We present various kinds of statistical convergence and
ℐ-convergence for sequences of functions with values in 2-normed spaces and obtain
a criterion for ℐ-convergence of sequences of functions in 2-normed spaces. We also
define the notion of ℐ-equistatistically convergence and study ℐ-equi-statistically convergence of sequences of functions.

1. Introduction

The concept of ideal convergence was introduced first by Kostyrko et al. [1] as an interesting generalization of statistical convergence [2–5].

Throughout this paper ℕ will denote the set of positive integers. Let (X,∥·∥) be a normed space. Let K be a subset of positive integers ℕ and j∈ℕ. The quotient dj(K)=card(K⋂{1,…,j})/j is called the j’th partial density of K and dj is a probability measure on 𝒫(ℕ), with support {1,…,j} [2, 3].

The limit d(K) =limj→∞dj(K) (if exists) is called the natural density of K. Clearly, finite subsets have natural density zero and d(Kc)=1-d(K) where Kc=K∖ℕ, that is, the complement of K. If K1⊆K2 and K1,K2 have natural densities then d(K1)≤d(K2). Furthermore, if d(K1)=d(K2)=1, then d(K1∩K2)=1 [6].

Recall that a sequence (xn)n∈ℕ of elements of X is called to be statistically convergent to x∈X if the set A(ϵ)={n∈ℕ:∥xn-x∥≥ϵ} has natural density zero for each ϵ>0. In this case we write st-limn→∞xn=x [2–4].

A family ℐ⊆𝒫(Y) of subsets a nonempty set Y is said to be an ideal in Y if

∅∈ℐ,

A,B∈ℐimpliesA⋃B∈ℐ,

A∈ℐ,B⊆AimpliesB∈ℐ,

while an admissible ideal ℐ of Y further satisfies {x}∈ℐ for each x∈Y [7, 8]. Let ℐ⊆𝒫(ℕ) be a nontrivial ideal in ℕ. The sequence (xn)n∈ℕ in X is said to be ℐ-convergent to x∈X, if for each ϵ>0 the set A(ϵ)={n∈ℕ:∥xn-x∥≥ϵ} belongs to ℐ [1, 9].

2. Preliminaries

The notion of linear 2-normed spaces has been investigated by Gähler in the 60’s [10, 11] and this has been developed extensively in different subjects by others [12–14]. Let X be a real linear space of dimension greater than 1, and ∥·,·∥ be a nonnegative real-valued function on X×X satisfying the following conditions:

∥x,y∥=0 if and only if x and y are linearly dependent vectors;

∥x,y∥=∥y,x∥ for all x,y in X;

∥αx,y∥=|α|∥x,y∥ where α is real,

∥x+y,z∥≤∥x,z∥+∥y,z∥ for all x,y,z in X

∥·,·∥ is called a 2-norm on X and the pair (X,∥·,·∥) is called a linear 2-normed space. In addition, for all scalars α and all x,y,z in X, we have the following properties:

∥·,·∥ is nonnegative;

∥x,y∥=∥x,y+αx∥;

∥x-y,y-z∥=∥x-y,x-z∥.

Some of the basic properties of 2-norm are introduced in [14]. Given a 2-normed space (X,∥·,·∥), one can derive a topology for it via the following definition of the limit of a sequence: a sequence (xn)n∈ℕ in X is said to be convergent to x in X if limn→∞∥xn-x,z∥=0 for every z∈X. This can be written by the formula (∀z∈Y)(∀ϵ>0)(∃n0∈N)(∀n≥n0)‖xn-x,z‖<ϵ.
We write it as xn→‖⋅,⋅‖Xx.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B9">13</xref>]).

Let v={v1,…,vk} be a basis of X. A sequence (xn)n∈ℕ in X is convergent to x in X if and only if limn→∞∥xn-x,vi∥=0 for every i=1,…,k. We can define the norm ∥·∥∞ on X by
‖x‖∞:=max{‖x,vi‖:i=1,…,d=k}.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B9">13</xref>]).

A sequence (xn)n∈ℕ in X is convergent to x in X if and only if limn→∞∥xn-x∥∞=0.

Example 2.3.

Let X=ℝ2 be equipped with the 2-norm ∥x,y∥:= the area of the parallelogram spanned by the vectors x and y, which may be given explicitly by the formula
‖x,y‖=|x1y2-x2y1|,x=(x1,x2),y=(y1,y2).
Take the standard basis {i,j} for ℝ2.

Then, ∥x,i∥=|x2| and ∥x,j∥=|x1|, and so the derived norm ∥·∥∞ with respect to {i,j} is‖x‖∞=max{|x1|,|x1|},x=(x1,x2).
Thus, here the derived norm ∥·∥∞ is exactly the same as the uniform norm on R2. Since the derived norm is a norm, it is equivalent to the Euclidean norm on R2.

Definition 2.4.

Let ℐ⊂2N be a nontrivial ideal in ℕ. The sequence (xn)n∈ℕ of X is said to be ℐ-convergent to x, if for each ϵ>0 and nonzero z in X the set A(ϵ)={n∈ℕ:∥xn-x,z∥≥ϵ} belongs to ℐ [9].

If (xn)n∈ℕ is ℐ-convergent to x then we write it as I-limn→∞‖xn-x,z‖=0orI-limn→∞‖xn,z‖=‖x,z‖.
The element x is ℐ-limit of the sequence (xn)n∈ℕ.

Remark 2.5.

If (xn)n∈ℕ is any sequence in X and x is any element of X, then the set
{n∈N:‖xn-x,z‖≥ϵ,∀z∈X}=∅
since if z=0, ∥xn-x,z∥=0<ϵ so the above set is empty.

Further we will give some examples of ideals and corresponding ℐ-convergences.

Now we give an example of ℐ-convergence in 2-normed spaces.

Example 2.6.

(i) Let ℐf be the family of all finite subsets of ℕ. Then ℐf is an admissible ideal in ℕ and ℐf-convergence coincides with usual convergence [11].

(ii) Put ℐd= {A⊂ℕ:d(A)=0}. Then ℐd is an admissible ideal in ℕ and ℐd-convergence coincides with the statistical convergence [15].

Example 2.7.

Let ℐ=ℐd. Define the (xn)n∈ℕ in 2-normed space (X,∥·,·∥) by
xn={(0,n),n=k2,k∈N,(0,0),otherwise
and let x=(0,0) and z=(z1,z2). Then for every ϵ>0 and z∈X{n∈N:‖xn-x,z‖>ϵ}⊆{1,4,9,16,…,n2,…}
we have that
d({n∈N:‖xn-x,z‖>ϵ})=0foreveryε>0andnonzeroz∈X.
This implies that ℐd-limn→∞∥xn,z∥=∥x,z∥. But, the sequence (xn)n∈ℕ is not convergent to x.

3. Convergence for Sequences of Functions in 2-Normed Spaces

We discuss various kinds of convergence and ℐ-convergence for sequences of functions with values in 2-normed spaces.

Let X,Y be 2-normed spaces and assume that functions f:X⟶Y,fn:X⟶Y,n∈N
are given.

Definition 3.1.

The sequence (fn)n∈N is said to be positive convergent to f (on X) if
fn(x)→‖⋅,⋅‖Yf(x)foreachx∈X.
We write
fn→‖⋅,⋅‖Yf.

This can be expressed by the formula (∀y∈Y)(∀x∈X)(∀ϵ>0)(∃n0∈N)(∀n>n0)‖fn(x)-f(x),y‖Y<ϵ.

Remark 3.2.

If functions f,fn are given as in Definition 3.1 and dimY<∞ then (fn) is pointwise convergent to f (on X) if and only if
(∀x∈X)(∀ϵ>0)(∃n0∈N)(∀n>n0)‖fn(x)-f(x),y‖∞<ϵ.

We introduce uniform convergent of (fn)n∈ℕ to f by the formula(∀y∈Y)(∀ϵ>0)(∃n0∈N)(∀n>n0)(∀x∈X)‖fn(x)-f(x),y‖Y<ϵ
and we write it asfn→‖⋅,⋅‖Yuniformf.

Example 3.3.

If X=Y=ℝ2 is introduced in Lemma 2.1 then define
f(x1,x2)={(0,0)if|x2|<1,(0,12)if|x2|=1(0,1)if|x2|>1,fn(x)=(0,x22n1+x22n),
then
fn→‖⋅,⋅‖Yuniformf,fn→‖⋅,⋅‖Yf.

Example 3.4.

Let X=Y=[0,1]×(0,1)⊆ℝ2 and define
fn(x1,x2)=(0,1nx2+1),f(x1,x2)=(0,0).
Then obviously fn→∥·,·∥Yf. But we show that fn does not converge uniformly to f in Y. Fix ε=1/2 and for all n0∈ℕ put n0=n+1,xn=(0,1/2n) then
‖fn(x1,x2)-0‖∞=|1nx2+1|=23>ε.

Definition 3.5.

Let X and Y be 2-normed spaces with dimY<∞ and let f:X→Y be a function. The function f is said to be sequentially continuous at x0∈X if for any sequence (xn)n∈ℕ of X converging to x0 one has
f(xn)→‖⋅,⋅‖Yf(x0).

Definition 3.6.

Let X and Y be two 2-normed spaces, and dimY <∞. If fn:X→Y is a sequence of functions, we say (fn)n∈ℕ is equi-continuous (on X) if
(∀z∈X)(∀ε>0)(∃δ>0)(∀x,x0∈X)‖x-x0,z‖X<δ⟹‖fn(x)-fn(x0)‖∞<ε.

Corollary 3.7.

Let X and Y be two 2-normed spaces, x0∈X with dimY<∞.

If f:X→Y is a function such that satisfying the following formula(∀z∈X)(∀ε>0)(∃δ>0)(∀x∈X)‖x-x0,z‖X<δ⟹‖f(x)-f(x0)‖∞<ε
then f is sequentially continuous at x0.

Proof.

Let (xn)n∈ℕ be a sequence in X such that xn→∥·,·∥Xx0. Let ε>0. There exists δ>0 such that ∥f(x)-f(x0)∥∞<ε for every x∈X where ∥x-x0,z∥X≤δ for each z∈X. On the other hand xn→∥·,·∥Xx0 hence for all z∈X there exist n0 such that ∥xn-x0,z∥X<δ for all n≥n0. Therefore f(xn)→∥·,·∥Yf(x0) and f is sequentially continuous at x0.

4. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M238"><mml:mrow><mml:mi>ℐ</mml:mi></mml:mrow></mml:math></inline-formula>-Convergence of Functions in 2-Normed Spaces

Let X,Y be 2-normed spaces. Fix an admissible ideal ℐ⊆𝒫(ℕ) and assume that functions f:X→Y,fn:X→Y,n∈ℕ are given.

Definition 4.1.

A sequence (fn)n∈ℕ of functions is said to be ℐ-pointwise convergent to f (on X) if ℐ- limn→∞∥fn(x)-f(x),z∥Y=0 (in (Y,∥·,·∥Y)) for each x∈X. We Write
fn→‖⋅,⋅‖YIf.

This can be expressed by the formula(∀z∈Y)(∀x∈X)(∀ε>0)(∃M∈I)(∀n∈N∖M)‖fn(x)-f(x),z‖Y<ε.

Definition 4.2.

A sequence, (fn)n∈ℕ is said to be ℐ-uniformly convergent to f (on X) if and only if
(∀z∈Y)(∀ε>0)(∃M∈I)(∀n∈N∖M)(∀x∈X)‖fn(x)-f(x),z‖Y≤ε.
We write fn→∥·,·∥Yℐuniformf.

Remark 4.3.

If ℐ=ℐd then →∥·,·∥Yℐd and →∥·,·∥Yℐduniform will be read (respectively) as ℐ-pointwise and ℐ-uniform statistically convergence. If fn→∥·,·∥Yℐdf, then for all x∈Xfn(x)→∥·,·∥Yℐdf(x) which may be given by the formula
(∀x∈X)(∀ε>0){n∈N:‖fn(x)-f(x)‖∞≥ε}∈Id
we have by [15]
(∀x∈X)(∀ε,δ>0)(∃n0∈N)(∀n≥n0)dj({n∈N:‖fn(x)-f(x)‖∞≥ε})<δ.

Remark 4.4.

We obviously have
fn→‖⋅,⋅‖YIuniformf⟹fn→‖⋅,⋅‖YIf,fn→‖⋅,⋅‖YIuniformf⟺supx∈X‖fn(x)-f(x),z‖Y→‖⋅,⋅‖YI0∀z∈Y.

Remark 4.5.

Let ℐ be such that ℐ-convergence of sequences of points in (Y,∥·,·∥Y) is strictly more general than the usual convergence. Then there is a sequence (yn)n∈ℕ⊆Y, such that
yn→‖⋅,⋅‖YIybutlimn→∞‖yn-y,z‖Y≠0foreachz∈Y.
Putting fn(x)=yn and f(x)=y for x∈X and n∈ℕ, we have
fn→‖⋅,⋅‖YIuniformfbut¬fn→‖⋅,⋅‖Yuniformf.
Thus, in this situation, ℐ-uniform convergence of sequences of functions is strictly more general than the usual uniform convergence.

Theorem 4.6.

Let ℐ⊆𝒫(ℕ) be an admissible ideal and X,Y be two 2-normed spaces with dimY<∞. Assume that fn→∥·,·∥Yℐf (on X) where functions fn:X→Y,n∈N are equi-continuous (on X) and f:X→Y. Then f is sequentially continuous (on X).

Proof.

Let z,x0∈X and ε>0. By equi-continuty of fn’s, there exist δ>0 such that ∥fn(x0)-fn(x)∥∞≤ε for every n∈ℕ whenever ∥x-x0,z∥<δ.

Fix x∈X such that ∥x-x0,z∥<δ. Since fn→∥·,·∥Yℐf, the set{n∈N:‖fn(x0)-f(x0)‖∞≥ε3}∪{n∈N:‖fn(x)-f(x)‖∞≥ε3}
is in ℐ and different from ℕ. Hence there exists n0∈N such that
∥fn0(x0)-f(x0)∥∞<ε3,∥fn0(x)-f(x)∥∞<ε3.
We have
‖f(x0)-f(x)‖∞≤‖f(x0)-fn0(x0)‖∞+‖fn0(x0)-fn0(x)‖∞+‖fn0(x)-f(x)‖∞≤ε3+ε3+ε3=ϵ
and by (Corollary 3.7) f is sequentially continuous at x0∈X.

Let X,Y be two 2-normed spaces with dimY<∞ and ℐ=ℐd⊆2X be admissible ideal on X.

Definition 5.1.

A (fn)n∈ℕ is called ℐ-equi-statistically convergent to f (we write it as fn⇝∥·,·∥Yℐdf) if for every ε>0 the sequence (gj,ϵ)j∈N of functionsgj,ε:X→ℝ given by
gj,ε(x)=dj({n∈N:‖fn(x)-f(x)‖∞≥ε}),x∈X
is uniformly convergent to the zero function (on X). Hence fn⇝∥·,·∥Yℐdf if and only if the following formula holds:
(∀ε,δ>0)(∃n0∈N)(∀j≥n0)(∀x∈X),dj({n∈N:‖fn(x)-f(x)‖∞≥ε})<δ.

Corollary 5.2.

The following properties hold:

fn⇝∥·,·∥Yℐdf implies fn→∥·,·∥Yℐdf,

fn→∥·,·∥Yℐduniformf implies fn⇝∥·,·∥Yℐdf.

Proof.

If fn⇝∥·,·∥Yℐdf by the monotonicity of operator dj, we take ε=δ in Definition 4.2. Thus it is obvious.

Assume fn→∥·,·∥Yℐduniformf and ε>0. By Definition 4.2 there exist a set M∈ℐd such that ∥fn(x)-f(x)∥∞<ε for all n∈ℐd∖M and x∈X. Since M∈ℐd. We can pick n0∈N such that dj(M)<ε for all j≥n0. Let x∈X and n∈ℕ. Thus ∥fn(x)-f(x)∥∞≥ε implies n∈M. Hence for each j≥n0, we have

dj({n∈N:‖fn(x)-f(x)‖∞≥ε})≤dj(M)<ε
by Definition 4.2 witnesses that fn⇝∥·,·∥Yℐdf. Example 5.3.

Define f:[0,1]×[0,1]→ℝ2,fn:[0,1]×[0,1]→ℝ2,n∈ℕfn(x1,x2)={(0,1n),ifx2=1n,(0,0),otherwise,f(x1,x2)=(0,0),
Then fn⇝∥·,·∥Yℐdf but ¬fn→∥·,·∥Yℐduniformf. Indeed, let ε>0 and k∈ℕ such that 1/k<ε. Then for all j≥k and x=(x1,x2)∈[0,1]×[0,1] we have
dj({n∈N:‖fn(x)-f(x)‖∞>ε})≤1j≤1k≤ε.
Hence fn⇝∥·,·∥Yℐdf.

Suppose that fn→∥·,·∥Yℐduniformf. Thus there is the set M∈ℐd such that for all n∈ℐd∖M and x∈[0,1]×[0,1] we have ∥fn(x)-f(x)∥∞<1.

Choose k∈ℐd∖M. Then fk must be the zero function, a contradiction.

Theorem 5.4.

Assume f:X→Y and fn:X→Y for n∈ℕ fix x0∈X. If fn⇝∥·,·∥Yℐdf and all functions fn,n∈ℕ, are sequentially continuous at x0 then f is sequentially continuous at x0.

Proof.

Let ε>0. Since fn⇝∥·,·∥Yℐdf, we can find n0∈ℕ such that
dn0({n∈N:‖fn(x)-f(x)‖∞≥ε3})<12∀x∈X.
Put E(x)={n≤K:∥fn(x)-f(x)∥∞<ε/3},x∈X. In other word dn0 is a probability measure on 𝒫(ℕ) with the support {1,…,n0}, it follows that dn0(E(x))>1/2 for all x∈X. By the sequentially continuity of f1,…,fn0 at x0, there exist δ>0 such that ∥fi(x)-fi(x0)∥∞<ε/3 for all 1≤i≤n0 and x∈X,∥x-x0,z∥<δ for each z∈X. Fix x such that x∈X, ∥x-x0,z∥<δ for each z∈X.

Since dn0(E(x))>1/2 and dn0(E(x0))>1/2, there exists p∈E(x)⋂E(x0) such that‖f(x)-f(x0)‖∞≤‖f(x)-fp(x)‖∞+‖fp(x)-fp(x0)‖∞+‖fp(x0)-f(x0)‖∞≤ε3+ε3+ε3=ϵ.

Thus we show that ∥f(x)-f(x0)∥∞<ε for all x∈X,∥x-x0,z∥<δ for each z∈X.

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