Sequence spaces $M(\phi)$ and $N(\phi)$ with application in clustering

Abstract

Distance measures play a central role in evolving the clustering technique. Due to the rich mathematical background and natural implementation of $l_{p}$ distance measures, researchers were motivated to use them in almost every clustering process. Beside $l_{p}$ distance measures, there exist several distance measures. Sargent introduced a special type of distance measures $m(\phi)$ and $n(\phi)$ which is closely related to $l_{p}$. In this paper, we generalized the Sargent sequence spaces through introduction of $M(\phi)$ and $N(\phi)$ sequence spaces. Moreover, it is shown that both spaces are BK-spaces, and one is a dual of another. Further, we have clustered the two-moon dataset by using an induced $M(\phi)$-distance measure (induced by the Sargent sequence space $M(\phi)$) in the k-means clustering algorithm. The clustering result established the efficacy of replacing the Euclidean distance measure by the $M(\phi)$-distance measure in the k-means algorithm.

1 Introduction

Clustering is a well-known procedure to deal with an unsupervised learning problem appearing in pattern recognition. Clustering is a process of organizing data into groups called clusters so that objects in the same cluster are similar to one another, but are dissimilar to objects in other clusters [1]. The main contribution in the field of clustering analysis was the pioneering work of MacQueen [1] and Bezdek [2]. They had introduced highly significant clustering algorithms such as k-means [1] and fuzzy c-means [2]. Among all clustering algorithms, k-means is the simplest unsupervised clustering algorithm that makes use of a minimum distance from the center, and it has many applications in scientific and industrial research [3–6] (for more information about the k-means clustering algorithm, see Section 5). K-means algorithm is distance dependent, so its outputs vary with changing distance measures. Among all distance measures, a clustering process was usually carried out through the Euclidean distance measure [7], but many times it failed to offer good results. In this paper, we define $M(\phi)$- and $N(\phi)$-distance measure. Further, $M(\phi )$-distance is used to cluster two-moon dataset. The output result is compared with the result of Euclidean distance measure to show the efficacy of $M(\phi)$-distance over the Euclidean distance measure. $M(\phi )$ and $N(\phi)$-distance measures are the generalization of $m(\phi)$- and $n(\phi)$-distance measures introduced by Sargent [8] and further studied by Mursaleen [9, 10] (to know more about $m(\phi )$ and $n(\phi)$, refer to [8–10]). The $M(\phi)$ and $N(\phi)$ spaces are closely related to $l_{p}$ distance measures. $l_{p}$ measures and its variance are mostly used to solve the problems evolving in the fields of Market prediction [11], Machine Learning [12], Pattern Recognition [13], Clustering [20] etc.

Throughout the paper, by ω we denote the set of all real or complex sequences. Moreover, by $l_{\infty}$, c and $c_{0}$ we denote the Banach spaces of bounded, convergent and null sequences, respectively; and let $l_{p}$ be the Banach space of absolutely p-summable sequences with p-norm ${\Vert \cdot \Vert } _{p}$. For the following notions, we refer to [14, 15]. A double sequence $x = (x_{jk})$ of real or complex numbers is said to be bounded if $\Vert x \Vert _{\infty} < \infty$, the space of all bounded double sequences is denoted by $\mathcal{L}_{\infty}$. A double sequence $x = (x_{jk})$ is said to converge to the limit L in Pringsheim’s sense (shortly, convergent to L) if for every $\varepsilon> 0$, there exists an integer N such that $\vert x_{jk} - L\vert < \varepsilon$ whenever $j,k > N$. In this case L is called the p-limit of x. If in addition $x \in\mathcal{L}_{\infty}$, then x is said to be boundedly convergent to L in Pringsheim’s sense (shortly, bp-convergent to L). A double sequence $x = (x_{jk})$ is said to converge regularly to L (shortly, r-convergent to L) if x is p-convergent and the limits $x_{j}: = \lim_{k}x_{jk}$ ($j \in\mathbb {N}$) and $x^{k}: = \lim_{j}x_{jk}$ ($k \in\mathbb{N}$) exist. Note that in this case the limits $\lim_{j}\lim_{k}x_{jk}$ and $\lim_{k}\lim_{j}x_{jk}$ exist and are equal to the p-limit of x. In general, for any notion of convergence ν, the space of all ν-convergent double sequences will be denoted by $\mathcal{C}_{\nu}$ and the limit of a ν-convergent double sequence x by $\nu\textrm{-} \lim_{j,k}x_{jk}$, where $\nu\in\{ p,\mathit{bp},r\}$.

Let Ω denote a vector space of all double sequences with the vector space operations defined coordinate-wise. Vector subspaces of Ω are called double sequence spaces. Let us consider a double sequence $x = \{ x_{mn}\}$ and define the sequence $s = \{ s_{mn}\}$ via x by

$$s_{mn}: = \sum_{i,j}^{m,n} x_{ij}\quad (m,n \in\mathbb{N} ). $$

Then the pair $(x,s)$ and the sequence $s = \{ s_{mn}\}$ are called a double series and a sequence of partial sums of the double series, respectively. Let λ be the space of double sequences converging with respect to some linear convergence rule $\mu\textrm{-} \lim:\lambda\to \mathbb{R}$. The sum of a double series $\sum_{i,j = 1}^{\infty,\infty} x_{ij}$ with respect to this rule is defined by $\mu\textrm{-} \sum_{i,j = 1}^{\infty,\infty} x_{ij}: = \mu\textrm{-} \lim s_{mn}$. Başar and Şever introduced the space $L_{p}$ in [16]

$$L_{p}: = \biggl\{ \{ x_{mn}\} \in\Omega:\sum _{m,n} \vert x_{mn}\vert ^{p} < \infty \biggr\} \quad (1 \le p < \infty) $$

corresponding to the space $l_{p}$ for $p \ge1$ and examined some of its properties. Altay and Başar [17] have generalized the spaces of double sequences $L_{\infty}$, $C_{p}$ and $C_{\mathit{bp}}$ to

$$\begin{aligned}& L_{\infty} (t) = \Bigl\{ \{ x_{mn}\} \in\Omega:\sup _{m,n \in\mathbb{N}} \vert x_{mn}\vert ^{t_{mn}} < \infty \Bigr\} , \\& C_{p}(t) = \Bigl\{ \{ x_{mn}\} \in\Omega:p \textrm{-} \lim _{m,n \to \infty} \vert x_{mn} - \ell \vert ^{t_{mn}} = 0 \Bigr\} , \end{aligned}$$

and

$$C_{\mathit{bp}}(t) = C_{p}(t) \cap L_{\infty} (t), $$

respectively, where $t = \{ t_{mn}\}$ is the sequence of strictly positive reals $t_{mn}$. In the case $t_{mn} = 1$, for all $m,n \in\mathbb{N}$, $L_{\infty} (t)$, $C_{p}(t)$ and $C_{\mathit{bp}}(t)$ reduce to the sets $L_{\infty}$, $C_{p}$ and $C_{\mathit{bp}}$, respectively. Further, let C be the space whose elements are finite sets of distinct positive integers. Given any element σ of C, we denote by $c(\sigma)$ the sequence $\{ c_{n}(\sigma)\}$ which is such that $c_{n}(\sigma) = 1$ if $n \in \sigma$, $c_{n}(\sigma) = 0$ otherwise. Further, let

$$C_{s} = \Biggl\{ \sigma\in C:\sum_{n = 1}^{\infty} c_{n}(\sigma) \le s \Biggr\} $$

be the set of those σ whose support has cardinality at most s, and

$$\Phi= \biggl\{ \phi= \{ \phi_{n}\} \in\omega: \phi_{1} > 0,\Delta\phi_{n} \ge0\mbox{ and }\Delta \biggl( \frac{\phi_{n}}{n} \biggr) \le0\ (n = 1,2, \ldots) \biggr\} , $$

where $\Delta\phi_{n} = \phi_{n} - \phi_{n - 1}$ and $\phi _{0} = 0$.

For $\phi\in\Phi$, the following sequence spaces were introduced and studied in [8] by Sargent and further studied by Mursaleen in [9, 10]:

$$m(\phi) = \biggl\{ x = \{ x_{n}\} \in\omega:\sup _{s \ge 1} \sup_{\sigma\in C_{s}} \biggl( \frac{1}{\phi_{s}} \sum_{n \in\sigma} \vert x_{n}\vert \biggr) < \infty \biggr\} , $$

and

$$n(\phi) = \Biggl\{ x = \{ x_{n}\} \in\omega:\sup _{u \in S(x)} \Biggl( \sum_{m,n = 1,1}^{\infty,\infty} \vert u_{n}\vert \Delta\phi_{n} \Biggr) < \infty \Biggr\} . $$

Remark 1.1

• (i) The spaces $m(\phi)$ and $n(\phi)$ are BK-spaces with their usual norms.

• (ii) If $\phi_{n} = 1$ ($n = 1,2,3,\ldots$), then $m(\phi) = l_{1}$ [$n(\phi) = l_{\infty}$], and if $\phi_{n} = n$ ($n = 1,2,3,\ldots$), then $m(\phi) = l_{\infty}$ [$n(\phi) = l_{1}$].

• (iii) $l_{1} \subseteq m(\phi) \subseteq l_{\infty}$ [$l_{1} \subseteq n(\phi) \subseteq l_{\infty}$] for all $\phi\in\Phi$.

• (iv) For any $\phi\in\Phi$, $m(\phi)\neq l_{p}$ [$n(\phi)\neq l_{q}$], $1 < p < \infty$.

In this paper, we define Sargent’s spaces for double sequences $x = \{ x_{mn}\}$. For this we first suppose U to be the set whose elements are finite sets of distinct elements of $\mathbb{N} \times\mathbb{N}$ obtained by $\sigma\times\varsigma$, where $\sigma\in C_{s}$ and $\varsigma\in C_{t}$ for each $s,t \ge1$. Therefore, any element ζ of U means $(m,n)$; $m \in\sigma$ and $n \in\varsigma$ having cardinality at most st, where s is the cardinality with respect to m and t is the cardinality with respect to n. Given any element ζ of U, we denote by $c(\zeta)$ the sequence $\{ c_{mn}(\zeta)\}$ such that

$$c_{mn}(\zeta) = \textstyle\begin{cases} 1; & \mbox{if } (m,n) \in\zeta,\\ 0; & \mbox{otherwise}. \end{cases} $$

Further, we write

$$U_{st} = \Biggl\{ \zeta\in U:\sum_{m,n = 1}^{\infty,\infty} c_{mn}(\zeta) \le st \Biggr\} $$

for the set of those ζ whose support has cardinality at most st; and

$$\Theta= \biggl\{ \phi= \{ \phi_{mn}\} \in\Omega: \phi_{11} > 0,\Delta_{11}\phi_{mn} \ge0\mbox{ and } \Delta_{11} \biggl( \frac{\phi_{mn}}{mn} \biggr) \le0\ (m,n = 1,2, \ldots) \biggr\} , $$

where $\Delta_{11}\phi_{mn} = \phi_{mn} - \phi_{m - 1,n} - \phi_{m,n - 1} + \phi_{m - 1,n - 1}$ and $\phi_{00}$, $\phi_{0m}$, $\phi_{n0} = 0$, $\forall m,n \in\mathbb{I}^{ +}$. Throughout the paper, we write $\sum_{m,n \in\zeta}$ for $\sum_{m \in \sigma} \sum_{n \in\varsigma}$, and $S(x)$ is used to denote the set of all double sequences that are rearrangements of $x = \{ x_{mn}\} \in \Omega$. For $\phi\in\Theta$, we define the following sequence spaces:

$$M(\phi) = \biggl\{ x = \{ x_{mn}\} \in\Omega: \Vert x \Vert _{M(\phi )} = \sup_{s,t \ge1}\sup_{\zeta\in U_{st}} \biggl( \frac{1}{\phi_{st}}\sum_{m,n \in\zeta} \vert x_{mn} \vert \biggr) < \infty \biggr\} $$

and

$$N(\phi) = \Biggl\{ x = \{ x_{mn}\} \in\Omega: \Vert x \Vert _{N(\phi)} = \sup_{u \in S(x)} \Biggl(\sum _{m,n = 1}^{\infty,\infty} \vert u_{mn}\vert \Delta_{11}\phi_{mn} \Biggr) < \infty \Biggr\} . $$

Then the distances between $x = \{ x_{mn}\}$ and $y = \{ y_{mn}\}$ induced by $M(\phi)$ and $N(\phi)$ can be expressed as

$$d_{M(\phi)} = \sup_{s,t \ge1}\sup_{\zeta \in U_{st}} \biggl( \frac{1}{\phi_{st}}\sum_{m,n \in\zeta} \vert x_{mn} - y_{mn}\vert \biggr) $$

and

$$d_{N(\phi)} = \sup_{u,v \in S(x)} \Biggl( \sum _{m,n = 1}^{\infty,\infty} \vert u_{mn} - v_{mn} \vert \Delta_{11}\phi_{mn} \Biggr). $$

Remark 1.2

If $\phi_{st} = 1$ ($s,t = 1,2,3,\ldots$), then $M(\phi) = L_{1}$ [$N(\phi) = L_{\infty}$], and if $\phi_{st} = st$ ($s,t = 1,2,3,\ldots$), then $M(\phi) = L_{\infty}$ [$N(\phi) = L_{1}$].

We now state the following known results of [18] for single sequences (series) which can also be proved easily for double sequences (series).

Lemma 1.1

If the series $\sum u_{n}x_{n}$ is convergent for every x of a BK-space E, then the functional $\sum_{n = 1}^{\infty} u_{n}x_{n}$ is linear and continuous in E.

Lemma 1.2

If E and F are BK-spaces, and if $E \subseteq F$, then there is a real number K such that, for all x of E,

$$\Vert x \Vert _{F} \le K\Vert x \Vert _{E}. $$

2 Properties of the spaces $M(\phi)$ and $N(\phi)$

Theorem 2.1

The space $M(\phi)$ is a BK-space with the norm.

$$ \Vert x \Vert _{M(\phi)} = \sup_{s,t \ge 1}\sup _{\zeta\in U_{st}} \frac{1}{\phi_{st}} \biggl( \sum _{m,n \in\zeta} \vert x_{mn}\vert \biggr). $$

(2.1)

Proof

It is a routine verification to show that $M(\phi)$ is a normed space with the given norm (2.1), and so we omit it. Now, we proceed to showing that $M(\phi)$ is complete. Let $\{ x^{l}\}$ be a Cauchy sequence in $M(\phi)$, where $x^{l} = \{ x_{mn}^{l}\}_{m,n = 1,1}^{\infty,\infty}$ for every fixed $l \in\mathbb{N}$. Then, for a given $\varepsilon> 0$, there exists a positive integer $n_{0}(\varepsilon) > 0$ such that

$$\bigl\Vert x^{l} - x^{r} \bigr\Vert _{M(\phi)} = \sup_{s,t \ge 1}\sup_{\zeta\in U_{st}}\frac{1}{\phi_{st}} \biggl( \sum_{m,n \in\zeta} \bigl\vert x_{mn}^{l} - x_{mn}^{r} \bigr\vert \biggr) < \varepsilon $$

for all $l,r > n_{0}(\varepsilon)$, which yields, for each fixed $s,t \ge 1$ and $\zeta\in U_{st}$,

$$ \sum_{m,n \in\zeta} \bigl\vert x_{mn}^{l} - x_{mn}^{r} \bigr\vert \le\varepsilon \phi_{11}\quad \mbox{for all }l,r > n_{0}(\varepsilon). $$

(2.2)

Therefore

$$ \biggl\vert \sum_{m,n \in\zeta} \bigl\vert x_{mn}^{l} \bigr\vert - \sum_{m,n \in\zeta} \bigl\vert x_{mn}^{r} \bigr\vert \biggr\vert < \varepsilon\phi_{11} \quad \mbox{for all }l,r > n_{ 0} ( \varepsilon). $$

(2.3)

This means that $\{ \sum_{m,n \in\zeta} \vert x_{mn}^{l}\vert \} _{l \in\mathbb{N}}$ is a Cauchy sequence in $\mathbb{R}$ for every fixed $s,t \ge1$ and $\zeta \in U_{st}$. Since $\mathbb{R}$ is complete, it converges, say

$$\sum_{m,n \in\zeta} \bigl\vert x_{mn}^{l} \bigr\vert \to\sum_{m,n \in\zeta} \vert x_{mn} \vert \quad \mbox{as }l \to\infty. $$

Since absolute convergence implies convergence in $\mathbb{R}$, hence

$$ \sum_{m,n \in\zeta} x_{mn}^{l} \to\sum _{m,n \in\zeta} x_{mn} \quad \mbox{as } l \to\infty. $$

(2.4)

Hence we have

$$ \lim_{l \to\infty} \biggl\Vert \sum_{m,n \in\zeta} x_{mn}^{l} - \sum_{m,n \in\zeta} x_{mn} \biggr\Vert _{M(\phi)} = 0. $$

(2.5)

Let $y^{l} = \sum_{m,n \in\zeta} \vert x_{mn}^{l}\vert$. Then $\{ y^{l}\} \in l_{\infty}$. Therefore

$$\sup_{l \in\mathbb{N}} \sum_{m,n \in\zeta} \bigl\vert x_{mn}^{l} \bigr\vert \le k. $$

Since $\sum_{m,n \in\zeta} \vert x_{mn}\vert \le\sum_{m,n \in \zeta} \vert x_{mn} - x_{mn}^{l}\vert + \sum_{m,n \in\zeta} \vert x_{mn}^{l}\vert \le\varepsilon \phi_{11} + k$, it follows that $x = \{ x_{mn}\} \in M(\phi)$. Since $\{ x^{l}\}_{l \in\mathbb{N}}$ was an arbitrary Cauchy sequence, the space $M(\phi)$ is complete. Now we prove that $M(\phi)$ has continuous coordinate projections $p_{mn}$, where $p_{mn}:\Omega\to K$ and $p_{mn}(x) = x_{mn}$. The coordinate projections $p_{mn}$ are continuous since $\vert x_{mn}\vert \le\sup_{s,t \ge1}\sup_{\zeta\in U_{st}}\phi_{st}\Vert x \Vert _{M(\phi)}$ for each $m,n \in \mathbb{N}$. □

Remark 2.1

The space $N(\phi)$ is a BK-space with the norm

$$\Vert x \Vert _{N(\phi)} = \sup_{u \in S(x)} \Biggl( \sum _{m,n = 1}^{\infty,\infty} \vert u_{mn}\vert \Delta_{11}\phi_{mn} \Biggr). $$

Lemma 2.1

• (i) If $x \in M(\phi)$ [$x \in N(\phi)$] and $u \in S(x)$, then $u \in M(\phi)$ [$u \in N(\phi)$] and $\Vert u\Vert = \Vert x\Vert$.

• (ii) If $x \in M(\phi)$ [$x \in N(\phi)$] and $\vert u_{mn}\vert \le \vert x_{mn}\vert$ for every positive integer m, n, then $u \in M(\phi)$ [$u \in N(\phi)$] and $\Vert u\Vert \le \Vert x\Vert$.

Proof

(i) Let $x \in M(\phi)$, then $\sup_{s,t \ge1} \sup_{\zeta\in U_{st}}\frac{1}{\phi_{st}}\sum_{m,n \in\zeta} \vert x_{mn}\vert < \infty$. So, we have

$$\frac{1}{\phi_{st}}\sum_{m,n \in\zeta} \vert x_{mn} \vert < \infty \quad \mbox{for each }\zeta\in U_{st}\mbox{ and }s,t \ge1. $$

Since the sum of a finite number of terms remains the same for all the rearrangements,

$$\frac{1}{\phi_{st}}\sum_{m,n \in\zeta} \vert u_{mn} \vert = \frac{1}{\phi_{st}}\sum_{m,n \in\zeta} \vert x_{mn}\vert \quad \mbox{for each }u \in S(x)\mbox{ and }\zeta\in U_{st}, s,t \ge1. $$

Hence

$$\sup_{s,t \ge1}\sup_{\zeta\in U_{st}}\frac{1}{\phi_{st}}\sum _{m,n \in\zeta} \vert u_{mn}\vert = \sup _{s,t \ge1} \sup_{\zeta\in U_{st}}\frac{1}{\phi_{st}}\sum _{m,n \in\zeta} \vert x_{mn}\vert < \infty, $$

thus $u \in M(\phi)$ and $\Vert u\Vert = \Vert x\Vert$.

(ii) By using the definition, easy to prove. □

Theorem 2.2

For arbitrary $\phi\in\Theta$, we have $\Delta_{11}\phi\in M(\phi)$ and $\Vert \Delta_{11}\phi \Vert _{M(\phi)} \le2$.

Proof

Let s and t be arbitrary positive integers, let $\sigma,\varsigma\in U_{st}$, and let $\tau_{1}$, $\tau_{2}$ constitute the element of σ and ς exceed by s and t respectively, also from the definition we have $\Delta_{11}\phi\ge0$ and $\Delta_{11} ( \frac{\phi_{mn}}{mn} ) \le0$. Then

$$\begin{aligned} \sum_{n \in\sigma,m \in\varsigma} \vert \Delta_{11} \phi_{mn} \vert \le& \sum_{n = 1,m = 1}^{s,t} \Delta_{11}\phi_{mn} + \sum_{n \in \tau_{1},m \in\tau_{2}} \Delta_{11}\phi_{mn} \\ \leq& \phi_{st} + \sum_{n \in\tau_{1},m \in\tau_{2}} \biggl( \frac{\phi_{m - 1,n - 1}}{(m - 1)(n - 1)} \biggr) \\ \leq & \phi_{st} + \left . \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \frac{\phi_{st}}{st} & + \frac{\phi_{s,t + 1}}{s(t + 1)} & + \frac{\phi_{s,t + 2}}{s(t + 2)} & +. &. \\ \frac{\phi_{s + 1,t}}{(s + 1)t} & + \frac{\phi_{s + 1,t + 1}}{(s + 1)(t + 1)} & + \frac{\phi_{s + 1,t + 2}}{(s + 1)(t + 2)} & +. &. \\ + & + & + &. & \\ . &. &. &. & \end{array}\displaystyle \right \} \quad \max(st) \textrm{-terms} \\ \leq& \phi_{st} + st\frac{\phi_{st}}{st} = 2\phi_{st}. \end{aligned}$$

 □

Lemma 2.2

If $x \in M(\phi)$ and $\{ c_{11},c_{12}, \ldots,c_{1n},c_{21},c_{22}, \ldots,c_{2n}, \ldots,c_{m1},c_{m2}, \ldots,c_{mn}\}$ is a rearrangement of $\{ b_{11},b_{12}, \ldots,b_{1n},b_{21},b_{22}, \ldots,b_{2n}, \ldots,b_{m1},b_{m2}, \ldots,b_{mn}\}$ such that $\vert c_{11}\vert \ge \vert c_{12}\vert \ge \cdots\ge \vert c_{1n}\vert$, $\vert c_{21}\vert \ge \vert c_{22}\vert \ge\cdots\ge \vert c_{2n}\vert ,\dots, \vert c_{m1}\vert \ge \vert c_{m2}\vert \ge\cdots\ge \vert c_{mn}\vert$ and $\vert c_{11}\vert \ge \vert c_{21}\vert \ge\cdots\ge \vert c_{m1}\vert$, $\vert c_{12}\vert \ge \vert c_{22}\vert \ge\cdots \ge \vert c_{m2}\vert ,\vert c_{n1}\vert \ge \vert c_{n2}\vert \ge\cdots\ge \vert c_{nm}\vert$, then

$$\sum_{i,j = 1,1}^{m,n} \vert b_{ij}x_{ij} \vert \le \Vert x\Vert _{M(\phi )}\sum_{i,j = 1,1}^{m,n} \vert c_{ij}\vert \Delta_{11}\phi_{ij}. $$

Proof

In view of Lemma 2.1(i), it is sufficient to consider the case when $b_{ij} = c_{ij}$ ($i = 1,2, \ldots,m$; $j = 1,2, \ldots,n$). Then writing $X_{mn} = \sum_{i,j = 1}^{m,n} \vert x_{ij}\vert$, we get

$$\begin{aligned} \sum_{i,j = 1,1}^{m,n} \vert b_{ij}x_{ij} \vert =& \sum_{i = 1}^{m - 1} \sum _{j = 1}^{n - 1} \bigl( \vert c_{ij}\vert - \vert c_{i,j + 1}\vert - \vert c_{i + 1,j}\vert +\vert c_{i + 1,j + 1}\vert \bigr)X_{ij} +\vert c_{mn}\vert X_{mn} \\ \le& \Vert x\Vert _{M(\phi)}\sum_{i = 1}^{m - 1} \sum_{j = 1}^{n - 1} \bigl( \vert c_{ij}\vert - \vert c_{i,j + 1}\vert - \vert c_{i + 1,j}\vert + \vert c_{i + 1,j + 1}\vert \bigr) \phi_{ij} + \Vert x\Vert _{M(\phi)}\vert c_{mn} \vert X_{mn} \\ =& \Vert x\Vert _{M(\phi)}\sum_{i,j = 1,1}^{m,n} \vert c_{ij}\vert \Delta_{11}\phi_{ij}. \end{aligned}$$

Hence we have $\sum_{i,j = 1,1}^{m,n} \vert b_{ij}x_{ij}\vert \le \Vert x\Vert _{M(\phi)}\sum_{i,j = 1,1}^{m,n} \vert c_{ij}\vert \Delta_{11}\phi_{ij}$. □

Theorem 2.3

In order that $\sum u_{ij}x_{ij}$ be convergent [absolutely convergent] whenever $x \in M(\phi)$, it is necessary and sufficient that $u \in N(\phi)$. Further, if $x \in M(\phi)$ and $u \in N(\phi)$, then

$$ \sum_{i,j = 1,1}^{\infty,\infty} \vert u_{ij}x_{ij} \vert \le \Vert u\Vert _{N(\phi)} \Vert x\Vert _{M(\phi)}. $$

(2.6)

Proof

Necessity. We now suppose that $\sum u_{ij}x_{ij}$ is convergent whenever $x \in M(\phi)$, then from Lemma 1.1 we have

$$\Biggl\vert \sum_{i,j = 1,1}^{\infty,\infty} u_{ij}x_{ij} \Biggr\vert \le K \Vert x\Vert _{M(\phi)} $$

for some real number K and all x of $M(\phi)$. In view of Lemma 2.1(ii), we may replace $x_{ij}$ by $x_{ij} \operatorname{sgn} \{ u_{ij}\}$, obtaining

$$ \sum_{i,j = 1,1}^{\infty,\infty} \vert u_{ij}x_{ij} \vert \le K \Vert x\Vert _{M(\phi)}. $$

(2.7)

Let $v \in S(u)$. Then taking x to be a suitable rearrangement of $\Delta_{11}\phi$, it follows from Eq. (2.7) and Theorem 2.2 and Lemma 2.1(i) that

$$\sum_{i,j = 1,1}^{\infty,\infty} \vert v_{ij} \vert \Delta_{11}\phi_{ij} \le4K, $$

and thus $u \in N(\phi)$.

Sufficiency. If $x \in M(\phi)$ and $u \in N(\phi)$, it follows from Lemma 2.2 that for every positive integer m and n,

$$\sum_{i,j = 1,1}^{\infty,\infty} \vert u_{ij}x_{ij} \vert \le \Vert u\Vert _{N(\phi)} \Vert x\Vert _{M(\phi)}. $$

 □

Theorem 2.4

In order that $\sum u_{mn}x_{mn}$ be convergent [absolutely convergent] whenever $x \in N(\phi)$, it is necessary and sufficient that $u \in M(\phi)$.

Proof

Since sufficiency is included in Theorem 2.3, we only consider necessity. We therefore suppose that $\sum u_{mn}x_{mn}$ is convergent whenever $x \in N(\phi)$. By arguments similar to those used in Theorem 2.3, we may therefore have that

$$ \sum_{m,n = 1,1}^{\infty,\infty} \vert u_{mn}x_{mn} \vert \le K \Vert x\Vert _{N(\phi)} $$

(2.8)

for some real number K and all x of $N(\phi)$. Let $x = c(\zeta)$, where $\zeta\in U_{st}$. Then $x \in N(\phi)$, and

$$\Vert x\Vert _{N(\phi)} = \sup_{\xi\in U_{st}}\sum _{m,n \in\xi} \Delta_{11}\phi_{mn} \le4 \phi_{st}, $$

from Theorem 2.2 and Eq. (2.8) we have

$$\sum_{m,n \in\zeta} \vert u_{mn}\vert \le4K \phi_{st}\quad (\zeta\in U_{st}; s,t = 1,2,3, \ldots), $$

and thus $u \in M(\phi)$. □

3 Inclusion relations for $M(\phi)$ and $N(\phi)$

Lemma 3.1

In order that $M(\phi) \subseteq M(\psi)$ [$N(\phi) \supseteq N(\psi)$], it is necessary and sufficient that

$$\sup_{s,t \ge1} \biggl( \frac{\phi_{st}}{\psi_{st}} \biggr) < \infty. $$

Proof

Since each of the spaces $M(\phi)$ and $N(\phi )$ is the dual of the other, by Theorems 2.3 and 2.4, the second version is equivalent to the first. Moreover, sufficiency follows from the definition of an $M(\phi)$ space. We therefore suppose that $M(\phi) \subseteq M(\psi)$. Since $\Delta\phi\in M(\phi)$, it follows that $\Delta \psi\in M(\psi)$, and hence we find that, for every positive integer $s,t \ge1$,

$$\phi_{st} = \sum_{i,j = 1,1}^{s,t} \Delta_{11}\phi_{ij} \le\psi_{st} \Vert \Delta \phi \Vert _{M(\psi)},\quad \mbox{where }\Delta= \Delta_{11}. $$

 □

Theorem 3.1

• (i) $L_{1} \subseteq M(\phi) \subseteq L_{\infty}$ [$L_{1} \subseteq N(\phi) \subseteq L_{\infty}$] for all ϕ of Θ.

• (ii) $M(\phi) = L_{1}$ [$N(\phi) = L_{\infty}$] if and only if $\mathit{bp}\textit{-}\lim_{s,t}\phi_{st} < \infty$.

• (iii) $M(\phi) = L_{\infty}$ [$N(\phi) = L_{1}$] if and only if $\mathit{bp}\textit{-}\lim_{s,t}(\phi_{st}/st) > 0$.

Proof

We prove here the first version, while the second version follows by Theorems 2.3 and 2.4. Since $\phi_{11} \le\phi_{mn} \le mn\phi_{mn}$ for all ϕ of Θ, we have by Lemma 3.1 that (i) is satisfied. Further, from Lemma 3.1, it follows that $M(\phi) \subseteq L_{1}$ if and only if $\sup_{s,t \ge 1}\phi_{st} < \infty$, while $L_{\infty} \subseteq M(\phi)$ if and only if $\sup_{s,t \ge1}(\phi_{st}/st) < \infty$; since the sequences $\{ \phi_{st}\}$ and $\{ st/\phi_{st}\}$ are monotonic, (ii) and (iii) are also satisfied. □

Theorem 3.2

Suppose that $1 < p < \infty$ and $\frac{1}{p} + \frac{1}{q} = 1$. Then

• (i) Given any ϕ of Θ, $M(\phi)\neq L_{p}$ [$N(\phi )\neq L_{q}$].

• (ii) In order that $L_{p} \subset M(\phi)$ [$N(\phi) \subset L_{q}$], it is necessary and sufficient that $\sup_{s,t \ge1} ( \frac{(st)^{1/q}}{\phi_{st}} ) < \infty$.

• (iii) In order that $M(\phi) \subset L_{p}$ [$N(\phi) \supset L_{q}$], it is necessary and sufficient that $\Delta\phi\in L_{p}$.

• (iv) $\bigcup_{\Delta\phi\in L_{p}} M(\phi) = L_{p}$ [$\bigcap_{\Delta\phi\in L_{p}} N(\phi) = L_{q}$].

Proof

(i) Let us suppose that $M(\phi) = L_{p}$.

Then, by Lemma 1.2, there exist real numbers $r_{1}$ and $r_{2}$ ($r_{1} > 0$, $r_{2} > 0$) such that, for all x of $M(\phi)$,

$$r_{1} \Vert x\Vert _{L_{p}} \le \Vert x\Vert _{M(\phi)} \le r_{2} \Vert x\Vert _{L_{p}}. $$

Taking $x = c(\zeta)$, where $\zeta\in U_{st}$, we have that

$$r_{1}(st)^{\frac{1}{p}} \le\frac{st}{\phi_{st}} \le r_{2}(st)^{\frac{1}{p}}\quad (s,t = 1,2,3, \ldots), $$

and hence that

$$r_{1} \le\frac{(st)^{\frac{1}{q}}}{\phi_{st}} \le r_{2} \quad (s,t = 1,2,3, \ldots). $$

In view of Lemma 3.1, this implies that $M(\phi) = M(\psi)$, where $\psi= \{ (mn)^{\frac{1}{q}}\}$. Since $\Delta\psi\in M(\psi)$ by Theorem 2.2, but $\Delta\psi\notin L_{q}$, this leads to a contradiction. Hence (i) follows.

(ii) If $L_{q} \subset M(\phi)$, arguments similar to those used in the proof of (i) show that

$$ (st)^{1/q} \le K\phi_{st} \quad (s,t = 1,2,3, \ldots). $$

(3.1)

For sufficiency, we suppose that (3.1) is satisfied. Then, whenever $x \in L_{p}$ and $\zeta\in U_{st}$,

$$\sum_{m,n \in\zeta} \vert x_{mn}\vert \le \biggl( \sum_{m,n \in\zeta} \vert x_{mn}\vert ^{p} \biggr)^{\frac{1}{p}} \biggl( \sum_{m,n \in\zeta} 1 \biggr)^{\frac{1}{q}} \le \Vert x\Vert _{L_{q}}(st)^{\frac{1}{q}} < K\phi_{st}\Vert x\Vert _{L_{q}}, $$

and hence $x \in M(\phi)$. In view of (i), it follows that $L_{q} \subset M(\phi)$.

(iii) By Theorem 2.2, we have $\Delta\phi\in M(\phi)$. For sufficiency, we suppose that $\Delta\phi\in L_{p}$ and that $x \in M(\phi)$. Then $\{ u_{mn}\Delta_{11}\phi_{mn}\} \in L_{1}$ whenever $u \in L_{q}$, and it therefore follows from Lemma 2.2 that $\{ u_{mn}x_{mn}\} \in L_{1}$ whenever $u \in L_{q}$. Since $L_{p}$ is the dual of $L_{q}$ and since $M(\phi)\neq L_{p}$, it follows that $M(\phi) \subset L_{q}$.

(iv) By using (iii) we have $\bigcup_{\Delta\phi\in L_{p}} M_{\phi} \subseteq L_{p}$. Now, for obtaining the complementary relation $L_{p} \subseteq\bigcup_{\Delta\phi\in L_{p}} M_{\phi}$, let us suppose that $x \in L_{p}$. Then $\lim_{m,n \to\infty} x_{mn} = 0$, and hence there is an element u of $S(x)$ such that $\{ \vert u_{mn}\vert \}$ is a non-increasing sequence. If we take $\psi= \{ \sum_{i,j = 1,1}^{m,n} \vert u_{ij}\vert \}$, then it is easy to verify that $\psi\in \Theta$ and that $x \in M(\phi)$. Since $\Delta\psi\in L_{p}$, the complementary relation is satisfied. □

4 Application of $M(\phi)$ and $N(\phi)$ in clustering

In this section, we implement a k-means clustering algorithm by using $M(\phi)$-distance measure. Further, we apply the k-means algorithm into clustering to cluster two-moon data. The clustering result obtained by the $M(\phi)$-distance measure is compared with the results derived by the existing Euclidean distance measures ($l_{2}$).

4.1 Algorithm to compute $M(\phi)$ distance

Let $x = [x_{1},x_{2},x_{3}, \ldots,x_{n}]_{1 \times n}$ and $y = [y_{1},y_{2},y_{3}, \ldots,y_{n}]_{1 \times n}$ be two matrices of size $1 \times n$, and let $\phi_{m,n} = \phi_{1,n} = n$.

• (1) Calculate $a_{i} = \frac{1}{\phi_{1,i}}\vert x_{i} - y_{i}\vert$, $i = 1,2,3, \ldots,n$.

• (2) The $M(\phi)$-distance between x and y is d, where

  $$d = \max\{ a_{1},a_{1} + a_{2}, \ldots,a_{1} + a_{2} + \cdots+ a_{n}\}. $$

4.2 K-means clustering algorithm for $M(\phi)$-distance measure

Let $X = [x_{1},x_{2},x_{3},\ldots,x_{n}]$ be the data set.

• (1) Randomly/judiciously select k cluster centers (in this paper we choose first k data points as the cluster center $y = [x_{1},x_{2},\ldots,x_{k}]$).

• (2) By using $M(\phi)$ or $N(\phi)$ distance measure (since both are dual of each other, in application point of view, we only consider $M(\phi)$), compute the distance between each data points and cluster centers.

• (3) Put data points into the cluster whose $M(\phi)$-distance with its center is minimum.

• (4) Define cluster centers for the new clusters evolved due to steps 1-3, the new cluster centers are computed as follows: $c_{i} = \frac{1}{k_{i}}\sum_{j = 1}^{k_{i}} x_{i}$, where $k_{i}$ denotes the number of points in the ith cluster.

• (5) Repeat the above process until the difference between two consecutive cluster centers reaches less than a small number ε.

4.3 Two-moon dataset clustering by using $M(\phi)$-distance measure in k-means algorithm

Two-moon dataset is a well-known nonconvex data set. It is an artificially designed two dimensional dataset consisting of 373 data points [19]. Two-moon dataset is visualized as moon-shaped clusters (see Figure 1).

Figure 1

Original shape of two-moon dataset.

By using $M(\phi)$-distance measure in the k-means clustering algorithm, the obtained result is represented in Figure 2. In Figure 3, we represent the result obtained by using the Euclidean distance measure in the k-means algorithm (we measure the accuracy of the cluster by using the formula, accuracy = (number of data points in the right cluster/total number of data points)). The experimental result shows that cluster accuracy of $M(\phi)$-distance measure is 84.72% while $l_{2}$-distance measure’s clustering accuracy is 78.55%. Thus, $M(\phi)$-distance measure substantially improves the clustering accuracy.

Figure 2

Clustering induced by $\pmb{M(\phi)}$ -distance measure.

Figure 3

Clustering induced by Euclidean distance $\pmb{l_{2}}$ .

5 Conclusions

In this paper, we defined Banach spaces $M(\phi)$ and $N(\phi)$ with discussion of their mathematical properties. Further, we proved some of their inclusion relation. Furthermore, we applied the distance measure induced by the Banach space $M(\phi)$ into clustering to cluster the two-moon data by using the k-means clustering algorithm; the result of the experiment shows that the $M(\phi)$-distance measure extensively improves the clustering accuracy.