Statistical Convergence of Double Sequences in Locally Solid Riesz Spaces

Abstract

Recently, the notion of statistical convergence is studied in a locally solid Riesz space by Albayrak and Pehlivan (2012). In this paper, we define and study statistical 𝜏-convergence, statistical 𝜏-Cauchy and 𝑆^∗(𝜏)-convergence of double sequences in a locally solid Riesz space.

1. Introduction and Preliminaries

The notion of statistical convergence was introduced by Fast [1] and Steinhaus [2] independently in the same year 1951. Actually, Henry Fast had heard about this concept from Steinhaus, but in fact it was Antoni Zygmund who proved theorems on the statistical convergence of Fourier series in the first edition of his book ([3], pp. 181–188) where he used the term “almost convergence” in place of statistical convergence and at that time this idea was not recognized much. Since the term “almost convergence” was already in use (see Lorentz [4]), Fast had to choose a different name for his concept and “statistical convergence” was mostly the suitable one. Active researches on this topic started after the papers of Fridy [5] and since then a huge amount of literature has appeared. It developed so rapidly like an explosion when the summability theory was at the dying stage. Various extensions, generalizations, variants, and applications have been given by several authors so far, for example [6–16]. This notion has also been defined and studied in different setups, for example in a locally convex space [17]; in topological groups [18, 19]; in probabilistic normed spaces [20]; in intuitionistic fuzzy normed spaces [21]; in fuzzy/random 2-normed space [22, 23]. Recently, Albayrak and Pehlivan [24] studied this notion in locally solid Riesz spaces. In this paper, we study statistically convergent, statistically bounded, and statistically Cauchy double sequences in locally solid Riesz spaces.

Let 𝑋 be a real vector space and ≤ a partial order on this space. Then, 𝑋 is said to be an ordered vector space if it satisfies the following properties:(i)if 𝑥,𝑦∈𝑋 and 𝑦≤𝑥, then 𝑦+𝑧≤𝑥+𝑧 for each 𝑧∈𝑋.(ii)if 𝑥,𝑦∈𝑋 and 𝑦≤𝑥, then 𝜆𝑦≤𝜆𝑥 for each 𝜆≥0.

If in addition 𝑋 is a lattice with respect to the partially order ≤, then 𝑋 is said to be a Riesz space (or a vector lattice) [25].

For an element 𝑥 of a Riesz space 𝑋, the positive part of 𝑥 is defined by 𝑥⁺=𝑥∨𝜃=sup{𝑥,𝜃}, the negative part of 𝑥 by 𝑥⁻=(−𝑥)∨𝜃 and the absolute value of 𝑥 by |𝑥|=𝑥∨(−𝑥), where 𝜃 is the zero element of 𝑋.

A subset 𝑆 of a Riesz space 𝑋 is said to be solid if 𝑦∈𝑆 and |𝑥|≤|𝑦| implies 𝑥∈𝑆.

A topological vector space (𝑋,𝜏) is a vector space 𝑋 which has a (linear) topology 𝜏 such that the algebraic operations of addition and scalar multiplication in 𝑋 are continuous. Continuity of addition means that the function 𝑓∶𝑋×𝑋→𝑋 defined by 𝑓(𝑥,𝑦)=𝑥+𝑦 is continuous on 𝑋×𝑋, and continuity of scalar multiplication means that the function 𝑓∶ℂ×𝑋→𝑋 defined by 𝑓(𝜆,𝑥)=𝜆𝑥 is continuous on ℂ×𝑋.

Every linear topology 𝜏 on a vector space 𝑋 has a base 𝒩 for the neighborhoods of 𝜃 satisfying the following properties:(C₁) each 𝑌∈𝒩 is a balanced set, that is, 𝜆𝑥∈𝑌 holds for all 𝑥∈𝑌 and every 𝜆∈ℝ with |𝜆|≤1.(C₂) each 𝑌∈𝒩 is an absorbing set, that is, for every 𝑥∈𝑋, there exists 𝜆>0 such that 𝜆𝑥∈𝑌.(C₃) for each 𝑌∈𝒩 there exists some 𝐸∈𝒩 with 𝐸+𝐸⊆𝑌.

A linear topology 𝜏 on a Riesz space 𝑋 is said to be locally solid [26] if 𝜏 has a base at zero consisting of solid sets. A locally solid Riesz space (𝑋,𝜏) is a Riesz space equipped with a locally solid topology 𝜏.

By the convergence of a double sequence we mean the convergence in the Pringsheim's sense [27]. A double sequence 𝑥=(𝑥_𝑗𝑘) is said to converge to the limit 𝐿 in Pringsheim's sense (shortly, 𝑃-convergent to 𝐿) if for every 𝜀>0 there exists an integer 𝑁 such that |𝑥_𝑗𝑘−𝐿|<𝜀 whenever 𝑗,𝑘>𝑁. In this case 𝐿 is called the 𝑃-limit of 𝑥.

A double sequence 𝑥=(𝑥_𝑗𝑘) of real or complex numbers is said to be bounded if ‖𝑥‖_∞=sup_𝑗,𝑘|𝑥_𝑗𝑘|<∞. The space of all bounded double sequences is denoted by ℳ_𝑢.

2. Statistical 𝜏-Convergence

Let 𝐾⊆ℕ×ℕ be a two-dimensional set of positive integers and let 𝐾(𝑚,𝑛)={(𝑗,𝑘)∶𝑗≤𝑚,𝑘≤𝑛}. Then the two-dimensional analogue of natural density can be defined as follows.

In case the sequence (𝐾(𝑚,𝑛)/𝑚𝑛) has a limit in Pringsheim's sense, then we say that 𝐾 has a double natural density and is defined as 𝑃-lim_𝑚,𝑛𝐾(𝑚,𝑛)𝑚𝑛=𝛿₂(𝐾). (2.1)

For example, let 𝐾={(𝑖²,𝑗²)∶𝑖,𝑗∈ℕ}. Then, 𝛿₂(𝐾)=𝑃-lim_𝑚,𝑛𝐾(𝑚,𝑛)𝑚𝑛≤𝑃-lim_𝑚,𝑛√𝑚√𝑛𝑚𝑛=0, (2.2) That is, the set 𝐾 has double natural density zero, while the set {(𝑖,2𝑗)∶𝑖,𝑗∈ℕ} has double natural density 1/2.

A real double sequence 𝑥=(𝑥_𝑗𝑘) is said to be statistically convergent (see [28–30]) to the number ℓ if for each 𝜖>0, the set 𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝑥_𝑗𝑘−ℓ≥𝜖 (2.3) has double natural density zero.

We shall assume throughout this paper that the symbol 𝒩_sol will denote any base at zero consisting of solid sets and satisfying the conditions (C₁), (C₂), and (C₃) in a locally solid topology.

Definition 2.1. Let (𝑋,𝜏) be a locally solid Riesz space. Then, a double sequence 𝑥=(𝑥_𝑗𝑘) in 𝑋 is said to be statistically 𝜏-convergent to the number 𝜉∈𝑋 if for every 𝜏-neighborhood 𝑈 of zero: 𝑃-lim_{𝑚,𝑛→∞}1𝑚𝑛𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝑥_𝑗𝑘−𝜉∉𝑈=0. (2.4) In this case, we write 𝒮(𝜏)-lim𝑥=𝜉 or 𝑥−−−−→_𝑗𝑘^𝒮(𝜏)𝜉.

Definition 2.2. Let (𝑋,𝜏) be a locally solid Riesz space. We say that a double sequence 𝑥=(𝑥_𝑗𝑘) in 𝑋 is statistically 𝜏-bounded if for every 𝜏-neighborhood 𝑈 of zero there exists some 𝜆>0, such that the set 𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝜆𝑥_𝑗𝑘∉𝑈 (2.5) has double natural density zero.

Theorem 2.3. Let (𝑋,𝜏) be a Hausdorff locally solid Riesz space and 𝑥=(𝑥_𝑗𝑘) and 𝑦=(𝑦_𝑗𝑘) be two double sequences in 𝑋. Then, the following hold:(i)if 𝒮(𝜏)-lim_𝑗,𝑘𝑥_𝑗𝑘=𝜉₁ and 𝒮(𝜏)-lim_𝑗,𝑘𝑥_𝑗𝑘=𝜉₂, then 𝜉₁=𝜉₂.(ii)if 𝒮(𝜏)-lim_𝑗,𝑘𝑥_𝑗𝑘=𝜉, then 𝒮(𝜏)-lim_𝑗,𝑘𝛼𝑥_𝑗𝑘=𝛼𝜉, 𝛼∈ℝ.(iii)if 𝒮(𝜏)-lim_𝑗,𝑘𝑥_𝑗𝑘=𝜉 and 𝒮(𝜏)-lim_𝑗,𝑘𝑦_𝑗𝑘=𝜂, then 𝒮(𝜏)-lim_𝑗,𝑘(𝑥_𝑗𝑘+𝑦_𝑗𝑘)=𝜉+𝜂.

Proof. (i) Suppose that 𝒮(𝜏)-lim_𝑗,𝑘𝑥_𝑗𝑘=𝜉₁ and 𝒮(𝜏)-lim_𝑗,𝑘𝑥_𝑗𝑘=𝜉₂. Let 𝑈 be any 𝜏-neighborhood of zero. Then, there exists 𝑌∈𝒩_sol such that 𝑌⊆𝑈. Choose any 𝐸∈𝒩_sol such that 𝐸+𝐸⊆𝑌. We define the following sets: 𝐾₁=𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝑥_𝑗𝑘−𝜉₁,𝐾∈𝐸₂=𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝑥_𝑗𝑘−𝜉₂∈𝐸. (2.6) Since 𝒮(𝜏)-lim_𝑗,𝑘𝑥_𝑗𝑘=𝜉₁ and 𝒮(𝜏)-lim_𝑗,𝑘𝑥_𝑗𝑘=𝜉₂, we have 𝛿₂(𝐾₁)=𝛿₂(𝐾₂)=1. Thus, 𝛿₂(𝐾₁∩𝐾₂)=1, and in particular 𝐾₁∩𝐾₂≠∅. Now, let (𝑗,𝑘)∈𝐾₁∩𝐾₂. Then 𝜉₁−𝜉₂=𝜉₁−𝑥_𝑗𝑘+𝑥_𝑗𝑘−𝜉₂∈𝐸+𝐸⊆𝑌⊆𝑈. (2.7) Hence for every 𝜏-neighborhood 𝑈 of zero, we have 𝜉₁−𝜉₂∈𝑈. Since (𝑋,𝜏) is Hausdorff, the intersection of all 𝜏-neighborhoods 𝑈 of zero is the singleton set {𝜃}. Thus, we get 𝜉₁−𝜉₂=𝜃, that is, 𝜉₁=𝜉₂.
(ii) Let 𝑈 be an arbitrary 𝜏-neighborhood of zero and 𝒮(𝜏)-lim_𝑗,𝑘𝑥_𝑗𝑘=𝜉. Then there exists 𝑌∈𝒩_sol such that 𝑌⊆𝑈 and also 𝑃-lim_{𝑚,𝑛→∞}1𝑚𝑛𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝑥_𝑗𝑘−𝜉∈𝑌=1. (2.8) Since 𝑌 is balanced, 𝑥_𝑗𝑘−𝜉∈𝑌 implies that 𝛼(𝑥_𝑗𝑘−𝜉)∈𝑌 for every 𝛼∈ℝ with |𝛼|≤1. Hence, 𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝑥_𝑗𝑘−𝜉∈𝑌⊆𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝛼𝑥_𝑗𝑘⊆−𝛼𝜉∈𝑌𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝛼𝑥_𝑗𝑘−𝛼𝜉∈𝑈. (2.9) Thus, we obtain 𝑃-lim_{𝑚,𝑛→∞}1𝑚𝑛𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝛼𝑥_𝑗𝑘−𝛼𝜉∈𝑈=1, (2.10) for each 𝜏-neighborhood 𝑈 of zero. Now let |𝛼|>1 and [|𝛼|] be the smallest integer greater than or equal to |𝛼|. There exists 𝐸∈𝒩_sol such that [|𝛼|]𝐸⊆𝑌. Since 𝒮(𝜏)-lim_𝑗,𝑘𝑥_𝑗𝑘=𝜉, the set 𝐾=𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝑥_𝑗𝑘−𝜉∈𝐸 (2.11) has double natural density zero. Therefore, 𝛼𝜉−𝛼𝑥_𝑗𝑘=|𝛼|𝜉−𝑥_𝑗𝑘≤[|𝛼|]𝜉−𝑥_𝑗𝑘∈[|𝛼|]𝐸⊆𝑌⊆𝑈. (2.12) Since the set 𝑌 is solid, we have 𝛼𝜉−𝛼𝑥_𝑗𝑘∈𝑌. This implies that 𝛼𝜉−𝛼𝑥_𝑗𝑘∈𝑈. Thus, 𝑃-lim_{𝑚,𝑛→∞}1𝑚𝑛𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝛼𝑥_𝑗𝑘−𝛼𝜉∈𝑈=1, (2.13) for each 𝜏-neighborhood 𝑈 of zero. Hence, 𝒮(𝜏)-lim_𝑗,𝑘𝛼𝑥_𝑗𝑘=𝛼𝜉.
(iii) Let 𝑈 be an arbitrary 𝜏-neighborhood of zero. Then there exists 𝑌∈𝒩_sol such that 𝑌⊆𝑈. Choose 𝐸 in 𝒩_sol such that 𝐸+𝐸⊆𝑌. Since 𝒮(𝜏)-lim_𝑗,𝑘𝑥_𝑗𝑘=𝜉 and 𝒮(𝜏)-lim_𝑗,𝑘𝑦_𝑗𝑘=𝜂, we have 𝛿₂(𝐻₁)=1=𝛿₂(𝐻₂), where 𝐻₁=𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝑥_𝑗𝑘,𝐻−𝜉∈𝐸₂=𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝑦_𝑗𝑘−𝜂∈𝐸. (2.14) Let 𝐻=𝐻₁∩𝐻₂. Hence, we have 𝛿₂(𝐻)=1 and 𝑥_𝑗𝑘+𝑦_𝑗𝑘−𝜉+𝑦=𝑥_𝑗𝑘−𝑥+𝑦_𝑗𝑘−𝜂∈𝐸+𝐸⊆𝑌⊆𝑈. (2.15) Therefore, 𝑃-lim_{𝑚,𝑛→∞}1𝑥𝑚𝑛𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶_𝑗𝑘+𝑦_𝑗𝑘−𝜉+𝜂∈𝑈=1. (2.16) Since 𝑈 is arbitrary, we have 𝒮(𝜏)-lim_𝑗,𝑘(𝑥_𝑗𝑘+𝑦_𝑗𝑘)=𝜉+𝜂.
This completes the proof of the theorem.

Theorem 2.4. Let (𝑋,𝜏) be a locally solid Riesz space. If a double sequence 𝑥=(𝑥_𝑗𝑘) is statistically 𝜏-convergent, then it is statistically 𝜏-bounded.

Proof. Suppose 𝑥=(𝑥_𝑗𝑘) is statistically 𝜏-convergent to the point 𝜉∈𝑋 and let 𝑈 be an arbitrary 𝜏-neighborhood of zero. Then, there exists 𝑌∈𝒩_sol such that 𝑌⊆𝑈. Let us choose 𝐸∈𝒩_sol such that 𝐸+𝐸⊆𝑌. Since 𝒮(𝜏)-lim_{𝑗,𝑘→∞}𝑥_𝑗𝑘=𝜉, the set 𝐾=𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝑥_𝑗𝑘−𝜉∉𝐸 (2.17) has double natural density zero. Since 𝐸 is absorbing, there exists 𝜆>0 such that 𝜆𝜉∈𝐸. Let 𝛼 be such that 𝛼≤1 and 𝛼≤𝜆. Since 𝐸 is solid and |𝛼𝜉|≤|𝜆𝜉|, we have 𝛼𝜉∈𝐸. Since 𝐸 is balanced, 𝑥_𝑗𝑘−𝜉∈𝐸 implies that 𝛼(𝑥_𝑗𝑘−𝜉)∈𝐸. Then, we have 𝛼𝑥_𝑗𝑘=𝛼𝑥_𝑗𝑘−𝜉+𝛼𝜉∈𝐸+𝐸⊆𝑌⊆𝑈, (2.18) for each (𝑗,𝑘)∈ℕ×ℕ⧵𝐾. Thus 𝑃-lim_{𝑚,𝑛→∞}1𝑚𝑛𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝛼𝑥_𝑗𝑘∉𝑈=0. (2.19) Hence, (𝑥_𝑗𝑘) is statistically 𝜏-bounded.
This completes the proof of the theorem.

Theorem 2.5. Let (𝑋,𝜏) be a locally solid Riesz space. If (𝑥_𝑗𝑘), (𝑦_𝑗𝑘), and (𝑧_𝑗𝑘) are three double sequences such that(i)𝑥_𝑗𝑘≤𝑦_𝑗𝑘≤𝑧_𝑗𝑘 for all 𝑗,𝑘∈ℕ;(ii)𝒮(𝜏)-lim_𝑗,𝑘𝑥_𝑗𝑘=𝜉=𝒮(𝜏)-lim_𝑗,𝑘𝑧_𝑗𝑘;
then 𝒮(𝜏)-lim_𝑗,𝑘𝑦_𝑗𝑘=𝜉.

Proof. Let 𝑈 be an arbitrary 𝜏-neighborhood of zero, there exists 𝑌∈𝒩_sol such that 𝑌⊆𝑈. Choose 𝐸∈𝒩_sol such that 𝐸+𝐸⊆𝑌. From condition (ii), we have 𝛿₂(𝐴)=1=𝛿₂(𝐵), where 𝐴=𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝑥_𝑗𝑘−𝜉∈𝐸𝐵=𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝑧_𝑗𝑘−𝜉∈𝐸. (2.20) Now, let 𝑄=𝐴∩𝐵. Then, from (i), we have 𝑥_𝑗𝑘−𝜉≤𝑦_𝑗𝑘−𝜉≤𝑧_𝑗𝑘−𝜉, (2.21) this implies 𝑦_𝑗𝑘−𝜉≤𝑥_𝑗𝑘−𝜉+𝑧_𝑗𝑘−𝜉∈𝐸+𝐸⊆𝑌 (2.22) for each (𝑗,𝑘)∈𝑄. Since 𝑌 is solid, we have 𝑦_𝑗𝑘−𝜉∈𝑌⊆𝑈. Thus, 𝑃-lim_{𝑚,𝑛→∞}1𝑚𝑛𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝑦_𝑗𝑘−𝜉∈𝑈=1, (2.23) for each 𝜏-neighborhood 𝑈 of zero. Hence 𝒮(𝜏)-lim_𝑗,𝑘𝑦_𝑗𝑘=𝜉.
This completes the proof of the theorem.

3. Statistically 𝜏-Cauchy and 𝒮^∗(𝜏)-Convergence

Definition 3.1. Let (𝑋,𝜏) be a locally solid Riesz space. A double sequence 𝑥=(𝑥_𝑗𝑘) in 𝑋 is statistically 𝜏-Cauchy if for every 𝜏-neighborhood 𝑈 of zero there exist 𝑁,𝑀∈ℕ such that for all 𝑗,𝑝≥𝑁, 𝑘,𝑞≥𝑀, the set 𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝑥_𝑗𝑘−𝑥_𝑝𝑞∉𝑈 (3.1) has double natural density zero.

Theorem 3.2. Let (𝑋,𝜏) be a locally solid Riesz space. If a double sequence 𝑥=(𝑥_𝑗𝑘) is statistically 𝜏-convergent, then it is statistically 𝜏-Cauchy.

Proof. Suppose that 𝒮(𝜏)-lim_{𝑗,𝑘→∞}𝑥_𝑗𝑘=𝜉. Let 𝑈 be an arbitrary 𝜏-neighborhood of zero, there exists 𝑌∈𝒩_sol such that 𝑌⊆𝑈. Choose 𝐸∈𝒩_sol such that 𝐸+𝐸⊆𝑌. Then, 𝑃-lim_{𝑚,𝑛→∞}1𝑚𝑛𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝑥_𝑗𝑘−𝜉∉𝐸=0. (3.2) Also, we have 𝑥_𝑗𝑘−𝑥_𝑝𝑞=𝑥_𝑗𝑘−𝜉+𝜉−𝑥_𝑝𝑞∈𝐸+𝐸⊆𝑌⊆𝑈, (3.3) for all (𝑗,𝑘),(𝑝,𝑞)∈ℕ×ℕ⧵𝐾, where 𝐾=𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝑥_𝑗𝑘−𝜉∉𝐸. (3.4) Therefore, the set 𝑗,𝑘,𝑗≤𝑚,𝑘≤𝑛∶𝑥_𝑗𝑘−𝑥_𝑝𝑞∉𝑈⊆𝐾. (3.5) For every 𝜏-neighborhood 𝑈 of zero there exist 𝑁,𝑀∈ℕ such that for all 𝑗,𝑝≥𝑁, 𝑘,𝑞≥𝑀, the set {(𝑗,𝑘),𝑗≤𝑚 and 𝑘≤𝑛∶𝑥_𝑗𝑘−𝑥_𝑝𝑞∉𝑈} has double natural density zero. Hence, (𝑥_𝑗𝑘) is statistically 𝜏-Cauchy.
This completes the proof of the theorem.

In [30], it was shown that a real double sequence 𝑥=(𝑥_𝑗𝑘) is statistically convergent to a number ℓ if and only if there exists a subset 𝐾={(𝑗,𝑘)}⊆ℕ×ℕ,𝑗,𝑘=1,2,… such that 𝛿₂(𝐾)=1 and lim_{𝑗,𝑘→∞}𝑥_𝑗𝑘=ℓ.

This fact suggests defining further another type of convergence in locally solid Riesz spaces.

Definition 3.3. A sequence (𝑥_𝑗𝑘) in a locally solid Riesz space (𝑋,𝜏) is said to 𝒮^∗(𝜏)-convergent to 𝜉∈𝑋 if there exists a set 𝐾={(𝑗,𝑘)}⊆ℕ×ℕ,𝑗,𝑘=1,2,…, with 𝛿₂(𝐾)=1 such that lim_{𝑗,𝑘→∞}𝑥_𝑗𝑘=𝜉. In this case, we write 𝜉=𝒮^∗(𝜏)-lim𝑥.

Theorem 3.4. A double sequence 𝑥=(𝑥_𝑗𝑘) is statistically 𝜏-convergent to a number 𝜉 if it is 𝒮^∗(𝜏)-convergent to 𝜉 in a locally solid Riesz space (𝑋,𝜏).

Proof. Let 𝑈 be an arbitrary 𝜏-neighborhood of 𝜉. Since 𝑥=(𝑥_𝑗𝑘) is 𝒮^∗(𝜏)-convergent to 𝜉, there is a set 𝐾={(𝑗,𝑘)}⊆ℕ×ℕ,𝑗,𝑘=1,2,…, with 𝛿₂(𝐾)=1 and 𝑗₀=𝑗₀(𝑈),  𝑘₀=𝑘₀(𝑈) such that 𝑗≥𝑗₀,  𝑘≥𝑘₀, and (𝑗,𝑘)∈𝐾 imply that 𝑥_𝑗𝑘−𝜉∈𝑈. Then, 𝐾_𝑈=𝑗,𝑘∈ℕ×ℕ∶𝑥_𝑗𝑘−𝜉∉𝑈⊆ℕ×ℕ−𝑗_𝑁+1,𝑘_𝑁+1,𝑗_𝑁+2,𝑘_𝑁+2,…. (3.6) Therefore, 𝛿₂𝐾_𝑈≤1−1=0. (3.7) Hence, 𝑥 is statistically 𝜏-convergent to 𝜉.
This completes the proof of the theorem.

Note that the converse holds for first countable space.

Recall that a first countable space is a topological space satisfying the “first axiom of countability.” Specifically, a space 𝑋 is said to be first countable if each point has a countable neighbourhood basis (local base), that is, for each point 𝑥 in 𝑋 there exists a sequence 𝑈₁,𝑈₂,… of open neighbourhoods of 𝑥 such that for any open neighbourhood 𝑉 of 𝑥 there exists an integer 𝑖 with 𝑈_𝑖 contained in 𝑉.

Theorem 3.5. Let (𝑋,𝜏) be a first countable locally solid Riesz space. If a double sequence 𝑥=(𝑥_𝑗𝑘) is statistically 𝜏-convergent to a number 𝜉, then it is 𝒮^∗(𝜏)-convergent to 𝜉.

Proof. Let 𝑥 be statistically 𝜏-convergent to a number 𝜉. Fix a countable local base 𝑈₁⊃𝑈₂⊃𝑈₃⊃⋯ at 𝜉. For every 𝑖∈ℕ, put 𝐾_𝑖=𝑗,𝑘∈ℕ×ℕ∶𝑥_𝑗𝑘−𝜉∉𝑈_𝑖,𝑀_𝑖=𝑗,𝑘∈ℕ×ℕ∶𝑥_𝑗𝑘−𝜉∈𝑈_𝑖(𝑟=1,2,3,…). (3.8) Then, 𝛿₂(𝐾_𝑖)=0 and(1)𝑀₁⊃𝑀₂⊃⋯⊃𝑀_𝑖⊃𝑀_𝑖+1⊃⋯; (2)𝛿₂(𝑀_𝑖)=1,𝑖=1,2,3,….
Now we have to show that for (𝑗,𝑘)∈𝑀_𝑖,(𝑥_𝑗𝑘) is convergent to 𝜉. Suppose that (𝑥_𝑗𝑘) is not convergent to 𝜉. Therefore, 𝑥_𝑗𝑘∉𝑈_𝑖 for infinitely many terms. Let 𝑀_𝑟=𝑗,𝑘∈ℕ×ℕ∶𝑥_𝑗𝑘−𝜉∈𝑈_𝑟(𝑟>𝑖). (3.9) Then,(3)𝛿₂(𝑀_𝑟)=0, and by (1), 𝑀_𝑖⊂𝑀_𝑟. Hence, 𝛿₂(𝑀_𝑖)=0 which contradicts (2). Therefore, (𝑥_𝑗𝑘) is convergent to 𝜉. Hence by Definition 3.3, 𝑥 is 𝒮^∗(𝜏)-convergent to 𝜉.
This completes the proof of the theorem.