Some Results on Interval-Valued Riesz-Summable Series

Abstract

In this study, the concept of Riesz summability for interval-valued series is introduced, and the corresponding interval-valued Riesz-summable series space $R_I^{cs}$ is constructed. In contrast to the sequence setting, Riesz summability of an interval-valued series is defined by applying the Riesz transformation to its sequence of partial sums. A metric based on the Hausdorff metric is defined on $R_I^{cs}$, and the resulting metric space is proved to be complete. It is also shown that every convergent interval-valued series is Riesz-summable to the same interval and that the inclusion of the convergent series space in $R_I^{cs}$ is proper. Furthermore, $R_I^{cs}$ is proved to be a quasilinear space and, when equipped with an appropriate norm, a normed quasilinear space. The metric associated with this norm is shown to coincide with the previously defined metric. In addition, the solid-floored property of $R_I^{cs}$ is established. These results provide a systematic theoretical framework for the metric, algebraic, topological, and order-related analysis of interval-valued Riesz-summable series and contribute to the further development of summability theory in interval-valued settings.

1. Introduction

Real-world problems frequently involve data that cannot be represented by precise numerical values. Owing to measurement errors, experimental uncertainties, approximation processes, and numerical rounding, it is often more appropriate to represent such data by intervals rather than by single real numbers. Motivated by this need, interval analysis has become an important research area with numerous applications in mathematics, engineering, optimization, control theory, and other branches of applied sciences. The foundations of interval arithmetic were first established by Dwyer [1]. Subsequently, Moore and Yang developed the theoretical framework of interval arithmetic and extended its applications to numerical analysis [2,3]. Moore’s monograph Interval Analysis became one of the fundamental references in the field [4], while the comprehensive work by Moore, Kearfott, and Cloud further established the theory by systematically presenting both its theoretical foundations and practical applications [5]. Furthermore, Chiao investigated the fundamental properties of the maximum norm for interval-valued sequences, thereby making a significant contribution to the metric structure of interval spaces [6].

Alongside the developments in interval analysis, the theory of quasilinear spaces has also emerged as an important area of research. The concept of a quasilinear space was first introduced by Aseev for the investigation of multivalued mappings [7]. Subsequently, the topological structures of quasilinear spaces were studied in detail [8], while the fundamental properties of normed quasilinear spaces were further developed, leading to important results such as Riesz’s lemma in these spaces [9,10]. In subsequent studies, orthonormal systems in inner product quasilinear spaces were investigated [11], and new inner product quasilinear space structures on interval numbers were established [12]. Furthermore, the theory of quasilinear spaces was extended to signal processing applications [13,14], the analysis of uncertain data through interval-valued functions was investigated [15], and a quasilinear algebraic approach for linear interval equation systems was proposed [16]. Consequently, the theory of quasilinear spaces has evolved not only from a theoretical perspective but also as a powerful framework for a variety of practical applications.

Parallel to the developments in interval analysis and the theory of quasilinear spaces, the study of the convergence behavior of sequences and series has remained one of the fundamental topics in mathematical analysis. Since the classical notion of convergence is insufficient to describe the behavior of certain sequences and series, summability theory emerged as a natural extension and has undergone substantial development over the years. The pioneering works of Hardy [17], Knopp [18], and Petersen [19] laid the foundations of modern summability theory. Subsequently, Boos [20], Aasma et al. [21], and Natarajan [22] presented systematic treatments of summability methods and investigated a wide range of summability techniques. In addition, Mazhar [23] and Bor [24] established several new results concerning various summability methods, while the Tauberian theorems of Hardy and Littlewood [25], the comprehensive treatment by Korevaar [26], and the results of Çanak and Totur on weighted mean methods [27] contributed significantly to a deeper understanding of the relationships among different summability methods. Among these methods, Riesz summability has attracted particular attention because its weighted mean structure extends the classical concept of convergence and provides an effective framework for the analysis of oscillatory or irregularly behaving sequences and series.

The developments in summability theory have also motivated extensive research on interval-valued sequence spaces. Şengönül and Eryılmaz introduced interval-valued sequence spaces, providing one of the first fundamental contributions in this area [28]. Subsequently, Kılınç and Yıldırım extended the Cesàro summability method to interval-valued sequences and investigated the corresponding interval-valued Cesàro convergent sequence space [29]. Nuray, Ulusu, and Dündar examined several structural properties of two-dimensional interval numbers [30], whereas Erdoğan and Osman introduced the interval-valued Abel convergent sequence space and established its fundamental properties [31]. More recently, Tuncer and Akgün introduced the interval-valued Riesz convergent sequence space and systematically investigated its algebraic, metric, topological, and normed quasilinear structures, together with its completeness and other fundamental properties [32]. This study constitutes an important step toward extending Riesz summability theory to interval-valued structures and provides a solid theoretical foundation for subsequent research.

Despite these developments, the existing literature has been devoted exclusively to interval-valued sequences, whereas the theory of Riesz summability for interval-valued series and the corresponding interval-valued Riesz-summable series space have not yet been investigated. However, the transition from the sequence setting to the series setting cannot be regarded as a straightforward or merely formal extension. In the sequence setting considered in [32], the Riesz transformation is applied directly to the terms of an interval-valued sequence. In contrast, the Riesz summability of an interval-valued series is determined by applying the Riesz transformation to its sequence of partial sums. Therefore, the behavior of the series depends not only on its individual terms but also on their cumulative behavior through the partial sums. Consequently, the corresponding Riesz-summable series space must be constructed independently, and its algebraic, metric, topological, completeness, normed quasilinear, and order-related properties require separate arguments and proofs. To the best of our knowledge, this study introduces, for the first time, the concept of Riesz summability for interval-valued series and establishes the corresponding interval-valued Riesz-summable series space. Furthermore, the algebraic, metric, and topological structures of this space are systematically investigated, while its completeness, normed quasilinear, and solid-floored properties are studied in detail. Consequently, the present work extends Riesz summability theory from interval-valued sequences to interval-valued series, fills an important gap in the existing literature, and provides a new and systematic theoretical framework for the structural analysis of interval-valued series.

2. Preliminaries

Definition 1 

([6]). An interval L is defined as $L=\left[\underline{L},\overline{L}\right]=\left\{x\in \mathbb{R}:\underline{L}\le x\le \overline{L}\right\}$ where $\underline{L}$ is the lower bound, $\overline{L}$ is the upper bound.

Definition 2 

([5]). An interval that coincides with a single real number can be written as $\left[a,a\right]$ for some $a\in \mathbb{R}$. Such an interval is called a degenerate (or derived) interval. This type of interval represents an ordinary real number, i.e., $a=\left[a,a\right]$.

Given two intervals $L=\left[\underline{L},\overline{L}\right]$ and $G=\left[\underline{G},\overline{G}\right]$, it holds that

$$L<G\iff \overline{L}<\underline{G.}$$

Furthermore

$$L\subseteq G\iff (\underline{G}\le \underline{L},\overline{L}\le \overline{G)}.$$

Also, arithmetic operations on intervals are defined by the endpoint formulas below [5].

$$L+G=\left[\underline{L}+\underline{G},\overline{L}+\overline{G}\right],L-G=\left[\underline{L}-\overline{G},\overline{L}-\underline{G}\right],$$

$$S=\left\{\underline{L}\underline{G},\underline{L}\overline{G},\overline{L}\underline{G},\overline{L}\overline{G}\right\}\mathit{then}L.G=\left[\min S,\max S\right],$$

$${\displaystyle \frac{L}{G}}=L·{\displaystyle \frac{1}{G}}=\left[\underline{L},\overline{L}\right]·\left[{\displaystyle \frac{1}{\overline{G}}},{\displaystyle \frac{1}{\underline{G}}}\right];0\notin G,$$

$$\lambda L=\lambda [\underline{L},\overline{L}]=\left\{\begin{array}{cc}[\lambda \underline{L},\lambda \overline{L}],\hfill & \lambda \ge 0,\hfill \\ [\lambda \overline{L},\lambda \underline{L}],\hfill & \lambda <0.\hfill \end{array}\right.$$

Definition 3 

([5]). Let $I=\left\{J\subset \mathbb{R}:J=\left[\underline{J},\overline{J}\right],\underline{J},\overline{J}\in \mathbb{R},\underline{J}\le \overline{J}\right\}$. An interval valued sequence is a function $f:\mathbb{N}\to I$, where the domain is the set of natural numbers $\mathbb{N}$, and the codomain is the set of all real intervals I.

The class of all interval sequence spaces is denoted by $\omega \left(I\right)=\left\{\left(J_n\right):n\in \mathbb{N},J_n\in I\right\}$.

Definition 4 

([5]). The set of all real valued interval numbers I is a metric space with the metric h called the Hausdorff metric, where h is defined as

$$h(J,K)=max\left\{\left|\underline{J}-\underline{K}\right|,\left|\overline{J}-\overline{K}\right|\right\}.$$

(1)

Definition 5 

([6]). Let $\left(J_n\right)$ be an interval-valued sequence and let $J_0=\left[\underline{J_0},\overline{J_0}\right]\in I$. The sequence $\left(J_n\right)$ is said to converge to $J_0$ with respect to the Hausdorff metric if, for every $\epsilon >0$, there exists $N=N\left(\epsilon \right)\in \mathbb{N}$ such that $h\left(J_n,J_0\right)<\epsilon$ for all $n\ge N$. This is denoted by $\underset{n\to \infty }{\lim }{J}_n=J_0$ or, equivalently, by $J_n\to J_0\left(n\to \infty \right)$.

Example 1. 

Let $J_n=\left[2+{\displaystyle \frac{{\left(-1\right)}^n}{n}},5+{\displaystyle \frac{1}{n^2}}\right].$ It can be easily seen that $\underset{n\to \infty }{\lim }{J}_n=\left[2,5\right]$.

Example 2. 

Let $J_n=\left[0,n\right]$. Since the upper endpoint tends to infinity, the sequence does not converge in I.

In [28], some sequence spaces are defined as follows.

$$\begin{array}{c}\hfill I_C=\left\{\left(J_n\right)\in \omega \left(I\right):\underset{n\to \infty }{\lim }{J}_n=J_0,J_0\in I\right\},\end{array}$$

Definition 6 

([7]). A set X is called a quasilinear space (QLS, for short), if a partial ordering relation “≼” algebraic sum operation and an operation of multiplication by real numbers are in it in such a way that the following conditions hold for any elements $x,y,z,t\in X$ and any real scalars $\lambda ,\phi$:

$$x\preccurlyeq x,$$

$$x\preccurlyeq y\mathit{if}(x\preccurlyeq z\mathit{and}z\preccurlyeq y),$$

$$x=z\mathit{if}(x\preccurlyeq z\mathit{and}z\preccurlyeq x),$$

$$x+z=z+x,$$

$$x+(z+y)=(x+z)+y,$$

$$\mathit{there}\mathit{exists}\mathbf{0}\in X\mathit{such}\mathit{that}x+\mathbf{0}=x,$$

$$\lambda ·(\phi ·x)=(\lambda ·\phi )·x,$$

$$\lambda ·(x+z)=\lambda ·x+\lambda ·z,$$

$$1·x=x,$$

$$0·x=\mathbf{0},$$

$$(\lambda +\phi )·x\preccurlyeq \lambda ·x+\phi ·x,$$

$$x+y\preccurlyeq z+t\mathit{if}x\preccurlyeq z\mathit{and}y\preccurlyeq t,$$

$$\lambda ·x\preccurlyeq \lambda ·z\mathit{whenever}x\preccurlyeq z.$$

A linear space is a QLS with the partial ordering relation “$x\preccurlyeq y$ if and only if $x=y$”.

Let X be a quasilinear space and suppose that $Y\subseteq X$. Y is a subspace of X if and only if Y itself is a quasilinear space with the same partial ordering as X. In other words, Y is a subspace of the quasilinear space X if and only if for all $x,y\in Y$ and all $\lambda ,\phi \in \mathbb{R}$, one has $\lambda x+\phi y\in Y$ (see [15]).

Definition 7 

([29]). The space $\omega \left(I\right)$ forms a quasilinear space under the operations defined below:

For any $J=\left(J_k\right)$ and $G=\left(G_k\right)$ in $\omega \left(I\right)$, where

$$\begin{array}{c}\hfill J_k=[\underline{J_k},\overline{J_k}],G_k=[\underline{G_k},\overline{G_k}],k\in \mathbb{N}\end{array}$$

Algebraic operations are defined as follows:

$$\begin{array}{c}\hfill J+G=\left([\underline{J_k}+\underline{G_k},\overline{J_k}+\overline{G_k}]\right)\mathit{and}\gamma J=\left(\gamma [\underline{J_k},\overline{J_k}]\right),\end{array}$$

where

$$\begin{array}{c}\hfill \gamma [\underline{J_k},\overline{J_k}]=\left\{\begin{array}{cc}[\gamma \underline{J_k},\gamma \overline{J_k}],\hfill & \gamma \ge 0,\hfill \\ [\gamma \overline{J_k},\gamma \underline{J_k}],\hfill & \gamma <0.\hfill \end{array}\right.\end{array}$$

Moreover, the partial order on $\omega \left(I\right)$ is defined by

$$\begin{array}{c}\hfill J⪯G\mathit{if} \mathit{and} \mathit{only} \mathit{if}[\underline{J_k},\overline{J_k}]\subseteq [\underline{G_k},\overline{G_k}],\mathit{for} \mathit{all}k\in \mathbb{N}.\end{array}$$

Definition 8 

([4]). Let X be a quasilinear space. An element $x^{\prime }\in X$ is called an inverse of $x\in X$ if $x+x^{\prime }=0$. If such an element exists, then it is unique and x is called a regular element of X; otherwise, is called a singular element. If X is also a linear space under the same operations, then every element of X has an additive inverse. Therefore, X contains no singular elements. More generally, an element $x\in X$ is regular if and only if $x+\left(-x\right)=0$, where $-x=\left(-1\right)x$. In this case, the inverse of x is $x^{\prime }=-x$.

Definition 9 

([11]). Let X be a quasilinear space. X is called a solid floored quasilinear space whenever

$$y=\underset{\preccurlyeq }{\sup }\left\{x\in X_r:x\preccurlyeq y\right\}$$

for each $y\in X$. The supremum considered here is taken with respect to the partial order relation ≼ defined on the quasilinear space X. Otherwise, X is called a non-solid-floored quasilinear space.

Definition 10 

([7]). Let X be a quasilinear space. A real function $\parallel .\parallel :X\to \mathbb{R}$ is called a norm, if the following conditions are satisfied for all $a,b\in X$ and $\rho \in \mathbb{R}$:

$$\begin{array}{cc}& \parallel a\parallel >0\mathit{for}a\ne \theta ,\hfill \\ & \parallel a+b\parallel \le \parallel a\parallel +\parallel b\parallel ,\hfill \\ & \parallel \rho a\parallel =\left|\rho \right|\parallel a\parallel ,\hfill \\ & a⪯b\Rightarrow \parallel a\parallel \le \parallel b\parallel ,\hfill \\ & \left[\forall \epsilon >0,\exists a_{\epsilon }\in X:∥a_{\epsilon }∥\le \epsilon anda\preccurlyeq b+a_{\epsilon }\right]\Rightarrow a\preccurlyeq b.\hfill \end{array}$$

A quasilinear space X with a norm defined on it is called a normed quasilinear space.

Definition 11 

([17]). Let $\sum a_n$ be a given infinite series with partial sums $s_n$. Let $\left(p_n\right)$ be a sequence of positive numbers such that

$$\begin{array}{c}\hfill P_n={\displaystyle \sum _{k=0}^n}{p}_k\to \infty ,(n\to \infty ),(P_{-i}=p_{-i}=0,i\ge 1.)\end{array}$$

(2)

The sequence-to-sequence transformation $T_n={\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{s}_k$ defines the sequence $\left(T_n\right)$ of the Riesz means or simply $\left(R,p_n\right)$ means of the sequence $\left(s_n\right)$ generated by the sequence of coefficients $\left(p_n\right).$ The series $\sum a_n$ is said to be Riesz-summable $\left(R,p_n\right)$-summable, for short) to the value s if $\underset{n\to \infty }{\lim }{T}_n=s$.

Definition 12 

([17]). Let $A=\left(a_{nk}\right)$ be an infinite matrix defining the matrix transformation

$${\left(Ax\right)}_n={\displaystyle \sum _{k=0}^{\infty }}{a}_{nk}{x}_k,n=0,1,2,\dots$$

The matrix transformation A is said to be regular (or to define a regular summability method) if, for every convergent sequence $x=\left(x_k\right)$ with ${\lim }_{k\to \infty }{x}_k=s,$ it follows that

$$\underset{n\to \infty }{\lim }{\left(Ax\right)}_n=\underset{n\to \infty }{\lim }{\displaystyle \sum _{k=0}^{\infty }}{a}_{nk}{x}_k=s.$$

In other words, a regular matrix transformation is a limit-preserving linear transformation that maps every convergent sequence to a sequence converging to the same limit.

The complete characterization of regular matrix transformations is given by the Silverman–Toeplitz Theorem [17]. According to this theorem, the matrix transformation $A=\left(a_{nk}\right)$ is regular if and only if the following conditions are satisfied:

$$\underset{n}{\sup }{\displaystyle \sum _{k=0}^{\infty }}\left|a_{nk}\right|<\infty ,$$

$$\underset{n\to \infty }{\lim }{a}_{nk}=0,(k=0,1,2,\dots ),$$

and

$$\underset{n\to \infty }{\lim }{\displaystyle \sum _{k=0}^{\infty }}{a}_{nk}=1.$$

Remark 1 

([17]). The means $\left(R,p_n\right)$ is regular if and only if $P_n\to \infty$ as $n\to \infty$.

3. Main Results

Definition 13. 

Let $J=\left(J_k\right)\in \omega \left(I\right)$ and let $\left(S_n\left(J\right)\right)$ be the sequence of partial sums of the interval valued series ${\displaystyle \sum _{k=0}^{\infty }}{J}_k$. That is, let

$$S_n\left(J\right)={\displaystyle \sum _{k=0}^n}{J}_k=J_0+J_1+J_2+\dots +J_n.$$

(3)

Let $\left(p_n\right)$ be a fixed sequence of positive real numbers satisfying $P_n={\displaystyle \sum _{k=0}^n}{p}_k\to \infty \left(n\to \infty \right),$ as specified in (2) and let $R_n^{cs}\left(J\right)$ denote the Riesz mean. That is,

$$R_n^{cs}\left(J\right)=\left[\underline{R_n^{cs}}\left(J\right),\overline{R_n^{cs}}\left(J\right)\right]={\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{S}_k\left(J\right)={\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k\left[\underline{S_k}\left(J\right),\overline{S_k}\left(J\right)\right].$$

(4)

If the condition

$$\underset{n\to \infty }{\lim }{R}_n^{cs}\left(J\right)=L=\left[\underline{L},\overline{L}\right]$$

is satisfied, then the interval series ${\displaystyle \sum _{k=0}^{\infty }}{J}_k$ is said to be Riesz-summable to the interval L.

Equivalently, the series ${\displaystyle \sum _{k=0}^{\infty }}{J}_k$ is Riesz-summable to $L\in I$ if

$$\underset{n\to \infty }{\lim }h\left(R_n^{cs}\left(J\right),L\right)=\underset{n\to \infty }{\lim }h\left({\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{S}_k\left(J\right),[\underline{L},\overline{L}]\right)=0.$$

(5)

Throughout the remainder of the paper, all definitions and results concerning $R_I^{cs}$ are understood with respect to this fixed weight sequence $\left(p_n\right)$. Here, I denotes the set of all closed and bounded real intervals, while $J=\left(J_k\right)$, $K=\left(K_k\right)$, and $U=\left(U_k\right)$ denote interval-valued series. The notation $R_n^{cs}\left(J\right)$ denotes the nth Riesz mean of the sequence of partial sums of J. The superscript $cs$ originates from the classical notation for “convergent series” and is retained here to distinguish the series setting from the sequence setting and to maintain consistency with our previous notation.

Motivated by the interval-valued convergent sequence space $I_C$ introduced by Şengönül and Eryılmaz [28], we define the interval-valued convergent series space $I^{cs}$ as the natural analogue of the classical convergent series space $cs$.

Definition 14. 

Let $\omega \left(I\right)$ denote the space of all interval-valued sequences and $I_C$ the space of interval-valued convergent sequences. Then the interval-valued convergent series space is defined by

$$I^{cs}=\left\{J=\left(J_k\right)\in \omega \left(I\right):{\left({\displaystyle \sum _{k=0}^n}{J}_k\right)}_{n=0}^{\infty }\in I_C\right\}$$

Equivalently, $I^{cs}$ consists of all interval-valued sequences whose sequence of partial sums converges in $I_C$.

In this work, $R_I^{cs}$ represents the Riesz-summable space of interval-valued series. That is

$$R_I^{cs}=\left\{\left(J_k\right)\in \omega \left(I\right):\exists L\in I϶\underset{n\to \infty }{\lim }h\left({\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{S}_k\left(J\right),[\underline{L},\overline{L}]\right)=0,S_n\left(J\right)={\displaystyle \sum _{k=0}^n}{J}_k\right\}$$

(6)

or

$$R_I^{cs}=\left\{\left(J_k\right)\in \omega \left(I\right):R^{cs}\left(J\right)\in I^{cs}\right\}.$$

Theorem 1. 

Let $J=\left(J_k\right)\in \omega \left(I\right)$, and let

$$R_n^{cs}\left(J\right)={\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{S}_k\left(J\right)$$

be the sequence of Riesz means of the interval-valued series ${\displaystyle \sum _{k=0}^{\infty }}{J}_k$. If

$$\underset{n\to \infty }{\lim }{R}_n^{cs}\left(J\right)=L\mathit{and}\underset{n\to \infty }{\lim }{R}_n^{cs}\left(J\right)=M$$

then $L=M.$ Therefore, the Riesz limit of an interval-valued series is unique.

Proof. 

By the hypothesis

$$\underset{n\to \infty }{\lim }{R}_n^{cs}\left(J\right)=L⟹\underset{n\to \infty }{\lim }h\left(R_n^{cs}\left(J\right),L\right)=0.$$

and

$$\underset{n\to \infty }{\lim }{R}_n^{cs}\left(J\right)=M⟹\underset{n\to \infty }{\lim }h\left(R_n^{cs}\left(J\right),M\right)=0.$$

is obtained. By the triangle inequality of the Hausdorff metric, for every $n\in \mathbb{N}$,

$$h(L,M)\le h\left(L,R_n^{cs}\left(J\right)\right)+h\left(R_n^{cs}\left(J\right),M\right)$$

holds. Taking the limit as $n\to \infty$, we obtain

$$h(L,M)\le \underset{n\to \infty }{\lim }h\left(L,R_n^{cs}\left(J\right)\right)+\underset{n\to \infty }{\lim }h\left(R_n^{cs}\left(J\right),M\right)=0.$$

Since the Hausdorff metric is nonnegative, $0\le h(L,M)\le 0$, it follows that $h(L,M)=0$. Since h is a metric,

$$h(L,M)=0⟺L=M$$

is obtained. Thus, it is shown that the Riesz limit of an interval-valued series is unique. □

Example 3. 

Let us examine the Riesz summability of the interval series ${\displaystyle \sum _{k=0}^{\infty }}{J}_k$ whose general term is $J_k=[\underline{J_k},\overline{J_k}]=[{(-1)}^k(k+1),{(-1)}^k(k+1)+1]$.

Let $p_n=1$ for all $n\in \mathbb{N}$. In this case, $P_n=p_0+p_1+p_2+\cdots +p_n=n+1$. On the other hand, since

$$J_0=\left[1,2\right],J_1=\left[-2,-1\right],J_2=\left[3,4\right],J_3=\left[-4,-3\right],\dots$$

and $\left(S_n\left(J\right)\right)$ is the sequence of partial sums of the interval series ${\displaystyle \sum _{k=0}^{\infty }}{J}_k,$ we have

$$\begin{array}{cc}\hfill \underline{S_n}\left(J\right)& =\underline{J_0}+\underline{J_1}+\underline{J_2}+\cdots +\underline{J_n}\hfill \\ & =1+(-2)+3+(-4)+5+(-6)+\dots +{(-1)}^n.(n+1)\hfill \\ & =\left\{\begin{array}{cc}-{\displaystyle \frac{n+1}{2}},\hfill & n\mathit{odd}\\ & \\ {\displaystyle \frac{n+2}{2}},\hfill & n\mathit{even}\end{array}\right.\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \overline{S_n}\left(J\right)& =\overline{J_0}+\overline{J_1}+\cdots +\overline{J_n}\hfill \\ & =2+(-1)+4+(-3)+\dots +{(-1)}^n.(n+1)+1\hfill \\ & =\left\{\begin{array}{cc}{\displaystyle \frac{n+1}{2}},\hfill & n\mathit{odd}\\ & \\ {\displaystyle \frac{3n+4}{2}},\hfill & n\mathit{even}.\end{array}\right.\hfill \end{array}$$

By (3), we get

$$\begin{array}{cc}\hfill S_n\left(J\right)=J_0+J_1+\cdots +J_n& =\left[\underline{S_n}\left(J\right),\overline{S_n}\left(J\right)\right]=\left\{\begin{array}{cc}\left[-{\displaystyle \frac{n+1}{2}},{\displaystyle \frac{n+1}{2}}\right],\hfill & n\mathit{odd}\\ & \\ \left[{\displaystyle \frac{n+2}{2}},{\displaystyle \frac{3n+4}{2}}\right],\hfill & n\mathit{even}.\end{array}\right.\hfill \end{array}$$

$$S_0=\left[1,2\right],S_1=\left[-1,1\right],S_2=\left[2,5\right],S_3=\left[-2,2\right],\dots$$

Therefore,

$$\begin{array}{cc}\hfill {\displaystyle \sum _{k=0}^n}\underline{S_k}\left(J\right)& =\underline{S_0}\left(J\right)+\underline{S_1}\left(J\right)+\dots +\underline{S_n}\left(J\right)\hfill \\ \hfill & =1+(-1)+2+(-2)+\dots +\underline{S_n}\left(J\right)=\left\{\begin{array}{cc}0,& n\mathit{odd}\\ & \\ {\displaystyle \frac{n+2}{2}},& n\mathit{even}.\end{array}\right.\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\displaystyle \sum _{k=0}^n}\overline{S_k}\left(J\right)& =\overline{S_0}\left(J\right)+\overline{S_1}\left(J\right)+\dots +\overline{S_n}\left(J\right)\hfill \\ \hfill & =2+1+5+2+8+3+\dots +\overline{S_n}\left(J\right)=\left\{\begin{array}{cc}{\displaystyle \frac{(n+1)·(n+2)}{2}},\hfill & n\mathit{odd}\\ & \\ {\displaystyle \frac{{(n+2)}^2}{2}},\hfill & n\mathit{even},\end{array}\right.\hfill \end{array}$$

we have that

$${\displaystyle \sum _{k=0}^n}{S}_k\left(J\right)=\left[{\displaystyle \sum _{k=0}^n}\underline{S_k}\left(J\right),{\displaystyle \sum _{k=0}^n}\overline{S_k}\left(J\right)\right]=\left\{\begin{array}{cc}\left[0,{\displaystyle \frac{(n+1)·(n+2)}{2}}\right],\hfill & n\mathit{odd}\\ & \\ \left[{\displaystyle \frac{n+2}{2}},{\displaystyle \frac{{(n+2)}^2}{2}}\right],\hfill & n\mathit{even}.\end{array}\right.$$

If these values found are substituted in the expression (4), we obtain

$$\begin{array}{c}\hfill R_n^{cs}\left(J\right)={\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{S}_k\left(J\right)={\displaystyle \frac{1}{n+1}}{\displaystyle \sum _{k=0}^n}{S}_k\left(J\right)=\left\{\begin{array}{cc}\left[0,{\displaystyle \frac{n+2}{2}}\right],\hfill & n\mathit{odd}\\ & \\ \left[{\displaystyle \frac{n+2}{2·(n+1)}},{\displaystyle \frac{{(n+2)}^2}{2·(n+1)}}\right],\hfill & n\mathit{even}.\end{array}\right.\end{array}$$

(7)

Taking the limit for $n\to \infty$, since the right endpoints of $\left(R_n^{cs}\left(J\right)\right)$ in equality (7) tend to infinity, $\left(R_n^{cs}\left(J\right)\right)$ is not convergent. In other words, the given interval series is not Riesz-summable.

Example 4. 

Let us examine the Riesz summability of the interval series ${\displaystyle \sum _{k=0}^{\infty }}{J}_k$ whose general term is $J_k=[\underline{J_k},\overline{J_k}]=\left[-{\displaystyle \frac{1}{2^k}},{\displaystyle \frac{1}{2^k}}\right].$

If we take $p_n=1$ for all $n\in \mathbb{N}$, then $P_n=p_0+p_1+p_2+\cdots +p_n=n+1.$ From (3),

$$S_n\left(J\right)={\displaystyle \sum _{k=0}^n}{J}_k=\left[{\displaystyle \sum _{k=0}^n}\left(-{\displaystyle \frac{1}{2^k}}\right),{\displaystyle \sum _{k=0}^n}{\displaystyle \frac{1}{2^k}}\right]=\left[-2+{\displaystyle \frac{1}{2^n}},2-{\displaystyle \frac{1}{2^n}}\right]$$

and therefore $\underset{n\to \infty }{\lim }{S}_n\left(J\right)=\left[-2,2\right].$

Since $\left(S_n\left(J\right)\right)$ converges to $\left[-2,2\right]$ and the Riesz method is regular, the interval-valued series is Riesz-summable to $\left[-2,2\right]$. Indeed, from (4),

$$\begin{array}{cc}\hfill R_n^{cs}\left(J\right)& ={\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{S}_k\left(J\right)={\displaystyle \frac{1}{n+1}}{\displaystyle \sum _{k=0}^n}\left[-2+{\displaystyle \frac{1}{2^k}},2-{\displaystyle \frac{1}{2^k}}\right]\hfill \\ \hfill & =\left[-2+{\displaystyle \frac{1}{n+1}}·(2-{\displaystyle \frac{1}{2^n}}),2-{\displaystyle \frac{1}{n+1}}·(2-{\displaystyle \frac{1}{2^n}})\right].\hfill \end{array}$$

From here, $\underset{n\to \infty }{\lim }{R}_n^{cs}\left(J\right)=\left[-2,2\right]$, and the interval series ${\displaystyle \sum _{k=0}^{\infty }}{J}_k$ is Riesz-summable to the interval $\left[-2,2\right]$.

Now, in the same example, if $p_0=1$ and $p_n=n$ is taken for $n\ge 1$, then $P_n=p_0+p_1+p_2+\cdots +p_n={\displaystyle \frac{n^2+n+2}{2}}.$ By (4), we get that

$$S_0\left(J\right)=\left[-1,1\right],$$

$$\begin{array}{cc}\hfill \underline{R_n^{cs}}\left(J\right)& ={\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k\underline{S_k}\left(J\right)={\displaystyle \frac{1}{P_n}}\left\{p_0\underline{S_0}\left(J\right)+{\displaystyle \sum _{k=1}^n}{p}_k\underline{S_k}\left(J\right)\right\}\hfill \\ \hfill & ={\displaystyle \frac{2}{n^2+n+2}}·\left\{(-1)+{\displaystyle \sum _{k=1}^n}k·(-2+{\displaystyle \frac{1}{2^k}})\right\}\hfill \\ \hfill & ={\displaystyle \frac{2}{n^2+n+2}}·\left\{-n^2-n+1-{\displaystyle \frac{n+2}{2^n}}\right\}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \overline{R_n^{cs}}\left(J\right)& ={\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k\overline{S_k}\left(J\right)={\displaystyle \frac{1}{P_n}}\left\{p_0\overline{S_0}\left(J\right)+{\displaystyle \sum _{k=1}^n}{p}_k\overline{S_k}\left(J\right)\right\}\hfill \\ \hfill & ={\displaystyle \frac{2}{n^2+n+2}}·\left\{1+{\displaystyle \sum _{k=1}^n}k·(2-{\displaystyle \frac{1}{2^k}})\right\}\hfill \\ \hfill & ={\displaystyle \frac{2}{n^2+n+2}}·\left\{n^2+n-1+{\displaystyle \frac{n+2}{2^n}}\right\}.\hfill \end{array}$$

From here, we get

$$\underset{n\to \infty }{\lim }{R}_n^{cs}\left(J\right)=\underset{n\to \infty }{\lim }\left[\underline{R_n^{cs}}\left(J\right),\overline{R_n^{cs}}\left(J\right)\right]=\left[-2,2\right].$$

Thus, the given interval series is Riesz-summable to the interval $\left[-2,2\right]$.

Example 5. 

Using the Definition 4, let us show that the interval-valued series ${\displaystyle \sum _{k=0}^{\infty }}{J}_k$, whose general term is $J_k=[\underline{J_k},\overline{J_k}]=\left[{(-1)}^k,{(-1)}^k\right]$ is Riesz-summable to $\left[{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right]$.

Let $p_n=1$ for all $n\in \mathbb{N}.$ Then $P_n=p_0+p_1+p_2+\cdots +p_n=n+1$. From (3) and (4), we obtain the following expressions:

$$S_n\left(J\right)={\displaystyle \sum _{k=0}^n}{J}_k=\left[{\displaystyle \sum _{k=0}^n}{\left(-1\right)}^k,{\displaystyle \sum _{k=0}^n}{\left(-1\right)}^k\right]=\left\{\begin{array}{ccc}\left[1,1\right],& & neven\\ & & \\ \left[0,0\right],& & nodd,\end{array}\right.$$

$${\displaystyle \sum _{k=0}^n}{p}_k{S}_k\left(J\right)={\displaystyle \sum _{k=0}^n}{S}_k\left(J\right)=\left\{\begin{array}{ccc}\left[{\displaystyle \frac{n}{2}}+1,{\displaystyle \frac{n}{2}}+1\right],& & neven\\ & & \\ \left[{\displaystyle \frac{n+1}{2}},{\displaystyle \frac{n+1}{2}}\right],& & nodd,\end{array}\right.$$

and

$$\begin{array}{cc}\hfill R_n^{cs}\left(J\right)& ={\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{S}_k\left(J\right)={\displaystyle \frac{1}{n+1}}·\left\{\begin{array}{cc}\left[{\displaystyle \frac{n}{2}}+1,{\displaystyle \frac{n}{2}}+1\right],& neven\\ & \\ \left[{\displaystyle \frac{n+1}{2}},{\displaystyle \frac{n+1}{2}}\right],& nodd\end{array}\right.\hfill \\ & \\ \hfill & =\left\{\begin{array}{c}\left[{\displaystyle \frac{n}{2\left(n+1\right)}}+{\displaystyle \frac{1}{n+1}},{\displaystyle \frac{n}{2\left(n+1\right)}}+{\displaystyle \frac{1}{n+1}}\right],neven\\ \\ \left[{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right],nodd.\end{array}\right.\hfill \end{array}$$

Now let us show that

$$\underset{n\to \infty }{\lim }h\left({\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{S}_k\left(J\right),\left[{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right]\right)=0.$$

For even n, by the definition of the Hausdorff metric given in (1), we obtain

$$\begin{array}{cc}\hfill \underset{n\to \infty }{\lim }h\left({\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{S}_k\left(J\right),\left[{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right]\right)& =\underset{n\to \infty }{\lim }h\left(\left[{\displaystyle \frac{n}{2\left(n+1\right)}}+{\displaystyle \frac{1}{n+1}},{\displaystyle \frac{n}{2\left(n+1\right)}}+{\displaystyle \frac{1}{n+1}}\right],\left[{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right]\right)\hfill \\ \hfill & =\underset{n\to \infty }{\lim }\max \left\{\left|{\displaystyle \frac{n}{2\left(n+1\right)}}+{\displaystyle \frac{1}{n+1}}-{\displaystyle \frac{1}{2}}\right|,\left|{\displaystyle \frac{n}{2\left(n+1\right)}}+{\displaystyle \frac{1}{n+1}}-{\displaystyle \frac{1}{2}}\right|\right\}\hfill \\ \hfill & =\underset{n\to \infty }{\lim }\left\{{\displaystyle \frac{1}{2\left(n+1\right)}}\right\}=0.\hfill \end{array}$$

For odd n, similarly,

$$\underset{n\to \infty }{\lim }h\left({\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{S}_k\left(J\right),\left[{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right]\right)=\underset{n\to \infty }{\lim }h\left(\left[{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right],\left[{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right]\right)=0.$$

Thus, the series is not convergent in the ordinary sense, whereas it is Riesz-summable to $\left[{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right]$.

Theorem 2. 

The function d, defined as

$$d:R_I^{cs}\times R_I^{cs}\to \mathbb{R}$$

$$(J,K)\mapsto d(J,K)=\underset{n\in \mathbb{N}}{\sup }\left\{\max \left\{\left|\underline{R_n^{cs}}\left(J\right)-\underline{R_n^{cs}}\left(K\right)\right|,\left|\overline{R_n^{cs}}\left(J\right)-\overline{R_n^{cs}}\left(K\right)\right|\right\}\right\}$$

is a metric, and the space $(R_I^{cs},d)$ is a metric space.

Proof. 

M1- Let $\left(J_k),(K_k\right)\in R_I^{cs}.$ Since h is a metric on I, $h\left(R_n^{cs}\left(J\right),R_n^{cs}\left(K\right)\right)\ge 0$ for every $n\in \mathbb{N}$. Therefore,

$$d\left(J,K\right)=\underset{n\in N}{\sup }h\left(R_n^{cs}\left(J\right),R_n^{cs}\left(K\right)\right)\ge 0.$$

We now show that $d\left(J,K\right)$ is finite. Since $\left(J_k\right),\left(K_k\right)\in R_I^{cs}$, there exist intervals $M=\left[\underline{M},\overline{M}\right]\in I$ and $N=\left[\underline{N},\overline{N}\right]\in I$ such that

$$R_n^{cs}\left(J\right)\to M\mathrm{and}{R}_n^{cs}\left(K\right)\to N\left(n\to \infty \right).$$

Every convergent sequence in a metric space is bounded. Hence, there exist finite constants $C_J,C_K\ge 0$ such that

$$h\left(R_n^{cs}\left(J\right),M\right)\le C_J\mathrm{and}h\left(R_n^{cs}\left(K\right),N\right)\le C_K$$

for every $n\in \mathbb{N}$. Moreover, since $M,N\in I,$

$$h\left(M,N\right)=\max \left\{\left|\underline{M}-\underline{N}\right|,\left|\overline{M}-\overline{N}\right|\right\}<\infty .$$

By the triangle inequality for the Hausdorff metric, for every $n\in \mathbb{N}$,

$$h\left(R_n^{cs}\left(J\right),R_n^{cs}\left(K\right)\right)\le h\left(R_n^{cs}\left(J\right),M\right)+h\left(M,N\right)+h\left(N,R_n^{cs}\left(K\right)\right)\le C_J+h\left(M,N\right)+C_K.$$

The right-hand side is a finite constant independent of n. Therefore,

$$d\left(J,K\right)=\underset{n\in \mathbb{N}}{\sup }h\left(R_n^{cs}\left(J\right),R_n^{cs}\left(K\right)\right)\le C_J+h\left(M,N\right)+C_K<\infty .$$

Thus, $0\le d\left(J,K\right)<\infty$. Consequently, d is a well-defined, nonnegative, real-valued function on $R_I^{cs}\times R_I^{cs}$.

M2- Clearly, $J=K$ implies $d\left(J,K\right)=0$.

Conversely, suppose that $d(J,K)=0$. Since the Hausdorff metric is nonnegative, $h\left(R_n^{cs}\left(J\right),R_n^{cs}\left(K\right)\right)=0$ for every $n\in \mathbb{N}$. Hence, $R_n^{cs}\left(J\right)=R_n^{cs}\left(K\right)$. Equality of the lower endpoints yields

$${\displaystyle \sum _{k=0}^n}{p}_k\underline{S_k}\left(J\right)={\displaystyle \sum _{k=0}^n}{p}_k\underline{S_k}\left(K\right).$$

Subtracting the corresponding equality for $n-1$, with the convention $P_{-1}=0,$ gives $p_n\underline{S_n}\left(J\right)=p_n\overline{S_n}\left(K\right)$. Since $p_n>0$, $\underline{S_n}\left(J\right)=\overline{S_n}\left(K\right)$. Applying the same argument to the upper endpoints gives $\overline{S_n}\left(J\right)=\overline{S_n}\left(K\right)$.Therefore, $S_n\left(J\right)=S_n\left(K\right)$ for every $n\in \mathbb{N}$. Taking $S_{-1}\left(J\right)=S_{-1}\left(K\right)=\left[0,0\right]$, we obtain

$$\underline{J_n}=\underline{S_n}\left(J\right)-\underline{S_{n-1}}\left(J\right)=\underline{S_n}\left(K\right)-\underline{S_{n-1}}\left(K\right)=\underline{K_n}$$

and

$$\overline{J_n}=\overline{S_n}\left(J\right)-\overline{S_{n-1}}\left(J\right)=\overline{S_n}\left(K\right)-\overline{S_{n-1}}\left(K\right)=\overline{K_n}.$$

Thus, $J_n=K_n$ for every $n\in \mathbb{N}$, and consequently $J=K$. Therefore, the condition $d(J,K)=0\iff J=K$ is satisfied.

M3- Let $\left(J_k\right),\left(K_k\right)\in R_I^{cs}$. Since the Hausdorff metric is symmetric, for every $n\in \mathbb{N}$,

$$h\left(R_n^{cs}\left(J\right),R_n^{cs}\left(K\right)\right)=h\left(R_n^{cs}\left(K\right),R_n^{cs}\left(J\right)\right).$$

Taking the supremum over $n\in \mathbb{N}$, we obtain

$$d\left(J,K\right)=\underset{n\in \mathbb{N}}{\sup }h\left(R_n^{cs}\left(J\right),R_n^{cs}\left(K\right)\right)=\underset{n\in \mathbb{N}}{\sup }h\left(R_n^{cs}\left(K\right),R_n^{cs}\left(J\right)\right)=d\left(K,J\right).$$

Thus, d is symmetric.

M4- Let $\left(J_k\right),\left(K_k\right),\left(E_k\right)\in R_I^{cs}$. By the triangle inequality for the Hausdorff metric, for every $n\in \mathbb{N}$,

$$h\left(R_n^{cs}\left(J\right),R_n^{cs}\left(K\right)\right)\le h\left(R_n^{cs}\left(J\right),R_n^{cs}\left(E\right)\right)+h\left(R_n^{cs}\left(E\right),R_n^{cs}\left(K\right)\right).$$

By the definition of d,

$$h\left(R_n^{cs}\left(J\right),R_n^{cs}\left(E\right)\right)\le d\left(J,E\right)\mathrm{and}h\left(R_n^{cs}\left(E\right),R_n^{cs}\left(K\right)\right)\le d\left(E,K\right).$$

Hence, for every $n\in \mathbb{N}$,

$$h\left(R_n^{cs}\left(J\right),R_n^{cs}\left(K\right)\right)\le d\left(J,E\right)+d\left(E,K\right).$$

Taking the supremum over $n\in \mathbb{N}$, we obtain

$$d\left(J,K\right)\le d\left(J,E\right)+d\left(E,K\right).$$

Thus, d satisfies the triangle inequality. Therefore, d satisfies all metric axioms and hence $(R_I^{cs},d)$ is a metric space. □

Theorem 3. 

The metric space $\left(R_I^{cs},d\right)$ is complete.

Proof. 

Let ${\left(J^r\right)}_{r\in \mathbb{N}}=(J^1,J^2,\dots )$ be a Cauchy sequence in $(R_I^{cs},d)$. For each $r\in \mathbb{N}$, write $J^r={\left(J_k^r\right)}_{k\in \mathbb{N}},$ and denote the nth partial sum of the interval-valued series $J^r$ by

$$S_n\left(J^r\right)={\displaystyle \sum _{k=0}^n}{J}_k^r.$$

The corresponding nth Riesz mean is

$$R_n^{cs}\left(J^r\right)={\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{S}_k\left(J^r\right).$$

First, we determine the pointwise limits of the Riesz means.

Since ${\left(J^r\right)}_{r\in \mathbb{N}}$ is a Cauchy sequence, for every $\epsilon >0$, there exists $r_0\in \mathbb{N}$ such that $d(J^r,J^m)<\epsilon$ whenever $r,m\ge r_0$. By the definition of d, for every $n\in \mathbb{N}$,

$$h\left(R_n^{cs}\left(J^r\right),R_n^{cs}\left(J^m\right)\right)\le d(J^r,J^m)<\epsilon .$$

Thus, for each fixed $n\in \mathbb{N}$, ${\left(R_n^{cs}{J}^r)\right)}_{r\in \mathbb{N}}$ is a Cauchy sequence in the complete metric space $(I,h)$. Therefore, for each $n\in \mathbb{N}$, there exists an interval $A_n\in I$ such that $R_n^{cs}\left(J^r\right)⟶A_n(r\to \infty ).$

Write $R_n^{cs}\left(J^r\right)=\left[\underline{R_n^{cs}}\left(J^r\right),\overline{R_n^{cs}}\left(J^r\right)\right]$ and $A_n=[a_n,b_n]$. By the definition of the Hausdorff metric,

$$\underline{R_n^{cs}}\left(J^r\right)⟶a_n\mathrm{and}\overline{R_n^{cs}}\left(J^r\right)⟶b_n(r\to \infty ).$$

Since $\underline{R_n^{cs}}\left(J^r\right)\le \overline{R_n^{cs}}\left(J^r\right)$ for every $r\in \mathbb{N}$, passing to the limit as $r\to \infty$ gives $a_n\le b_n$. Hence, $A_n=[a_n,b_n]$ is a well-defined interval for every $n\in \mathbb{N}$.

We now construct the partial sums of the limit series.

For every $r\in \mathbb{N}$ and $n\ge 1$, the definition of the Riesz mean gives

$$P_n{R}_n^{cs}\left(J^r\right)={\displaystyle \sum _{k=0}^n}{p}_k{S}_k\left(J^r\right)$$

and

$$P_{n-1}{R}_{n-1}^{cs}\left(J^r\right)={\displaystyle \sum _{k=0}^{n-1}}{p}_k{S}_k\left(J^r\right).$$

Considering the corresponding real-valued equalities for the lower endpoints and subtracting them, we obtain

$$\underline{S_n}\left(J^r\right)={\displaystyle \frac{P_n\underline{R_n^{cs}}\left(J^r\right)-P_{n-1}\underline{R_{n-1}^{cs}}\left(J^r\right)}{p_n}}.$$

Similarly,

$$\overline{S_n}\left(J^r\right)={\displaystyle \frac{P_n\overline{R_n^{cs}}\left(J^r\right)-P_{n-1}\overline{R_{n-1}^{cs}}\left(J^r\right)}{p_n}}.$$

Here, the assumption $p_n>0$ is used. Define ${\alpha }_0=a_0,{\beta }_0=b_0,$ and, for $n\ge 1$, define

$${\alpha }_n={\displaystyle \frac{P_n{a}_n-P_{n-1}{a}_{n-1}}{p_n}},{\beta }_n={\displaystyle \frac{P_n{b}_n-P_{n-1}{b}_{n-1}}{p_n}}.$$

Since $P_0=p_0$, the definitions ${\alpha }_0=a_0$ and ${\beta }_0=b_0$ are consistent with the corresponding formulas for $n=0$. For each fixed n, passing to the limit as $r\to \infty$ yields

$$\underline{S_n}\left(J^r\right)⟶{\alpha }_n\mathrm{and}\overline{S_n}\left(J^r\right)⟶{\beta }_n.$$

Since $S_n\left(J^r\right)$ is an interval for every $r\in \mathbb{N}$, $\underline{S_n}\left(J^r\right)\le \overline{S_n}\left(J^r\right)$. Passing to the limit as $r\to \infty$, we obtain ${\alpha }_n\le {\beta }_n.$ Therefore, $S_n\left(J\right)=[{\alpha }_n,{\beta }_n]$ is a well-defined interval for every $n\in \mathbb{N}$.

We now construct the terms of the limit series.

Define $J_0=S_0\left(J\right)=[{\alpha }_0,{\beta }_0].$ For $n\ge 1$, let ${\gamma }_n={\alpha }_n-{\alpha }_{n-1}$, ${\delta }_n={\beta }_n-{\beta }_{n-1}$ and define $J_n=[{\gamma }_n,{\delta }_n].$

We verify that $J_n$ is an interval. For every $r\in \mathbb{N}$ and $n\ge 1$,

$$\underline{J_n^r}=\underline{S_n}\left(J^r\right)-\underline{S_{n-1}}\left(J^r\right)\mathrm{and}\overline{J_n^r}=\overline{S_n}\left(J^r\right)-\overline{S_{n-1}}\left(J^r\right).$$

Passing to the limit as $r\to \infty$, we obtain $\underline{J_n^r}⟶{\gamma }_n\mathrm{and}\overline{J_n^r}⟶{\delta }_n.$ Since $J_n^r$ is an interval for every $r\in \mathbb{N}$, $\underline{J_n^r}\le \overline{J_n^r}.$ Passing to the limit gives ${\gamma }_n\le {\delta }_n.$ Hence, $J_n=[{\gamma }_n,{\delta }_n]$ is a well-defined interval for every $n\ge 1$. Consequently, $J={\left(J_n\right)}_{n=0}^{\infty }\in \omega \left(I\right)$ defines an interval-valued series.

By the definitions of ${\gamma }_k$ and ${\delta }_k$,

$${\displaystyle \sum _{k=0}^n}\underline{J_k}={\alpha }_0+{\displaystyle \sum _{k=1}^n}({\alpha }_k-{\alpha }_{k-1})={\alpha }_n\mathrm{and}{\displaystyle \sum _{k=0}^n}\overline{J_k}={\beta }_0+{\displaystyle \sum _{k=1}^n}({\beta }_k-{\beta }_{k-1})={\beta }_n.$$

Therefore, ${\sum }_{k=0}^n{J}_k=[{\alpha }_n,{\beta }_n]=S_n\left(J\right)$. Thus, the intervals $[{\alpha }_n,{\beta }_n]$ are indeed the partial sums of the interval-valued series J.

We now identify the Riesz means of the limit series.

For the lower endpoint, we have

$$\underline{R_n^{cs}}\left(J\right)={\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{\alpha }_k.$$

For $k\ge 1$, the definition of ${\alpha }_k$ gives $p_k{\alpha }_k=P_k{a}_k-P_{k-1}{a}_{k-1},$ whereas $p_0{\alpha }_0=P_0{a}_0.$ Therefore,

$${\displaystyle \sum _{k=0}^n}{p}_k{\alpha }_k=P_0{a}_0+{\displaystyle \sum _{k=1}^n}\left(P_k{a}_k-P_{k-1}{a}_{k-1}\right)=P_n{a}_n.$$

It follows that $\underline{R_n^{cs}}\left(J\right)=a_n.$ Similarly,

$$\overline{R_n^{cs}}\left(J\right)={\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{\beta }_k=b_n.$$

Consequently, $R_n^{cs}\left(J\right)=[a_n,b_n]=A_n.$

Since $J^r\in R_I^{cs}$ for every $r\in \mathbb{N}$, for each $r\in \mathbb{N}$ there exists an interval $L_r\in I$ such that $R_n^{cs}\left(J^r\right)⟶L_r(n\to \infty ).$

We now prove that ${\left(L_r\right)}_{r\in \mathbb{N}}$ is a Cauchy sequence in $(I,h)$. Let $\epsilon >0$ be given. Since $\left(J^r\right)$ is a Cauchy sequence, there exists $r_0\in \mathbb{N}$ such that

$$d(J^r,J^m)<{\displaystyle \frac{\epsilon }{2}}$$

whenever $r,m\ge r_0$. Hence, for every $n\in \mathbb{N}$,

$$h\left(R_n^{cs}\left(J^r\right),R_n^{cs}\left(J^m\right)\right)\le d(J^r,J^m)<{\displaystyle \frac{\epsilon }{2}}.$$

Letting $n\to \infty$ and using the continuity of the Hausdorff metric, we obtain

$$h(L_r,L_m)\le {\displaystyle \frac{\epsilon }{2}}<\epsilon .$$

Thus, $\left(L_r\right)$ is a Cauchy sequence in $(I,h)$. Since $(I,h)$ is complete, there exists an interval $L\in I$ such that $L_r⟶L(r\to \infty )$ [4].

We now prove that $A_n⟶L(n\to \infty ).$ Let $\epsilon >0$ be given. Since $\left(J^r\right)$ is a Cauchy sequence, there exists $r_0\in \mathbb{N}$ such that

$$d(J^r,J^m)<{\displaystyle \frac{\epsilon }{3}}$$

whenever $r,m\ge r_0$. Moreover, since $L_r\to L$, we may choose and fix an index $r\ge r_0$ sufficiently large so that

$$h(L_r,L)<{\displaystyle \frac{\epsilon }{3}}.$$

For this fixed r, every $m\ge r_0$, and every $n\in \mathbb{N}$, we have

$$h\left(R_n^{cs}\left(J^r\right),R_n^{cs}\left(J^m\right)\right)<{\displaystyle \frac{\epsilon }{3}}.$$

For each fixed n, letting $m\to \infty$ and using $R_n^{cs}\left(J^m\right)⟶A_n,$ we obtain

$$h\left(R_n^{cs}\left(J^r\right),A_n\right)\le {\displaystyle \frac{\epsilon }{3}}.$$

This inequality holds for every $n\in \mathbb{N}$.

On the other hand, for the fixed r chosen above, $R_n^{cs}\left(J^r\right)⟶L_r(n\to \infty )$. Hence, for all sufficiently large n,

$$h\left(R_n^{cs}\left(J^r\right),L_r\right)<{\displaystyle \frac{\epsilon }{3}}.$$

By the triangle inequality, for all sufficiently large n,

$$h(A_n,L)\le h\left(A_n,R_n^{cs}\left(J^r\right)\right)+h\left(R_n^{cs}\left(J^r\right),L_r\right)+h(L_r,L)<\epsilon .$$

Therefore, $A_n⟶L(n\to \infty )$. Since $R_n^{cs}\left(J\right)=A_n$, it follows that $R_n^{cs}\left(J\right)⟶L$. Hence, $J\in R_I^{cs}.$

Finally, we prove that $d(J^r,J)⟶0(r\to \infty ).$ Let $\epsilon >0$ be given. Since $\left(J^r\right)$ is a Cauchy sequence, there exists $r_0\in \mathbb{N}$ such that

$$d(J^r,J^m)<{\displaystyle \frac{\epsilon }{2}}$$

whenever $r,m\ge r_0$. Therefore, for every $n\in \mathbb{N}$,

$$h\left(R_n^{cs}\left(J^r\right),R_n^{cs}\left(J^m\right)\right)<{\displaystyle \frac{\epsilon }{2}}.$$

For fixed $r\ge r_0$ and fixed n, letting $m\to \infty$ and using $R_n^{cs}\left(J^m\right)⟶A_n=R_n^{cs}\left(J\right),$ we obtain

$$h\left(R_n^{cs}\left(J^r\right),R_n^{cs}\left(J\right)\right)\le {\displaystyle \frac{\epsilon }{2}}.$$

This inequality holds for every $n\in \mathbb{N}$. Hence,

$$d(J^r,J)=\underset{n\in \mathbb{N}}{\sup }h\left(R_n^{cs}\left(J^r\right),R_n^{cs}\left(J\right)\right)\le {\displaystyle \frac{\epsilon }{2}}<\epsilon .$$

Therefore, $d(J^r,J)⟶0(r\to \infty ).$ Thus, every Cauchy sequence in $(R_I^{cs},d)$ converges to an element of $R_I^{cs}$. Consequently, the metric space $(R_I^{cs},d)$ is complete. □

Theorem 4. 

Every convergent interval-valued series is Riesz-summable to the same interval. Moreover, the inclusion is proper; that is, $I^{cs}⊊R_I^{cs}$.

Proof. 

Let $J=\left(J_k\right)\in I^{cs}$, and define the nth partial sum of the interval-valued series ${\displaystyle \sum _{k=0}^{\infty }}{J}_k$ by $S_n\left(J\right)={\displaystyle \sum _{k=0}^n}{J}_k$. By the definition of $I^{cs}$, the sequence of partial sums ${\left(S_n\left(J\right)\right)}_{n=0}^{\infty }$ belongs to $I_C$. Therefore, there exists an interval $L=\left[\underline{L},\overline{L}\right]\in I$ such that $S_n\left(J\right)\to L\left(n\to \infty \right)$ with respect to the Hausdorff metric. Writing $S_n\left(J\right)=\left[\underline{S_n}\left(J\right),\overline{S_n}\left(J\right)\right]$, we obtain

$$\underline{S_n}\left(J\right)\to \underline{L}\mathrm{and}\overline{S_n}\left(J\right)\to \overline{L}\left(n\to \infty \right).$$

Let $\left(p_n\right)$ be a sequence of positive real numbers such that $P_n={\displaystyle \sum _{k=0}^n}{p}_k\to \infty$. By the regularity of the classical Riesz method,

$${\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k\underline{S_k}\left(J\right)\to \underline{L}\mathrm{and}{\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k\overline{S_k}\left(J\right)\to \overline{L}.$$

Since $p_k>0$, interval arithmetic gives

$$R_n^{cs}\left(J\right)={\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{S}_k\left(J\right)=\left[{\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k\underline{S_k}\left(J\right),{\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k\overline{S_k}\left(J\right)\right].$$

Consequently,

$$h\left(R_n^{cs}\left(J\right),L\right)=\max \left\{\left|{\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k\underline{S_k}\left(J\right)-\underline{L}\right|,\left|{\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k\overline{S_k}\left(J\right)-\overline{L}\right|\right\}.$$

Taking the limit as $n\to \infty$, we obtain $\underset{n\to \infty }{\lim }h\left(R_n^{cs}\left(J\right),L\right)=0$. Hence, $R_n^{cs}\left(J\right)\to L\left(n\to \infty \right)$. Thus, the interval-valued series ${\displaystyle \sum _{k=0}^{\infty }}{J}_k$ is Riesz-summable to the same interval L. Since $J\in I^{cs}$ was arbitrary, it follows that $I^{cs}\subseteq R_I^{cs}$.

It remains to show that the inclusion is proper. In Example 5, the interval-valued series with general term $J_k=\left[{\left(-1\right)}^k,{\left(-1\right)}^k\right]$ is not convergent, because its sequence of partial sums alternates between $\left[1,1\right]$ and $\left[0,0\right]$. However, for $p_k=1$, its Riesz means converge to $\left[{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right]$. Consequently, $I^{cs}⊊R_I^{cs}$. This completes the proof. □

Theorem 5. 

$R_I^{cs}$ is a quasilinear space.

Proof. 

Let $J=\left(J_k\right)$ and $K=\left(K_k\right)$ be arbitrary elements of $R_I^{cs}$.Since these interval-valued series are Riesz-summable, there exist intervals $A_1=\left[\underline{A_1},\overline{A_1}\right]$ and $A_2=\left[\underline{A_2},\overline{A_2}\right]$ such that

$$\underset{n\to \infty }{\lim }h\left(R_n^{cs}\left(J\right),A_1\right)=0\mathrm{and}\underset{n\to \infty }{\lim }h\left(R_n^{cs}\left(K\right),A_2\right)=0.$$

By the definition of the Hausdorff metric, these convergences imply

$$\underline{R_n^{cs}}\left(J\right)\to \underline{A_1},\overline{R_n^{cs}}\left(J\right)\to \overline{A_1}$$

and

$$\underline{R_n^{cs}}\left(K\right)\to \underline{A_2},\overline{R_n^{cs}}\left(K\right)\to \overline{A_2}\left(n\to \infty \right).$$

We first show that $R_I^{cs}$ is closed under addition. For every $n\in \mathbb{N}$,

$$S_n\left(J+K\right)={\displaystyle \sum _{k=0}^n}\left(J_k+K_k\right)={\displaystyle \sum _{k=0}^n}{J}_k+{\displaystyle \sum _{k=0}^n}{K}_k=S_n\left(J\right)+S_n\left(K\right).$$

Applying the Riesz transformation, we obtain

$$\begin{array}{cc}\hfill & R_n^{cs}\left(J+K\right)={\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{S}_k\left(J+K\right)={\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k\left(S_k\left(J\right)+S_k\left(K\right)\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{S}_k\left(J\right)+{\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{S}_k\left(K\right)=R_n^{cs}\left(J\right)+R_n^{cs}\left(K\right).\hfill \end{array}$$

Consequently,

$$\underline{R_n^{cs}}\left(J+K\right)=\underline{R_n^{cs}}\left(J\right)+\underline{R_n^{cs}}\left(K\right)\to \underline{A_1}+\underline{A_2}$$

and

$$\overline{R_n^{cs}}\left(J+K\right)=\overline{R_n^{cs}}\left(J\right)+\overline{R_n^{cs}}\left(K\right)\to \overline{A_1}+\overline{A_2}\left(n\to \infty \right).$$

Therefore, $R_n^{cs}\left(J+K\right)\to A_1+A_2\left(n\to \infty \right)$. Equivalently, by the triangle inequality for the Hausdorff metric,

$$h\left(R_n^{cs}\left(J+K\right),A_1+A_2\right)\le h\left(R_n^{cs}\left(J\right),A_1\right)+h\left(R_n^{cs}\left(K\right),A_2\right)\to 0\left(n\to \infty \right).$$

Hence, ${\left(J_k+K_k\right)}_{k=0}^{\infty }\in R_I^{cs}$ and $R_I^{cs}$ is closed under addition.

We next show that $R_I^{cs}$ is closed under real scalar multiplication. Let $\mu \in \mathbb{R}$. For every $n\in \mathbb{N}$,

$$S_n\left(\mu J\right)={\displaystyle \sum _{k=0}^n}\mu J_k=\mu {\displaystyle \sum _{k=0}^n}{J}_k=\mu S_n\left(J\right).$$

It follows that

$$R_n^{cs}\left(\mu J\right)={\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{S}_k\left(\mu J\right)=\mu {\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{S}_k\left(J\right)=\mu R_n^{cs}\left(J\right).$$

If $\mu >0$, then

$$\underline{R_n^{cs}}\left(\mu J\right)=\mu \underline{R_n^{cs}}\left(J\right)\to \mu \underline{A_1}\mathrm{and}\overline{R_n^{cs}}\left(\mu J\right)=\mu \overline{R_n^{cs}}\left(J\right)\to \mu \overline{A_1}\left(n\to \infty \right).$$

If $\mu <0$, multiplication by $\mu$ reverses the endpoints, and hence

$$\underline{R_n^{cs}}\left(\mu J\right)=\mu \overline{R_n^{cs}}\left(J\right)\to \mu \overline{A_1}\mathrm{and}\overline{R_n^{cs}}\left(\mu J\right)=\mu \underline{R_n^{cs}}\left(J\right)\to \mu \underline{A_1}\left(n\to \infty \right).$$

Thus, in both cases, $R_n^{cs}\left(\mu J\right)=\mu A_1$. This also follows directly from the homogeneity of the Hausdorff metric:

$$h\left(R_n^{cs}\left(\mu J\right),\mu A_1\right)=h\left(\mu R_n^{cs}\left(J\right),\mu A_1\right)=\left|\mu \right|h\left(R_n^{cs}\left(J\right),A_1\right)\to 0\left(n\to \infty \right).$$

Therefore, ${\left(\mu J_k\right)}_{k=0}^{\infty }\in R_I^{cs}$ for every $\mu \in \mathbb{R}$, so $R_I^{cs}$ is closed under real scalar multiplication.

Moreover, the zero sequence $\theta =\left(\left[0,0\right],\left[0,0\right],\dots \right)$ belongs to $R_I^{cs}$, since $R_n^{cs}\left(\theta \right)=\left[0,0\right]$ for every $n\in \mathbb{N}$.

The addition, real scalar multiplication, and partial order on $R_I^{cs}$ are the restrictions of the corresponding operations and partial order on the quasilinear space $\omega \left(I\right).$ Since $R_I^{cs}$ is a nonempty subset of $\omega \left(I\right)$ that contains the zero element and is closed under addition and real scalar multiplication, all the quasilinear space axioms stated in Definition 6 are inherited from $\omega \left(I\right).$ In particular, associativity and commutativity of addition, the zero-element property, compatibility of successive scalar multiplications, the unit-scalar property, the distributive and quasilinear inclusion properties, and compatibility with the partial order hold coordinatewise in $R_I^{cs}$.

Consequently, $R_I^{cs}$ is a quasilinear subspace of $\omega \left(I\right).$ Therefore, $R_I^{cs}$ is a quasilinear space. □

Theorem 6. 

The space $R_I^{cs}$, equipped with the function

$${\parallel ·\parallel }_R:R_I^{cs}⟶\mathbb{R},$$

defined by

$${\parallel J\parallel }_R=\underset{n\in \mathbb{N}}{\sup }\max \left\{\left|\underline{R_n^{cs}}\left(J\right)\right|,\left|\overline{R_n^{cs}}\left(J\right)\right|\right\},$$

is a normed quasilinear space.

Proof. 

N1- Let $J=\left(J_k\right)\in R_I^{cs}$. Since $J\in R_I^{cs}$, there exists $L=\left[\underline{L},\overline{L}\right]\in I$ such that $R_n^{cs}\left(J\right)\to L\left(n\to \infty \right).$ Therefore, the real sequences $\left(\underline{R_n^{cs}}\left(J\right)\right)$ and $\left(\overline{R_n^{cs}}\left(J\right)\right)$ are convergent and hence bounded. Consequently,

$${∥J∥}_R=\underset{n\in \mathbb{N}}{\sup }\max \left\{\left|\underline{R_n^{cs}}\left(J\right)\right|,\left|\overline{R_n^{cs}}\left(J\right)\right|\right\}<\infty .$$

Thus, ${∥J∥}_R$ is a nonnegative, finite real number. Now, suppose that ${∥J∥}_R=0$. Then, for every $n\in \mathbb{N}$,

$$\max \left\{\left|\underline{R_n^{cs}}\left(J\right)\right|,\left|\overline{R_n^{cs}}\left(J\right)\right|\right\}=0$$

and hence

$$R_n^{cs}\left(J\right)=\left[0,0\right]=R_n^{cs}\left(\theta \right),$$

where $\theta =\left(\left[0,0\right],\left[0,0\right],\dots \right)$ is the zero element of $R_I^{cs}$. Since the Riesz transformation was shown to be injective in the proof of (M2), it follows that $J=\theta$. Conversely, if $J=\theta$, then $R_n^{cs}\left(J\right)=\left[0,0\right]$ for every n, and therefore ${∥J∥}_R=0$. Hence,

$${∥J∥}_R=0\iff J=\theta .$$

N2- Let $\left(J_k\right),\left(K_k\right)\in R_I^{cs}.$ Here, $J_k=[\underline{J_k},\overline{J_k}]\mathrm{and}{K}_k=[\underline{K_k},\overline{K_k}].$ Then, for every $n\in \mathbb{N}$, $S_n(J+K)=S_n\left(J\right)+S_n\left(K\right),$ and hence

$$\underline{R_n^{cs}}(J+K)=\underline{R_n^{cs}}\left(J\right)+\underline{R_n^{cs}}\left(K\right)\mathrm{and}\overline{R_n^{cs}}(J+K)=\overline{R_n^{cs}}\left(J\right)+\overline{R_n^{cs}}\left(K\right).$$

Applying the triangle inequality for real numbers, we obtain

$$\left|\underline{R_n^{cs}}(J+K)\right|\le \left|\underline{R_n^{cs}}\left(J\right)\right|+\left|\underline{R_n^{cs}}\left(K\right)\right|\mathrm{and}\left|\overline{R_n^{cs}}(J+K)\right|\le \left|\overline{R_n^{cs}}\left(J\right)\right|+\left|\overline{R_n^{cs}}\left(K\right)\right|.$$

Thus,

$$\max \left\{\left|\underline{R_n^{cs}}(J+K)\right|,\left|\overline{R_n^{cs}}(J+K)\right|\right\}\le \max \left\{\left|\underline{R_n^{cs}}\left(J\right)\right|,\left|\overline{R_n^{cs}}\left(J\right)\right|\right\}+\max \left\{\left|\underline{R_n^{cs}}\left(K\right)\right|,\left|\overline{R_n^{cs}}\left(K\right)\right|\right\}.$$

Taking the supremum over $n\in \mathbb{N}$, we obtain

$${\parallel J+K\parallel }_R\le {\parallel J\parallel }_R+{\parallel K\parallel }_R.$$

N3- Let $\lambda \in \mathbb{R}$ and $J=\left(J_k\right)\in R_I^{cs}$. Since $R_n^{cs}\left(\lambda J\right)=\lambda R_n^{cs}\left(J\right)$, for every $n\in \mathbb{N}$, we consider two cases.

If $\lambda \ge 0$, then

$$\underline{R_n^{cs}}\left(\lambda J\right)=\lambda \underline{R_n^{cs}}\left(J\right)\mathrm{and}\overline{R_n^{cs}}\left(\lambda J\right)=\lambda \overline{R_n^{cs}}\left(J\right).$$

If $\lambda <0$, the endpoints are interchanged, and hence

$$\underline{R_n^{cs}}\left(\lambda J\right)=\lambda \overline{R_n^{cs}}\left(J\right)\mathrm{and}\overline{R_n^{cs}}\left(\lambda J\right)=\lambda \underline{R_n^{cs}}\left(J\right).$$

Therefore, in both cases,

$$\begin{array}{cc}\hfill {\parallel \lambda J\parallel }_R=& \underset{n\in \mathbb{N}}{\sup }\max \left\{\left|\underline{R_n^{cs}}\left(\lambda J\right)\right|,\left|\overline{R_n^{cs}}\left(\lambda J\right)\right|\right\}=\underset{n\in \mathbb{N}}{\sup }\max \left\{\left|\lambda \right|\left|\underline{R_n^{cs}}\left(J\right)\right|,\left|\lambda \right|\left|\overline{R_n^{cs}}\left(J\right)\right|\right\}\hfill \\ & =\left|\lambda \right|\underset{n\in \mathbb{N}}{\sup }\max \left\{\left|\underline{R_n^{cs}}\left(J\right)\right|,\left|\overline{R_n^{cs}}\left(J\right)\right|\right\}=\left|\lambda \right|{\parallel J\parallel }_R.\hfill \end{array}$$

Consequently, ${∥\lambda J∥}_R=\left|\lambda \right|{∥J∥}_R$.

N4- Let $\left(J_k\right),\left(K_k\right)\in R_I^{cs},$ and suppose that $J_k⪯K_k.$ By the definition of the partial order, this means that $J_k\subseteq K_k$ for every $k\in \mathbb{N}$. Writing $J_k=\left[\underline{J_k},\overline{J_k}\right]$ and $K_k=\left[\underline{K_k},\overline{K_k}\right]$, we obtain $\underline{K_k}\le \underline{J_k}$ and $\overline{J_k}\le \overline{K_k}.$ That is,

$$\underline{S_n}\left(K\right)\le \underline{S_n}\left(J\right)\mathrm{and}\overline{S_n}\left(J\right)\le \overline{S_n}\left(K\right).$$

Therefore, for every $n\in \mathbb{N}$, $S_n\left(J\right)\subseteq S_n\left(K\right)$. Since $p_k>0$, these inequalities are preserved under the Riesz means. Therefore,

$$\underline{R_n^{cs}}\left(K\right)\le \underline{R_n^{cs}}\left(J\right)\mathrm{and}\overline{R_n^{cs}}\left(J\right)\le \overline{R_n^{cs}}\left(K\right).$$

That is $R_n^{cs}\left(J\right)\subseteq R_n^{cs}\left(K\right)$. Since, for any intervals $A\subseteq B,$

$$\max \left\{\left|\underline{A}\right|,\left|\overline{A}\right|\right\}\le \max \left\{\left|\underline{B}\right|,\left|\overline{B}\right|\right\}.$$

It follows that

$$\max \left\{\left|\underline{R_n^{cs}}\left(J\right)\right|,\left|\overline{R_n^{cs}}\left(J\right)\right|\right\}\le \max \left\{\left|\underline{R_n^{cs}}\left(K\right)\right|,\left|\overline{R_n^{cs}}\left(K\right)\right|\right\}.$$

Taking the supremum over $n\in \mathbb{N}$, we obtain ${∥J∥}_R\le {∥K∥}_R$.

N5- Let $\left(J_k\right),\left(K_k\right)\in R_I^{cs}$. Assume that for every $\epsilon >0$, there exists $\left(J_{\epsilon ,k}\right)\in R_I^{cs}$ such that $J\preccurlyeq K+J_{\epsilon }$ and ${∥J_{\epsilon }∥}_R\le \epsilon .$

According to the defined partial order relation, $J\preccurlyeq K+J_{\epsilon }$ means that $J_k\subseteq K_k+J_{\epsilon ,k}$ for every $k\in \mathbb{N}$. If $J_k=\left[\underline{J_k},\overline{J_k}\right]$, $K_k=\left[\underline{K_k},\overline{K_k}\right]$ and $J_{\epsilon ,k}=\left[\underline{J_{\epsilon ,k}},\overline{J_{\epsilon ,k}}\right]$, then

$$\left[\underline{J_k},\overline{J_k}\right]\subseteq \left[\underline{K_k},\overline{K_k}\right]+\left[\underline{J_{\epsilon ,k}},\overline{J_{\epsilon ,k}}\right]\Rightarrow \left[\underline{J_k},\overline{J_k}\right]\subseteq \left[\underline{K_k}+\underline{J_{\epsilon ,k}},\overline{K_k}+\overline{J_{\epsilon ,k}}\right].$$

Therefore, $\underline{K_k}+\underline{J_{\epsilon ,k}}\le \underline{J_k}\mathrm{and}\overline{J_k}\le \overline{K_k}+\overline{J_{\epsilon ,k}}$ are obtained. Since ${∥J_{\epsilon }∥}_R\le \epsilon$, for each fixed $k\in \mathbb{N}$, $J_{\epsilon ,k}\to \left[0,0\right]\left(\epsilon \to 0^{+}\right).$

By the definition of the max norm on I,

$$J_{\epsilon ,k}\to \left[0,0\right]\mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if}\underline{J_{\epsilon ,k}}\to 0\mathrm{and}\overline{J_{\epsilon ,k}}\to 0.$$

Therefore, from the above inequalities, $\underline{K_k}\le \underline{J_k}$ and $\overline{J_k}\le \overline{K_k}$ are obtained. Consequently, $J_k\subseteq K_k.$ For all $k\in \mathbb{N}$, and according to the defined partial order relation, $J\preccurlyeq K$ is obtained. □

Corollary 1. 

The metric associated with the norm ${∥.∥}_R$ coincides with the metric d defined in Theorem 2. More precisely, for every $\left(J_k\right),\left(K_k\right)\in R_I^{cs}$,

$$\begin{array}{cc}\hfill & d_R\left(J,K\right)=\underset{n\in \mathbb{N}}{\sup }\max \left\{\left|\underline{R_n^{cs}}\left(J\right)-\underline{R_n^{cs}}\left(K\right)\right|,\left|\overline{R_n^{cs}}\left(J\right)-\overline{R_n^{cs}}\left(K\right)\right|\right\}\hfill \\ \hfill & =\underset{n\in \mathbb{N}}{\sup }h\left(R_n^{cs}\left(J\right),R_n^{cs}\left(K\right)\right)=d\left(J,K\right).\hfill \end{array}$$

In particular, if $\theta =\left(\left[0,0\right],\left[0,0\right],\dots \right)$ denotes the zero element of $R_I^{cs}$, then

$$d_R\left(J,\theta \right)=\underset{n\in \mathbb{N}}{\sup }\max \left\{\left|\underline{R_n^{cs}}\left(J\right)\right|,\left|\overline{R_n^{cs}}\left(J\right)\right|\right\}={∥J∥}_R.$$

Therefore, the metric $d_R$ associated with the norm agrees exactly with the metric d previously defined in Theorem 2. Since $\left(R_I^{cs},d\right)$ is complete and $\left(R_I^{cs},{∥.∥}_R\right)$ is a normed quasilinear space, $R_I^{cs}$ is a Banach quasilinear space.

Theorem 7. 

$R_I^{cs}$ is a solid floored quasilinear space.

Proof. 

Let $U={\left(U_k\right)}_{k\in \mathbb{N}}={\left(\left[\underline{U_k},\overline{U_k}\right]\right)}_{k\in \mathbb{N}}\in R_I^{cs}$ be arbitrary. Since $U\in R_I^{cs}$, there exists an interval $L=\left[\underline{L},\overline{L}\right]\in I$ such that $h\left(R_n^{cs}\left(U\right),L\right)\to 0\left(n\to \infty \right)$. Since $p_k>0$, the lower and upper endpoints of $R_I^{cs}\left(U\right)$ are given by

$$R_n^{cs}\left(U\right)=\left[{\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{\displaystyle \sum _{i=0}^k}\underline{U_i},{\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{\displaystyle \sum _{i=0}^k}\overline{U_i}\right].$$

Therefore, by the definition of the Hausdorff metric,

$$h\left(R_n^{cs}\left(U\right),L\right)=\max \left\{\left|{\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{\displaystyle \sum _{i=0}^k}\underline{U_i}-\underline{L}\right|,\left|{\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{\displaystyle \sum _{i=0}^k}\overline{U_i}-\overline{L}\right|\right\}.$$

Since this maximum tends to zero, both terms tend to zero. Hence,

$${\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{\displaystyle \sum _{i=0}^k}\underline{U_i}=\underline{L}\mathrm{and}{\displaystyle \frac{1}{P_n}}{\displaystyle \sum _{k=0}^n}{p}_k{\displaystyle \sum _{i=0}^k}\overline{U_i}=\overline{L}.$$

Thus, the real series with terms $\underline{U_k}$ and $\overline{U_k}$ are Riesz-summable to $\underline{L}$ and $\overline{L}$, respectively. Now define the degenerate interval sequences

$$\underline{U}={\left(\left[\underline{U_k},\underline{U_k}\right]\right)}_{k\in \mathbb{N}}\mathrm{and}\overline{U}={\left(\left[\overline{U_k},\overline{U_k}\right]\right)}_{k\in \mathbb{N}}.$$

The preceding limits show that $\underline{U},\overline{U}\in R_I^{cs}$. Moreover, since all their components are degenerate intervals, $\underline{U}$ and $\overline{U}$ are regular elements of $R_I^{cs}$. In addition, for every $k\in \mathbb{N}$,

$$\left[\underline{U_k},\underline{U_k}\right]\subseteq U_k\mathrm{and}\left[\overline{U_k},\overline{U_k}\right]\subseteq U_k.$$

Hence, $\underline{U}\preccurlyeq U$ and $\overline{U}\preccurlyeq U.$ For each $k\in \mathbb{N}$, the least interval containing both endpoint intervals is $U_k$; that is,

$$\underset{\preccurlyeq }{\sup }\left\{\left[\underline{U_k},\underline{U_k}\right],\left[\overline{U_k},\overline{U_k}\right]\right\}=\left[\underline{U_k},\overline{U_k}\right]=U_k.$$

Since the order and the supremum are considered coordinatewise, it follows that $\underset{\preccurlyeq }{\sup }\left\{\underline{U},\overline{U}\right\}=U.$

Let $A_U=\left\{A\in {\left(R_I^{cs}\right)}_r:A\preccurlyeq U\right\}.$ Since $\underline{U},\overline{U}\in A_U$, we have

$$U=\underset{\preccurlyeq }{\sup }\left\{\underline{U},\overline{U}\right\}\preccurlyeq \underset{\preccurlyeq }{\sup }{A}_U.$$

On the other hand, every $A\in A_U$ satisfied $A\preccurlyeq U$, and therefore $\underset{\preccurlyeq }{\sup }{A}_U\preccurlyeq U$.

Consequently, $U=\underset{\preccurlyeq }{\sup }\left\{A\in {\left(R_I^{cs}\right)}_r:A\preccurlyeq U\right\}.$

Since $U\in R_I^{cs}$ was arbitrary, $R_I^{cs}$ is a solid-floored quasilinear space. Thus the proof is complete. □

4. Conclusions

In this study, Riesz summability was extended to interval-valued series by applying the Riesz transformation to their sequences of partial sums, and the corresponding interval-valued Riesz-summable series space $R_I^{cs}$ was introduced. This passage from sequences to series is not merely formal, since the cumulative behavior represented by the partial sums requires an independent construction and analysis of the resulting series space. The uniqueness of the Riesz sum was established, and the relationship between ordinary convergence and Riesz summability was clarified. In particular, every convergent interval-valued series was shown to be Riesz-summable to the same interval. Moreover, an example of a nonconvergent but Riesz-summable interval-valued series demonstrated that $I^{cs}⊊R_I^{cs}$.

A metric defined through the Hausdorff distances between the Riesz means was introduced on $R_I^{cs}$, and the resulting metric space was proved to be complete. The space $R_I^{cs}$ was also shown to be a quasilinear space and, when equipped with the introduced norm, a normed quasilinear space. Furthermore, the metric associated with this norm was proved to coincide with the previously defined metric. The solid-floored property of $R_I^{cs}$ was also established through the lower and upper endpoint sequences of its elements. Together, these results describe the principal metric, algebraic, topological, normed, and order-related properties of the interval-valued Riesz-summable series space.

The present work therefore provides a systematic theoretical framework for Riesz summability in the setting of interval-valued series and clarifies its distinction from the corresponding sequence setting. The results contribute to the interaction among interval analysis, summability theory, and quasilinear space theory and provide a basis for further investigations of interval-valued series involving uncertainty.

Possible directions for future research include the construction and structural analysis of interval-valued series spaces associated with Abel, Cesàro, matrix, statistical, ideal, and lacunary summability methods. Extensions to double interval-valued series and other generalized interval-valued series spaces may also be considered. Such developments may broaden the theoretical foundations needed for the analysis of mathematical models involving interval-valued or uncertain data.