Statistical Convergence and Ideal Convergence of Sequences of Functions in 2-Normed Spaces

Abstract

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 𝑑_𝑗(𝐾)=card(𝐾⋂{1,…,𝑗})/𝑗 is called the j’th partial density of 𝐾 and 𝑑_𝑗 is a probability measure on 𝒫(ℕ), with support {1,…,𝑗} [2, 3].

The limit 𝑑(𝐾) =lim_𝑗→∞𝑑_𝑗(𝐾) (if exists) is called the natural density of 𝐾. Clearly, finite subsets have natural density zero and 𝑑(𝐾^𝑐)=1−𝑑(𝐾) where 𝐾^𝑐=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 (𝑥_𝑛)_𝑛∈ℕ of elements of 𝑋 is called to be statistically convergent to 𝑥∈𝑋 if the set 𝐴(𝜖)={𝑛∈ℕ∶‖𝑥_𝑛−𝑥‖≥𝜖} has natural density zero for each 𝜖>0. In this case we write 𝑠𝑡-lim_𝑛→∞𝑥_𝑛=𝑥 [2–4].

A family ℐ⊆𝒫(𝑌) of subsets a nonempty set 𝑌 is said to be an ideal in 𝑌 if(i)∅∈ℐ, (ii)𝐴,𝐵∈ℐimplies𝐴⋃𝐵∈ℐ, (iii)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 (𝑥_𝑛)_𝑛∈ℕ in 𝑋 is said to be ℐ-convergent to 𝑥∈𝑋, if for each 𝜖>0 the set 𝐴(𝜖)={𝑛∈ℕ∶‖𝑥_𝑛−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:(G1)‖𝑥,𝑦‖=0 if and only if 𝑥 and 𝑦 are linearly dependent vectors;(G2)‖𝑥,𝑦‖=‖𝑦,𝑥‖ for all 𝑥,𝑦 in 𝑋;(G3)‖𝛼𝑥,𝑦‖=|𝛼|‖𝑥,𝑦‖ where 𝛼 is real,(G4)‖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:(1)‖⋅,⋅‖ is nonnegative;(2)‖𝑥,𝑦‖=‖𝑥,𝑦+𝛼𝑥‖; (3)‖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 (𝑥_𝑛)_𝑛∈ℕ in 𝑋 is said to be convergent to 𝑥 in 𝑋 if lim_𝑛→∞‖𝑥_𝑛−𝑥,𝑧‖=0 for every 𝑧∈𝑋. This can be written by the formula (∀𝑧∈𝑌)(∀𝜖>0)∃𝑛₀∈ℕ∀𝑛≥𝑛₀𝑥_𝑛−𝑥,𝑧<𝜖. (2.1) We write it as 𝑥−−−−→_𝑛^{‖‖⋅,⋅𝑋}𝑥. (2.2)

Lemma 2.1 (see [13]). Let 𝑣={𝑣₁,…,𝑣_𝑘} be a basis of 𝑋. A sequence (𝑥_𝑛)_𝑛∈ℕ in X is convergent to 𝑥 in 𝑋 if and only if lim_𝑛→∞‖𝑥_𝑛−𝑥,𝑣_𝑖‖=0 for every 𝑖=1,…,𝑘. We can define the norm ‖⋅‖_∞ on 𝑋 by ‖𝑥‖_∞∶=max𝑥,𝑣_𝑖∶𝑖=1,…,𝑑=𝑘. (2.3)

Lemma 2.2 (see [13]). A sequence (𝑥_𝑛)_𝑛∈ℕ in 𝑋 is convergent to 𝑥 in 𝑋 if and only if lim_𝑛→∞‖𝑥_𝑛−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. (2.4) 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‖_∞=max𝑥₁,𝑥₁,𝑥=𝑥₁,𝑥₂. (2.5) 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⊂2^𝑁 be a nontrivial ideal in ℕ. The sequence (𝑥_𝑛)_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∶‖𝑥_𝑛−𝑥,𝑧‖≥𝜖} belongs to ℐ [9].

If (𝑥_𝑛)_𝑛∈ℕ is ℐ-convergent to 𝑥 then we write it as ℐ−lim_𝑛→∞𝑥_𝑛−𝑥,𝑧=0orℐ−lim_𝑛→∞𝑥_𝑛,𝑧=‖𝑥,𝑧‖. (2.6) The element 𝑥 is ℐ-limit of the sequence (𝑥_𝑛)_𝑛∈ℕ.

Remark 2.5. If (𝑥_𝑛)_𝑛∈ℕ is any sequence in 𝑋 and 𝑥 is any element of 𝑋, then the set 𝑛∈ℕ∶𝑥_𝑛−𝑥,𝑧≥𝜖,∀𝑧∈𝑋=∅ (2.7) since if 𝑧=0, ‖𝑥_𝑛−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 ℐ_𝑓 be the family of all finite subsets of ℕ. Then ℐ_𝑓 is an admissible ideal in ℕ and ℐ_𝑓-convergence coincides with usual convergence [11].
(ii) Put ℐ_𝑑= {𝐴⊂ℕ∶𝑑(𝐴)=0}. Then ℐ_𝑑 is an admissible ideal in ℕ and ℐ_𝑑-convergence coincides with the statistical convergence [15].

Example 2.7. Let ℐ=ℐ_𝑑. Define the (𝑥_𝑛)_𝑛∈ℕ in 2-normed space (𝑋,‖⋅,⋅‖) by 𝑥_𝑛=(0,𝑛),𝑛=𝑘²(),𝑘∈ℕ,0,0,otherwise (2.8) and let 𝑥=(0,0) and 𝑧=(𝑧₁,𝑧₂). Then for every 𝜖>0 and 𝑧∈𝑋𝑛∈ℕ∶𝑥_𝑛−𝑥,𝑧>𝜖⊆1,4,9,16,…,𝑛²,… (2.9) we have that 𝑑𝑛∈ℕ∶𝑥_𝑛−𝑥,𝑧>𝜖=0forevery𝜀>0andnonzero𝑧∈𝑋. (2.10) This implies that ℐ_𝑑−lim_𝑛→∞‖𝑥_𝑛,𝑧‖=‖𝑥,𝑧‖. But, the sequence (𝑥_𝑛)_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,𝑓_𝑛∶𝑋⟶𝑌,𝑛∈𝑁 (3.1) are given.

Definition 3.1. The sequence (𝑓_𝑛)_𝑛∈𝑁 is said to be positive convergent to 𝑓 (on 𝑋) if 𝑓_𝑛(𝑥)−−−−→^{‖‖⋅,⋅𝑌}𝑓(𝑥)foreach𝑥∈𝑋. (3.2) We write 𝑓−−−−→_𝑛^{‖‖⋅,⋅𝑌}𝑓. (3.3)

This can be expressed by the formula ∀𝑦∈𝑌(∀𝑥∈𝑋)(∀𝜖>0)∃𝑛₀∈ℕ∀𝑛>𝑛₀𝑓_𝑛(𝑥)−𝑓(𝑥),𝑦_𝑌<𝜖. (3.4)

Remark 3.2. If functions 𝑓,𝑓_𝑛 are given as in Definition 3.1 and dim𝑌<∞ then (𝑓_𝑛) is pointwise convergent to 𝑓 (on 𝑋) if and only if (∀𝑥∈𝑋)(∀𝜖>0)∃𝑛₀∈ℕ∀𝑛>𝑛₀𝑓_𝑛(𝑥)−𝑓(𝑥),𝑦_∞<𝜖. (3.5)

We introduce uniform convergent of (𝑓_𝑛)_𝑛∈ℕ to 𝑓 by the formula∀𝑦∈𝑌(∀𝜖>0)∃𝑛₀∈ℕ∀𝑛>𝑛₀(∀𝑥∈𝑋)𝑓_𝑛(𝑥)−𝑓(𝑥),𝑦_𝑌<𝜖 (3.6) and we write it as𝑓−−−−−−_𝑛^{‖‖⋅,⋅𝑌}→_uniform𝑓. (3.7)

Example 3.3. If 𝑋=𝑌=ℝ² is introduced in Lemma 2.1 then define 𝑓𝑥₁,𝑥₂=(0,0)if𝑥₂1<1,0,2if𝑥₂()𝑥=10,1if₂>1,𝑓_𝑛(𝑥)=𝑥0,^22𝑛1+𝑥₂^2𝑛, (3.8) then 𝑓−−−−−−_𝑛^{‖‖⋅,⋅𝑌}→_uniform𝑓,𝑓−−−−→_𝑛^{‖‖⋅,⋅𝑌}𝑓. (3.9)

Example 3.4. Let 𝑋=𝑌=[0,1]×(0,1)⊆ℝ² and define 𝑓_𝑛𝑥₁,𝑥₂=10,𝑛𝑥₂𝑥+1,𝑓₁,𝑥₂=(0,0). (3.10) Then obviously 𝑓−−−−→_𝑛^{‖⋅,⋅‖𝑌}𝑓. But we show that 𝑓_𝑛 does not converge uniformly to 𝑓 in 𝑌. Fix 𝜀=1/2 and for all 𝑛₀∈ℕ put 𝑛₀=𝑛+1,𝑥_𝑛=(0,1/2𝑛) then 𝑓_𝑛𝑥₁,𝑥₂−0_∞=1𝑛𝑥₂=2+13>𝜀. (3.11)

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 (𝑥_𝑛)_𝑛∈ℕ of 𝑋 converging to 𝑥₀ one has 𝑓𝑥−−−−→_𝑛^{‖‖⋅,⋅𝑌}𝑓𝑥₀. (3.12)

Definition 3.6. Let 𝑋 and 𝑌 be two 2-normed spaces, and dimY <∞. If 𝑓_𝑛∶𝑋→𝑌 is a sequence of functions, we say (𝑓_𝑛)_𝑛∈ℕ is equi-continuous (on 𝑋) if (∀𝑧∈𝑋)(∀𝜀>0)(∃𝛿>0)∀𝑥,𝑥₀∈𝑋𝑥−𝑥₀,𝑧_𝑋<𝛿⟹𝑓_𝑛(𝑥)−𝑓_𝑛𝑥_0∞<𝜀. (3.13)

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)−𝑓𝑥_0∞<𝜀 (3.14) then 𝑓 is sequentially continuous at 𝑥₀.

Proof. Let (𝑥_𝑛)_𝑛∈ℕ be a sequence in 𝑋 such that 𝑥−−−−→_𝑛^{‖⋅,⋅‖𝑋}𝑥₀. 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 𝑥−−−−→_𝑛^{‖⋅,⋅‖𝑋}𝑥₀ hence for all 𝑧∈𝑋 there exist 𝑛₀ such that ‖𝑥_𝑛−𝑥₀,𝑧‖_𝑋<𝛿 for all 𝑛≥𝑛₀. Therefore 𝑓(𝑥_𝑛)−−−−→^{‖⋅,⋅‖𝑌}𝑓(𝑥₀) and 𝑓 is sequentially continuous at 𝑥₀.

4. ℐ-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,𝑓_𝑛∶𝑋→𝑌,𝑛∈ℕ are given.

Definition 4.1. A sequence (𝑓_𝑛)_𝑛∈ℕ of functions is said to be ℐ-pointwise convergent to f  (on 𝑋) if ℐ- lim_𝑛→∞‖𝑓_𝑛(𝑥)−𝑓(𝑥),𝑧‖_𝑌=0 (in (𝑌,‖⋅,⋅‖_𝑌)) for each 𝑥∈𝑋. We Write 𝑓−−−−→_𝑛^{‖‖⋅,⋅𝑌}_ℐ𝑓. (4.1)
This can be expressed by the formula(∀𝑧∈𝑌)(∀𝑥∈𝑋)(∀𝜀>0)(∃𝑀∈ℐ)(∀𝑛∈ℕ⧵𝑀)𝑓_𝑛(𝑥)−𝑓(𝑥),𝑧_𝑌<𝜀. (4.2)

Definition 4.2. A sequence, (𝑓_𝑛)_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)𝑓_𝑛(𝑥)−𝑓(𝑥),𝑧_𝑌≤𝜀. (4.3) We write 𝑓−−−−→_𝑛^{‖⋅,⋅‖𝑌}_ℐuniform𝑓.

Remark 4.3. If ℐ=ℐ_𝑑 then −−−−→^{‖⋅,⋅‖𝑌}_ℐ𝑑 and −−−−→^{‖⋅,⋅‖𝑌}_{ℐ𝑑uniform} will be read (respectively) as ℐ-pointwise and ℐ-uniform statistically convergence. If 𝑓−−−−→_𝑛^{‖⋅,⋅‖𝑌}_ℐ𝑑𝑓, then for all 𝑥∈𝑋𝑓_𝑛(𝑥)−−−−→^{‖⋅,⋅‖𝑌}_ℐ𝑑𝑓(𝑥) which may be given by the formula (∀𝑥∈𝑋)(∀𝜀>0)𝑛∈ℕ∶𝑓_𝑛(𝑥)−𝑓(𝑥)_∞≥𝜀∈ℐ_𝑑 (4.4) we have by [15] (∀𝑥∈𝑋)(∀𝜀,𝛿>0)∃𝑛₀∈ℕ∀𝑛≥𝑛₀𝑑_𝑗𝑛∈ℕ∶𝑓_𝑛(𝑥)−𝑓(𝑥)_∞≥𝜀<𝛿. (4.5)

Remark 4.4. We obviously have 𝑓−−−−→_𝑛^{‖‖⋅,⋅𝑌}_ℐuniform𝑓⟹𝑓−−−−→_𝑛^{‖‖⋅,⋅𝑌}_ℐ𝑓𝑓,_{𝑛‖‖⋅,⋅𝑌}−−−−→_ℐuniform𝑓⟺sup_𝑥∈𝑋𝑓_𝑛(𝑥)−𝑓(𝑥),𝑧−−−−→_𝑌^{‖‖⋅,⋅𝑌}_ℐ0∀𝑧∈𝑌. (4.6)

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 (𝑦_𝑛)_𝑛∈ℕ⊆𝑌, such that 𝑦−−−−→_𝑛^{‖‖⋅,⋅𝑌}_ℐ𝑦butlim_𝑛→∞𝑦_𝑛−𝑦,𝑧_𝑌≠0foreach𝑧∈𝑌. (4.7) Putting 𝑓_𝑛(𝑥)=𝑦_𝑛 and 𝑓(𝑥)=𝑦 for 𝑥∈𝑋 and 𝑛∈ℕ, we have 𝑓−−−−→_𝑛^{‖‖⋅,⋅𝑌}_ℐuniform𝑓but¬𝑓−−−−−−_𝑛^{‖‖⋅,⋅𝑌}→_uniform𝑓. (4.8) 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 𝑓−−−−→_𝑛^{‖⋅,⋅‖𝑌}_ℐ𝑓 (on 𝑋) where functions 𝑓_𝑛∶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 𝑓_𝑛’s, there exist 𝛿>0 such that ‖𝑓_𝑛(𝑥₀)−𝑓_𝑛(𝑥)‖_∞≤𝜀 for every 𝑛∈ℕ whenever ‖𝑥−𝑥₀,𝑧‖<𝛿.
Fix 𝑥∈𝑋 such that ‖𝑥−𝑥₀,𝑧‖<𝛿. Since 𝑓−−−−→_𝑛^{‖⋅,⋅‖𝑌}_ℐ𝑓, the set𝑛∈𝑁∶𝑓_𝑛𝑥₀−𝑓𝑥_0∞≥𝜀3∪𝑛∈𝑁∶𝑓_𝑛(𝑥)−𝑓(𝑥)_∞≥𝜀3 (4.9) is in ℐ and different from ℕ. Hence there exists 𝑛₀∈𝑁 such that ‖𝑓_𝑛0𝑥₀−𝑓𝑥₀‖_∞<𝜀3,‖𝑓_𝑛0(𝑥)−𝑓(𝑥)‖_∞<𝜀3. (4.10) We have 𝑓𝑥₀−𝑓(𝑥)_∞≤𝑓𝑥₀−𝑓_𝑛0𝑥_0∞+𝑓_𝑛0𝑥₀−𝑓_𝑛0(𝑥)_∞+𝑓_𝑛0(𝑥)−𝑓(𝑥)_∞≤𝜀3+𝜀3+𝜀3=𝜖 (4.11) and by (Corollary 3.7) 𝑓 is sequentially continuous at 𝑥₀∈𝑋.

5. ℐ-Equistatistically Convergent

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

Definition 5.1. A (𝑓_𝑛)_𝑛∈ℕ is called ℐ-equi-statistically convergent to 𝑓 (we write it as 𝑓⇝_𝑛^{‖⋅,⋅‖𝑌}_ℐ𝑑𝑓) if for every 𝜀>0 the sequence (𝑔_𝑗,𝜖)_𝑗∈𝑁 of functions𝑔_𝑗,𝜀∶𝑋→ℝ given by 𝑔_𝑗,𝜀(𝑥)=𝑑_𝑗𝑛∈ℕ∶𝑓_𝑛(𝑥)−𝑓(𝑥)_∞≥𝜀,𝑥∈𝑋 (5.1) is uniformly convergent to the zero function (on 𝑋). Hence 𝑓⇝_𝑛^{‖⋅,⋅‖𝑌}_ℐ𝑑𝑓 if and only if the following formula holds: (∀𝜀,𝛿>0)∃𝑛₀∈ℕ∀𝑗≥𝑛₀(∀𝑥∈𝑋),𝑑_𝑗𝑛∈ℕ∶𝑓_𝑛(𝑥)−𝑓(𝑥)_∞≥𝜀<𝛿. (5.2)

Corollary 5.2. The following properties hold:(i)𝑓⇝_𝑛^{‖⋅,⋅‖𝑌}_ℐ𝑑𝑓 implies 𝑓−−−−→_𝑛^{‖⋅,⋅‖𝑌}_ℐ𝑑𝑓,(ii)𝑓−−−−→_𝑛^{‖⋅,⋅‖𝑌}_{ℐ𝑑uniform}𝑓 implies 𝑓⇝_𝑛^{‖⋅,⋅‖𝑌}_ℐ𝑑𝑓.

Proof. (i)If 𝑓⇝_𝑛^{‖⋅,⋅‖𝑌}_ℐ𝑑𝑓 by the monotonicity of operator 𝑑_𝑗, we take 𝜀=𝛿 in Definition 4.2. Thus it is obvious.(ii)Assume 𝑓−−−−→_𝑛^{‖⋅,⋅‖𝑌}_ℐduniform𝑓 and 𝜀>0. By Definition 4.2 there exist a set 𝑀∈ℐ𝑑 such that ‖𝑓_𝑛(𝑥)−𝑓(𝑥)‖_∞<𝜀 for all 𝑛∈ℐ_𝑑⧵𝑀 and 𝑥∈𝑋. Since 𝑀∈ℐ_𝑑. We can pick 𝑛₀∈𝑁 such that 𝑑_𝑗(𝑀)<𝜀 for all 𝑗≥𝑛₀. Let 𝑥∈𝑋 and 𝑛∈ℕ. Thus ‖𝑓_𝑛(𝑥)−𝑓(𝑥)‖_∞≥𝜀 implies 𝑛∈𝑀. Hence for each 𝑗≥𝑛₀, we have 𝑑_𝑗𝑛∈ℕ∶𝑓_𝑛(𝑥)−𝑓(𝑥)_∞≥𝜀≤𝑑_𝑗(𝑀)<𝜀 (5.3) by Definition 4.2 witnesses that 𝑓⇝_𝑛^{‖⋅,⋅‖𝑌}_ℐ𝑑𝑓.

Example 5.3. Define 𝑓:[0,1]×[0,1]→ℝ²,𝑓_𝑛∶[0,1]×[0,1]→ℝ²,𝑛∈ℕ𝑓_𝑛𝑥₁,𝑥₂=10,𝑛,if𝑥₂=1𝑛,()𝑓𝑥0,0,otherwise,¹,𝑥₂=(0,0), (5.4) Then 𝑓⇝_𝑛^{‖⋅,⋅‖𝑌}_ℐ𝑑𝑓 but ¬𝑓−−−−→_𝑛^{‖⋅,⋅‖𝑌}_{ℐ𝑑uniform}𝑓. 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 𝑑_𝑗𝑛∈ℕ∶𝑓_𝑛(𝑥)−𝑓(𝑥)_∞≤1>𝜀𝑗≤1𝑘≤𝜀. (5.5) Hence 𝑓⇝_𝑛^{‖⋅,⋅‖𝑌}_ℐ𝑑𝑓.
Suppose that 𝑓−−−−→_𝑛^{‖⋅,⋅‖𝑌}_{ℐ𝑑uniform}𝑓. Thus there is the set 𝑀∈ℐ_𝑑 such that for all 𝑛∈ℐ_𝑑⧵𝑀 and 𝑥∈[0,1]×[0,1] we have ‖𝑓_𝑛(𝑥)−𝑓(𝑥)‖_∞<1.
Choose 𝑘∈ℐ_𝑑⧵𝑀. Then 𝑓_𝑘 must be the zero function, a contradiction.

Theorem 5.4. Assume 𝑓∶𝑋→𝑌 and 𝑓_𝑛∶𝑋→𝑌 for 𝑛∈ℕ fix 𝑥₀∈𝑋. If 𝑓⇝_𝑛^{‖⋅,⋅‖𝑌}_ℐ𝑑𝑓 and all functions 𝑓_𝑛,𝑛∈ℕ, are sequentially continuous at 𝑥₀ then f is sequentially continuous at 𝑥₀.

Proof. Let 𝜀>0. Since 𝑓⇝_𝑛^{‖⋅,⋅‖𝑌}_ℐ𝑑𝑓, we can find 𝑛₀∈ℕ such that 𝑑_𝑛0𝑛∈ℕ∶𝑓_𝑛(𝑥)−𝑓(𝑥)_∞≥𝜀3<12∀𝑥∈𝑋. (5.6) Put 𝐸(𝑥)={𝑛≤𝐾∶‖𝑓_𝑛(𝑥)−𝑓(𝑥)‖_∞<𝜀/3},𝑥∈𝑋. In other word 𝑑_𝑛0 is a probability measure on 𝒫(ℕ) with the support {1,…,𝑛₀}, it follows that 𝑑_𝑛0(𝐸(𝑥))>1/2 for all 𝑥∈𝑋. By the sequentially continuity of 𝑓₁,…,𝑓_𝑛0 at 𝑥₀, there exist 𝛿>0 such that ‖𝑓_𝑖(𝑥)−𝑓_𝑖(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 𝑑_𝑛0(𝐸(𝑥))>1/2 and 𝑑_𝑛0(𝐸(𝑥₀))>1/2, there exists 𝑝∈𝐸(𝑥)⋂𝐸(𝑥₀) such that𝑓(𝑥)−𝑓𝑥_0∞≤𝑓(𝑥)−𝑓_𝑝(𝑥)_∞+𝑓_𝑝(𝑥)−𝑓_𝑝𝑥_0∞+𝑓_𝑝𝑥₀−𝑓𝑥_0∞≤𝜀3+𝜀3+𝜀3=𝜖. (5.7)
Thus we show that ‖𝑓(𝑥)−𝑓(𝑥₀)‖_∞<𝜀 for all 𝑥∈𝑋,‖𝑥−𝑥₀,𝑧‖<𝛿 for each 𝑧∈𝑋.