# Statistical convergence in a paranormed space

- Abdullah Alotaibi
^{1}Email author and - Abdullah M Alroqi
^{1}

**2012**:39

https://doi.org/10.1186/1029-242X-2012-39

© Alotaibi and Altheabi et al; licensee Springer. 2012

**Received: **14 August 2011

**Accepted: **21 February 2012

**Published: **21 February 2012

## Abstract

In this article, we define the notion of statistical convergence, statistical Cauchy and strongly *p*-Cesàro summability in a paranormed space. We establish some relations between them.

**AMS subject classification (2000)**: 41A10; 41A25; 41A36; 40A05; 40A30.

## Keywords

*p*-Cesàro summability.

## 1 Introduction and preliminaries

The concept of statistical convergence for sequences of real numbers was introduced by Fast [1] and Steinhaus [2] independently in the same year 1951 and since then several generalizations and applications of this notion have been investigated by various authors, namely [3–11]. This notion was defined in normed spaces by Kolk [12] and in locally convex Hausdorff topological spaces by Maddox [13]. Çakalli [14] extended this notation to topological Hausdorff groups. Recently, in [15, 16], the concept of statistical convergence is studied in probabilistic normed space and in intuitionistic fuzzy normed spaces, respectively. In this article, we shall study the concept of statistical convergence, statistical Cauchy, and strongly *p*-Cesàro summability in a paranormed space.

Let *K* be a subset of the set of natural numbers ℕ. Then the *asymptotic density* of *K* denoted by *δ*(*K*), is defined as $\delta \left(K\right)={\text{lim}}_{n}\frac{1}{n}\left|\left\{k\le n:k\in K\right\}\right|$, where the vertical bars denote the cardinality of the enclosed set.

*x*= (

*x*

_{ k }) is said to be

*statistically convergent*to the number

*L*if for each

*ϵ*> 0, the set

*K*(

*ϵ*) = {

*k*≤

*n*: |

*x*

_{ k }-

*L*| >

*ϵ*} has asymptotic density zero, i.e.,

In this case we write *st*-lim *x* = *L*.

*x*= (

*x*

_{ k }) is said to be

*statistically Cauchy*sequence if for every

*ϵ*> 0, there exists a number

*N*=

*N*(

*ϵ*) such that

The concept of paranorm is a generalization of absolute value (see [17]).

A *paranorm* is a function *g*: *X* → ℝ defined on a linear space *X* such that for all *x, y, z* ∈ *X*

(*P* 1) *g*(*x*) = 0 if *x* = *θ*

(*P* 2) *g*(*-x*) = *g*(*x*)

(*P* 3) *g*(*x* + *y*) ≤ *g*(*x*) + *g*(*y*)

(*P* 4) If (*α*_{
n
}) is a sequence of scalars with *α*_{
n
}→ *α*_{0} (*n* → ∞) and *x*_{
n
}, *a* ∈ *X* with *x*_{
n
}→ *a* (*n* → ∞) in the sense that *g*(*x*_{
n
}- *a*) → 0 (*n* → ∞), then *α*_{
n
}*x*_{
n
}→ *α*_{0}*a* (*n* → ∞), in the sense that *g*(*α*_{
n
}*x*_{
n
}- *α*_{0}*a*) → 0 (*n* → ∞).

A paranorm *g* for which *g*(*x*) = 0 implies *x* = *θ* is called a *total paranorm* on *X*, and the pair (*X, g*) is called a *total paranormed space*.

Note that each seminorm (norm) *p* on *X* is a paranorm (total) but converse need not be true.

In this article, we define and study the notion of convergence, statistical convergence, statistical Cauchy, and strong summability by a modulus function in a paranormed space.

Let (*X, g*) be a paranormed space.

**Definition 1.1**. A sequence *x* = (*x*_{
k
}) is said to be *convergent* (or *g-convergent*) to the number *ξ* in (*X, g*) if for every *ε* > 0, there exists a positive integer *k*_{0} such that *g*(*x*_{
k
}- *ξ*) < *ε* whenever *k* ≥ *k*_{0}. In this we write *g*-lim *x* = *ξ*, and *ξ* is called the *g-limit* of *x*.

**Definition 1.2**. A sequence

*x*= (

*x*

_{ k }) is said to be

*statistically convergent to the number ξ in*(

*X, g*) (or

*g*(

*st*)-

*convergent*) if for each

*ϵ*> 0,

In this case, we write *g*(*st*)-lim *x* = *ξ*. We denote the set of all *g*(*st*)-convergent sequences by *S*_{
g
}.

**Definition 1.3**. A number sequence

*x*= (

*x*

_{ k }) is said to be

*statistically Cauchy*sequence in (

*X, g*) (or

*g*(

*st*)-

*Cauchy*) if for every

*ϵ*> 0 there exists a number

*N*=

*N*(

*ϵ*) such that

## 2 Main results

**Theorem 2.1**. If a sequence *x* = (*x*_{
k
}) is statistically convergent in (*X, g*) then *g*(*st*)-limit is unique.

**Proof**. Suppose that

*g*(

*st*)-lim

*x*=

*ξ*

_{1}and

*g*(

*st*)-lim

*x*=

*ξ*

_{2}. Given

*ε*> 0, define the following sets as:

Since *g*(*st*)-lim *x* = *ξ*_{1}, we have *δ*(*K*_{1}(*ε*)) = 0. Similarly, *g*(*st*)-lim *x* = *ξ*_{2} implies that *δ*(*K*_{2}(*ε*)) = 0. Now, let *K*(*ε*) = *K*_{1}(*ε*)∪*K*_{2}(*ε*). Then *δ*(*K*(*ε*)) = 0 and hence the compliment *K*^{
C
}(*ε*) is a nonempty set and *δ*(*K*^{
C
}(*ε*)) = 1. Now if *k* ∈ ℕ\*K*(*ε*), then we have *g*(*ξ*_{1}-*ξ*_{2}) ≤ *g*(*x*_{
n
}-*ξ*_{1})+*g*(*x*_{
n
}-*ξ*_{2}) < *ε*/2+*ε*/2 = *ε*.

Since *ε* > 0 was arbitrary, we get *g*(*ξ*_{1} - *ξ*_{2}) = 0 and hence *ξ*_{1} = *ξ*_{2}.

**Theorem** 2.2. If *g*-lim *x* = *ξ* then *g*(*st*)-lim *x* = *ξ* but converse need not be true in general.

**Proof**. Let

*g*-lim

*x*=

*ξ*. Then for every

*ε*> 0, there is a positive integer

*N*such that

for all *n* ≥ *N*. Since the set *A*(*ϵ*):= {*k* ∈ ℕ: *g*(*x*_{
k
}- *ξ*) ≥ *ε*} ⊂ {1, 2, 3, ...}, *δ*(*A*(*ϵ*)) = 0. Hence *g*(*st*)-lim *x* = *ξ*.

The following examle shows that the converse need not be true.

**Example 3.1**. Let

*X*= ℓ(1/

*k*): = {

*x*= (

*x*

_{ k }): ∑

_{ k }|

*x*

_{ k }|

^{1/k}< ∞} with the paranorm

*g*(

*x*) = (∑

_{ k }|

*x*

_{ k }|

^{1/k}). Define a sequence

*x*= (

*x*

_{ k }) by

Therefore *g*-lim *x* does not exist. On the other hand *δ*(*K*(*ε*)) = 0, that is, *g*(*st*)-lim *x* = 0.

**Theorem 2.3**. Let *g*(*st*)-lim *x* = *ξ*_{1} and *g*(*st*) - lim *y* = ξ_{2}. Then

(*i*) *g*(*st*)-lim(*x* ± *y*) = ξ_{1} ± ξ_{2},

(*ii*) *g*(*st*)-lim *αx* = αξ_{1}, α ∈ ℝ.

**Proof**. It is easy to prove.

**Theorem 2.4**. A sequence *x* = (*x*_{
k
}) in (*X, g*) is statistically convergent to *ξ* if and only if there exists a set *K* = {*k*_{1} < *k*_{2} < ⋯ < *k*_{
n
}< ⋯ } ⊆ ℕ with *δ*(*K*) = 1 such that $g\left({x}_{{k}_{n}}-\xi \right)\to 0\left(n\to \infty \right)$.

**Proof**. Suppose that

*g*(

*st*)-lim

*x*=

*ξ*. Now, write for

*r*= 1, 2, ....

*δ*(

*K*

_{ r }) = 0,

*n*∈

*M*

_{ r }, $\left({x}_{{k}_{n}}\right)$ is

*g*-convergent to

*ξ*. On contrary suppose that $\left({x}_{{k}_{n}}\right)$ is not

*g*-convergent to

*ξ*. Therefore there is

*ε*> 0 such that $g\left({x}_{{k}_{n}}-\xi \right)\le \epsilon $ for infinitely many terms. Let ${M}_{\epsilon}:=\left\{n\in \mathbb{N}:g\left({x}_{{k}_{n}}-\xi \right)>\epsilon \right\}$ and $\epsilon >\frac{1}{r},r\in \mathbb{N}$. Then

and by (2.4.1), *M*_{
r
}⊂ *M*_{
ε
}. Hence *δ*(*M*_{
r
}) = 0, which contradicts (2.4.2) and we get that $\left({x}_{{k}_{n}}\right)$ is *g*-convergent to *ξ*.

Conversely, suppose that there exists a set *K* = {*k*_{1} < *k*_{2} < *k*_{3} < ⋯ < *k*_{
n
}< ⋯} with *δ*(*K*) = 1 such that $g-{\text{lim}}_{n\to \infty}{x}_{{k}_{n}}=\xi $. Then there is a positive integer *N* such that *g*(*x*_{
n
}- *ξ*) < *ε* for *n* > *N*. Put *K*_{
ε
}(*t*): = {*n* ∈ ℕ: *g*(*x*_{
n
}-*ξ*) ≥ *ε*} and *K'*: = {*k*_{N+1}, *k*_{N+2}, ...}. Then *δ*(*K'*) = 1 and *K*_{
ε
}⊆ ℕ-*K'* which implies that *δ*(*K*_{
ε
}) = 0. Hence *g*(*st*)-lim *x* = *ξ*.

**Theorem 2.5**. Let (*X, g*) be a complete paranormed space. Then a sequence *x* = (*x*_{
k
}) of points in (*X, g*) is statistically convergent if and only if it is statistically Cauchy.

**Proof**. Suppose that

*g*(

*st*)-lim

*x*=

*ξ*. Then, we get

*A*(

*ε*): = {

*n*∈ ℕ:

*g*(

*x*

_{ n }-

*ξ*) ≥

*ε*/2}. This implies that

*m*∈

*A*

^{ C }(

*ε*). Then

*g*(

*x*

_{ m }-

*ξ*) <

*ε*/2. Now, let

*B*(

*ε*): = {

*n*∈ ℕ:

*g*(

*x*

_{ m }-

*x*

_{ n }) ≥

*ε*}. We need to show that

*B*(

*ε*) ⊂

*A*(

*ε*). Let

*n*∈

*B*(

*ε*). Then

*g*(

*x*

_{ n }-

*x*

_{ m }) ≥

*ε*and hence

*g*(

*x*

_{ n }-

*ξ*) ≥

*ε*/2, i.e.

*n*∈

*A*(

*ε*). Otherwise, if

*g*(

*x*

_{ n }-

*ξ*) <

*ε*then

which is not possible. Hence *B*(*ε*) ⊂ *A*(*ε*), which implies that *x* = (*x*_{
k
}) is *g*(*st*)-convergent.

Conversely, suppose that *x* = (*x*_{
k
}) is *g*(*st*)-Cauchy but not *g*(*st*)-convergent. Then there exists *M* ∈ ℕ such that *δ*(*G*(*ε*) = 0,

where *G*(*ε*): = {*n* ∈ ℕ: *g*(*x*_{
n
}- *x*_{
M
}) ≥ *ε*}, and *δ*(*D*(*ε*)) = 0, where *D*(*ε*): ={*n* ∈ ℕ: *g*(*x*_{
n
}- *ξ*) < *ε*/2}, i.e., *δ*(*D*^{
C
}(*ε*)) = 1. Since *g*(*x*_{
n
}- *x*_{
m
}) ≤ 2*g*(*x*_{
n
}- *ξ*) < *ε*,

if *g*(*x*_{
n
}- *ξ*) < *ε*/2. Therefore *δ*(*G*^{
C
}(*ε*)) = 0, i.e., *δ*(*G*(*ε*) = 1, which leads to a contradiction, since *x* = (*x*_{
k
}) was *g*(*st*)-Cauchy. Hence *x* = (*x*_{
k
}) must be *g*(*st*)-convergent.

## 3 Strong summability

In this section, we define the notion of strong summability by a modulus function and establish its relation with statistical convergence in a paranormed space.

**Definition 3.1**. A sequence

*x*= (

*x*

_{ k }) is said to be

*strongly p-Cesàro summable*(0 <

*p*< ∞) to the limit

*ξ*in (

*X, g*) if

and we write it as *x*_{
k
}→ *ξ*[*C*_{1}, *g*]_{
p
}. In this case *ξ* is called the [*C*_{1}, *g*]_{
p
}-limit of *x*.

**Theorem 3.1**. (*a*) If 0 < *p* < ∞ and *x*_{
k
}→ *ξ*[*C*_{1}, *g*]_{
p
}, then *x* = (*x*_{
k
}) is statistically convergent to *ξ* in (*X, g*).

(*b*) If *x* = (*x*_{
k
}) is bounded and statistically convergent to *ξ* in (*X, g*) then *x*_{
k
}→ *ξ*[*C*_{1}, *g*]_{
p
}.

**Proof**. (

*a*) Let

*x*

_{ k }→

*ξ*[

*C*

_{1},

*g*]

_{ p }, then

as *n* → ∞. That is, ${\text{lim}}_{n\to \infty}\frac{1}{n}\left|{K}_{\epsilon}\right|=0$ and so *δ*(*K*_{
ε
}) = 0, where *K*_{
ε
}: = {*k* ≤ *n*: (*g*(*x*_{
k
}- *ξ*))^{
p
}≥ *ε*} . Hence *x* = (*x*_{
k
}) is statistically convergent to *ξ* in (*X, g*).

*b*) Suppose that

*x*= (

*x*

_{ k }) is bounded and statistically convergent to

*ξ*in (

*X, g*). Then for

*ε*> 0, we have

*δ*(

*K*

_{ ε }) = 0. Since

*x*∈

*l*

_{∞}, there exists

*M*> 0 such that

*g*(

*x*

_{ k }-

*ξ*) ≤

*M*(

*k*= 1, 2, ...). We have

*k*∉

*K*

_{ ε }then

*S*

_{1}(

*n*) <

*ε*

^{ q }. For

*k*∈

*K*

_{ ε }, we have

as *n* → ∞, since *δ*(*K*_{
ε
}) = 0. Hence *x*_{
k
}→ *ξ*[*C*_{1}, *g*]_{
p
}.

This completes the proof of the theorem.

Recall that a *modulus f* is a function from [0, ∞) to [0, ∞) such that (*i*) *f*(*x*) = 0 if and only if *x* = 0, (*ii*) *f*(*x* + *y*) ≤ *f*(*x*) + *f*(*y*) for all *x, y* ≥ 0, (*iii*) *f* is increasing, and (*iv*) *f* is continuous from the right at 0.

Now we define the following:

**Definition 3.2**. Let

*f*be a modulus. we say that a sequence

*x*= (

*x*

_{ k }) is

*strongly p-Cesàro summable with respect to f*to the limit

*ξ*in (

*X, g*) if

(0 < *p* < ∞). In this case we write *x*_{
k
}→ *ξ*(*w*(*f, g, p*)).

As in [13], it is easy to prove the following:

**Theorem 3.2**. (*a*) Let *f* be any modulus and *x*_{
k
}→ *ξ*(*w*(*f, g, p*)). Then *x* = (*x*_{
k
}) is statistically convergent to *ξ* in (*X, g*).

(*b*) *S*_{
g
}= *w*(*f, g, p*) If and only if *f* is bounded.

## Declarations

### Acknowledgements

The authors would like to thank the Deanship of Scientific Research at King Abdulaziz University for its financial support under a grant with number 156/130/1431.

## Authors’ Affiliations

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