of all complex sequences x=(xk)k=0∞. Let ℓ∞, c, and c0 denote the sets of all bounded, convergent, and null sequences, respectively. We write ℓp:={x∈w:∑k=0∞|xk|p<∞} for 1≤p<∞. By e and e(n)(n∈ℕ), we denote the sequences such that ek=1 for k=0,1,…,en(n)=1, and ek(n)=0(k≠n). Note that c0, c, and ℓ∞ are Banach spaces with the sup-norm ∥x∥∞=supk|xk|, and ℓp(1≤p<∞) are Banach spaces with the norm ∥x∥p=(∑|xk|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 (βn)n=0∞ of scalars such that x=∑n=0∞βnb(n), that is, limm→∞∥x-∑n=0mβnb(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 ℓp(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 continuoust-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 continuoust-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 continuoust-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)
limt→∞μ(x,t)=1 and limt→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)
limt→∞ν(x,t)=0 andlimt→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
(1.1)μ(x,t)={tt+‖x‖ift>0,0ift≤0;ν(x,t)={‖x‖t+|x|ift>0,1ift≤0.
Then (X,μ,ν,*,♢) is an IFNS.Remark 1.5 (see [3]).
Let(X,μ,ν,*,♢) be an IFNS with the condition
(1.2)μ(x,t)>0,ν(x,t)<1impliesx=0∀t∈R.
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 (xk) 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(xk-x,ϵ)≥1-α and ν(xk-x,ϵ)≤αfor all k≥k0. In this case we write xk→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(xk-x,t)≥1-α and ν(xk-x,t)≤α for all t>0. In this case we write xk→sifx.The following result characterizes the (wif)- and (sif)-limit throughα-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→x0f(x)=L if and only if, for eachα∈(0,1)(2.1)lim⁡‖x-x0‖α(1)→0⁡‖f(x)-L‖α(2)=0;(ii)
(sif)-limx→x0f(x)=L if and only if
(2.2)lim⁡‖x-x0‖α(1)→0‖f(x)-L‖α(2)=0uniformlyinα,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)-limx→x0f(x)=L. For a given ϵ>0, there exists some δ=δ(ϵ)>0 such that
(2.3)μ2(f(x)-L,ϵ)≥μ1(x-x0,δ),ν2(f(x)-L,ϵ)≥ν1(x-x0,δ),
for all x∈X. For each α∈(0,1), if
(2.4)‖x-x0‖α(1)=inf⁡{t>0:μ1(x-x0,t)≥α,ν1(x-x0,t)≤1-α}<δ,
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
(2.5)lim⁡∥x-x0∥α(1)→0‖f(x)-L‖α(2)=0uniformlyinα.
Conversely, letlim∥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 that(2.6)‖x-x0‖α(1)<δimplies‖f(x)-L‖α(2)<ϵ
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
(2.7)μ1(x-x0,δ)=sup⁡{α∈(0,1):‖x-x0‖α(1)<δ},
there exists some α0∈(0,1) such that λ<α0 and ∥x-x0∥α0(1)<δ. Hence ∥f(x)-L∥α0(2)<ϵ by the hypothesis, that is,
(2.8)μ2(f(x)-L,ϵ)≥α0>λ,ν2(f(x)-L,ϵ)≤1-α0<1-λ.
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(xk) be a sequence in the IFNS (X,μ1,ν1,*,♢) satisfying (1.2). Then(i)
xk→wifx if and only if, for each α∈(0,1)(2.9)lim⁡k→∞‖xk-x‖α=0.(ii)
xk→sifx if and only if
(2.10)lim⁡k→∞‖xk-x‖α=0uniformlyinα,where ∥·∥α are α-norms of the intuitionistic fuzzy norms (μ,ν).The proof of the above theorem directly follows from Propositions2.2.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 B̅sif(B̅wif). B is said to be sif(wif)-closed whenever B̅sif(B̅wif)=B.It is easy to see thatB̅sif⊆(B̅wif). The following example shows that inclusion is strict.Example 2.5.
LetX=ℝ and
(2.11)μ(x,t)={t-|x|t+|x|ift>|x|,0ift≤|x|;ν(x,t)={2|x|t+|x|ift>|x|,1ift≤|x|,
on X. Let UX={x∈X:|x|<1} and we show that U̅Xwif=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
(2.12)‖xn-x‖α=1+α1-α|xn-x|<(1+α1-α)1n+1⟶0asn⟶∞,
for each α∈(0,1). However, U̅Xsif=UX. Indeed for x∈U̅Xsif, there exists (xn)⊂UX such that ∥xn-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∘,
(2.13)‖xn-x‖α<ϵ.
On the other hand,
(2.14)|x|≤|xn-x|+|xn|<|xn-x|+1=(1-α1+α)‖xn-x‖α+1<(1-α1+α)ϵ+1,
for all α∈(0,1) and n≥n∘. Letting ϵ→0 (and hence α→1), we get |x|<1. Therefore, U̅Xsif⊆UX.Definition 2.6.
A subsetS of an IFNS (X,μ,ν,*,♢) is said to be dense in (X,μ,ν,*,♢) if and only if S̅(sif)(S̅(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 {xk} with the following property: for each ϵ>0 and each x∈X, there is at least one xn with
(2.15)μ(xn-x,ϵ)≥1-α,ν(xn-x,ϵ)≤α
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,…,un} a basis of X. Since Q𝕂 is a countable subset of 𝕂, it follows that
(2.16)Dn={∑j=1nαjuj:αj∈QK}
is countable subset of span{u1,u2,…,un}=X. Also, D is dense in (X,μ,ν,*,♢). To see this, let x∈X and ϵ>0. Let β1,…,βn be scalars such that x=∑j=1nβjuj. By the denseness of Q𝕂 in 𝕂, there exist α1,α2,…,αn in Q𝕂 such that
(2.17)μ(βj-αj,ϵ|∑j=1nuj|)≥1-α,ν(βj-αj,ϵ|∑k=1nuj|)≤α,
for all j∈{1,2,…,n}. Then it follows that
(2.18)μ(x-∑j=1nαjuj,ϵ)≥μ((βj-αj)∑j=1nuj,ϵ)=μ(βj-αj,ϵ|∑j=1nuj|)≥1-α.
Similarly,
(2.19)ν(x-∑j=1nαjuj,ϵ)≤α.
This implies that D is dense in (X,μ,ν,*,♢).Theorem 2.9.
Every IFNS having wif-basis is separable.Proof.
LetY be IFNS with wif-basis {u1,u2,…}. Since Y=⋃n=1∞Yn with Yn=span{u1,u2,…,un} for all n∈ℕ is dense in Y, it is enough to show that Y has a countable dense subset. Let
(2.20)Dn={∑j=1nαjuj:αj∈QK},
for all n∈ℕ. Then Dn will be a countable dense subset of Yn (see Theorem 2.8). Thus ⋃n=1∞Dn 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 (xn) 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 (an) of scalars such that
(3.1)∑k=1nakxk→sifx,
this means that for each α∈(0,1) there exists n∘=n∘(α) such that n≥n∘ implies
(3.2)μ(x-∑k=1nakxk,t)≥1-α,ν(x-∑k=1nakxk,t)≤α,
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 (an) of scalars such that
(3.3)∑k=1nakxk→wifx,
this means, for each α∈(0,1) and ϵ>0, there exists n∘=n∘(α,ϵ) such that n≥n∘ implies
(3.4)μ(x-∑k=1nakxk,ϵ)≥1-α,ν(x-∑k=1nakxk,ϵ)≤α.Proposition 3.2.
Let(xk) be a sequence in the IFNS (X,μ,ν,*,♢) satisfying (1.2). Then(i)
(xk) is a wif-basis of X if and only if for every x∈X there exists a unique sequence (an) of scalars such that for each α∈(0,1),
(3.5)lim⁡n→∞‖x-∑k=1nakxk‖α=0;(ii)
(xk) is a sif-basis of X if and only if for every x∈X there exists a unique sequence (an) of scalars such that
(3.6)lim⁡n→∞‖x-∑k=1nakxk‖α=0uniformlyinα,where ∥·∥α are α-norms of the intuitionistic fuzzy norms (μ,ν).The proof of the above theorem is similar to Propositions2.2.Definition 3.3.
By the notations in Definition3.1., the mappings
(3.7)gn:X⟶Rsuchthatgn(x)=gn(∑k=1∞akxk)=an,hn:X⟶Xsuchthathn(x)=hn(∑k=1∞akxk)=∑k=1nakxk,n=1,2,…
are called coordinate functionals and natural projections, respectively, associated to the sif(wif)-basis (xn) in X.Proposition 3.4.
Let(xn) be a basis in wif-complete IFNS (X,μ,ν,*,♢) satisfying (1.2). Then each gn and hn is wif-continuous.Proof.
By Proposition3.2, (xn) is also a Schauder basis in the Banach space (X,∥·∥α) for each α∈(0,1). Thus
(3.8)gn:(X,‖⋅‖α)⟶Rsuchthatgn(x)=gn(∑k=1∞akxk)=an,hn:(X,‖⋅‖α)⟶(X,‖⋅‖α)suchthathn(x)=hn(∑k=1∞akxk)=∑k=1nakxk
are continuous. Therefore, the mappings are wif-continuous for each n.Remark 3.5.
It is obvious that, if(xn) 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.
LetX=ℓp(1≤p<∞), the Banach space of all absolutely p-summable sequences with the norm ∥x∥p=(∑k|xk|p)1/p, and consider the intuitionistic fuzzy norm
(3.9)μ(x,t)={t-‖x‖pt+‖x‖pift>‖x‖p,0ift≤‖x‖p;ν(x,t)={2‖x‖pt+‖x‖pift>‖x‖p,1ift≤‖x‖p,
on X. We can find α-norms of intuitionistic fuzzy norm (μ,ν) since it satisfies condition (1.2). Thus
(3.10)μ(x,t)≥α⟺t-‖x‖pt+‖x‖p≥α⟺1+α1-α‖x‖p≤t,ν(x,t)≤1-α⟺2‖x‖pt+‖x‖p≤1-α⟺1+α1-α‖x‖p≤t.
This shows that
(3.11)‖x‖α=inf⁡{t>0:μ(x,t)≥α,ν(x,t)≤1-α}=1+α1-α‖x‖p.
Now, we show that the sequence (ek)k=1∞ is wif-basis but not sif-basis. Take any x=(xk)∈ℓp. Put
(3.12)yn=x-(x1,x2,…,xn,0,0,…),
then
(3.13)yn=(0,0,…,xn+1,xn+2,…).
Hence
(3.14)lim⁡n‖yn‖α=1+α1-αlim⁡n‖yn‖p=1+α1-αlim⁡n(∑k=n+1∞|xk|p)1/p=0
and by Proposition 3.2(ek) is wif-basis for ℓp. However, this convergence is not uniform in α since (1+α)/(1-α)ϵ→∞ as α→1.
However, if we put(3.15)μ1(x,t)={1ift>‖x‖p,0ift≤‖x‖p;ν1(x,t)={0ift>‖x‖p,1ift≤‖x‖p,
on X, then (ℓp,μ1,ν1,*,♢) is an IFNS satisfying (1.2), and (ek) is a sif-basis for ℓp 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 topologyτ 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 (3.16)B(x,α,t):={y:μ(x-y,t)≥1-α,ν(x-y,t)≤α}.Proposition 3.8.
τ is a vector topology for X; that is, the vector space operations are continuous in this topology.Proof.
Since the familyB(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 xn→x and yn→y in the topological space (X,τ). This means μ(xn-x,t/2),μ(yn-y,t/2),ν(xn-x,t/2), and ν(yn-y,t/2)→1 as n→∞, for all t>0. Now
(3.17)μ(xn+yn-(x+y),t)≥μ(xn-x,t2)*μ(yn-y,t2)⟶1asn⟶∞,ν(xn+yn-(x+y),t)≤ν(xn-x,t2)♢ν(yn-y,t2)⟶1asn⟶∞,
for all t>0. Further, if λn→λ in ℝ or ℂ, the scalar field of X, then
(3.18)μ(λnxn-λx,t)=μ(λnxn-λxn+λxn-λx,t)=μ((λn-λ)xn+λ(xn-x),t)≥μ(xn,t2|λn-λ|)*μ(xn-x,t2|λ|)⟶1asn⟶∞,ν(λnxn-λx,t)=ν(λnxn-λxn+λxn-λx,t)=ν((λn-λ)xn+λ(xn-x),t)≤ν(xn,t2|λn-λ|)♢ν(xn-x,t2|λ|)⟶1asn⟶∞.
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 (xn). Then the topological space (X,τ) is separable.Proof.
LetE denotes the set of all finite linear combinations ∑k=1nbkxk, where each bk 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 (an) of scalars such that for each ϵ>0 and α∈(0,1), we can find some integer n∘=n∘(α,ϵ) such that
(3.19)μ(x-∑k=1nakxk,ϵ)≥1-α,ν(x-∑k=1nakxk,ϵ)≤α.
That is, for all n≥n∘,
(3.20)∑k=1nakxk∈B(x,α,ϵ).
On the other hand, one can constitute a sequence (bki)i=1∞ of scalars converging to ak, for each k. Hence the sequence (∑k=1nbkixk)i=1∞ converges to ∑k=1nakxk 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 ∑k=1nbkixk of E.Theorem 3.10.
Let(X,∥·∥) be a normed space and (xn) a basis in X. Then (xn) is a wif-basis for IFNS (X,μ,ν,*,♢), where
(3.21)μ(x,t)={tt+‖x‖ift>0,0ift≤0;ν(x,t)={‖x‖t+‖x‖ift>0,1ift≤0.Proof.
By the hypothesis, for eachx∈X, there exists a unique sequence (an) of scalars with ∑k=1nakxk→0 in the norm topology as n→∞. Explicitly, for each δ>0, there exists an integer n∘=n∘(δ) such that n≥n∘ implies
(3.22)‖x-∑k=1nakxk‖≤δ.
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
(3.23)‖x-∑k=1nakxk‖≤αϵ1-α
if and only if
(3.24)μ(x-∑k=1nakxk,ϵ)=ϵϵ+‖x-∑k=1nakxk‖≥1-α,ν(x-∑k=1nakxk,ϵ)=ϵϵ+‖x-∑k=1nakxk‖≤α.


## 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
(4.1)μ(T(x)-x,t)≥1-α,ν(T(x)-x,t)≤α,
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 Tα:X→X of finite rank such that
(4.2)μ(Tα(x)-x,ϵ)≥1-α,ν(Tα(x)-x,ϵ)≤α,
for all x∈K.Remark 4.3.
The operatorT 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 Tα:X→X of finite rank such that
(4.3)‖Tα(x)-x‖α<ϵ,
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
(4.4)‖T(x)-x‖α<ϵ,
for all x∈K.The proof of the above theorem directly follows from Propositions2.2.Theorem 4.5.
Let(X,μ,ν,*,♢) be an IFNS possessing a wif-basis (xn). Then X has the wif-AP.Proof.
LetK⊂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 (an) of scalars such that
(4.5)Pn(x)=∑k=1nakxk→wifxasn⟶∞.
Then, there exists some n∘(α,ϵ) such that
(4.6)μ(Pn(x)-x,ϵ)≥1-α,ν(Pn(x)-x,ϵ)≤α
for all n≥n∘. Further, each Pn has a finite rank in the linear space X since dimPn(x)=n. Hence, each Pn such that n≥n∘ can be taken as a desired finite rank operator in the definition.Remark 4.6.
Theorem4.5 can also proved for sif-basis.Example 4.7.
LetX=ℓ∞, the Banach space of all bounded sequence with sup-norm ∥x∥∞=supn|xn|. Also, ∥x∥∘=supn|xn/n| is another norm on ℓ∞. Define the function
(4.7)μ(x,t)={1ift>‖x‖∞,12if‖x‖∘<t≤‖x‖∞,0ift≤‖x‖∘;ν(x,t)={0ift>‖x‖∞,12if‖x‖∞≤t<‖x‖∘,1ift≤‖x‖∘.
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
(4.8)‖x‖α=‖x‖∞for1>α>12,‖x‖α=‖x‖∘for1<α≤12.
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,…,βn) of natural numbers is a directed set by the relation pa≪pb which means that each βi∈pa is included in some βj′∈pb. Now, for each p∈D(4.9)Λp(x)=∑i=1nxhiχβi,
where hi is the distinguished point in βi and χβi is the characteristic function of βi for 1≤i≤n. Then Λp is a projection on ℓ∞ of finite rank. It is well known that the set (Λp(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)≪p(4.10)‖Λp(x)-x‖∞<ϵ.
But
(4.11)‖Λp(x)-x‖∘<ϵ,
for p∘(ϵ)≪p since ∥x∥∘≤∥x∥∞ for every x∈ℓ∞. That is, for p∘(ϵ)≪p,
(4.12)‖Λp(x)-x‖α<ϵ,
for all α∈(0,1). Hence for some Λp(p∘≪p) meets all requirements for sif-AP in Proposition 4.4.



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*Source: 131868-2011-01-12.xml*