2.3. The spaces ω, l∞, c, c0, lp

A sequence space is a set of scalar sequences (real or complex) which is closed under coordinate-wise addition and scalar multiplication. In other words, a sequence space is a linear subspace of the space ω of all complex sequences, that is,

$$
\omega = \{ \mathbf{x} = (\mathbf{x}\_k) : \mathbf{x}\_k \in \mathbb{R} \text{ or } \mathbb{C} \}.
$$

The space l∞: The space l<sup>∞</sup> of bounded sequences is defined by

$$\left\{ x = (\mathfrak{x}\_k) : \sup\_k |\mathfrak{x}\_k| < \infty \right\}$$

The spaces c: The spaces c and c<sup>0</sup> of convergent and null sequences are given by

$$\left\{\mathbf{x} = (\mathbf{x}\_k) : \lim\_{k} \mathbf{x}\_k = l, l \in \mathbb{C}\right\}$$

The space c0: The space c<sup>0</sup> of all sequences converging to 0 is given by

$$\left\{\mathbf{x} = (\mathbf{x}\_k) : \lim\_{k} \mathbf{x}\_k = \mathbf{0} \right\}$$

The space lp: The space lp of absolutely p-summable sequences is defined by

$$\left\{\mathbf{x} = (\mathbf{x}\_k) : \sum\_k \left| \mathbf{x}\_k \right|^p < \infty \right\} , (0 < p < \infty)$$

The spaces l∞, c, and c<sup>0</sup> are Banach spaces with the norm,

$$\|\mathfrak{x}\|\_{\ast} = \sup\_{k} |\mathfrak{x}\_{k}|$$

The space lp is a Banach space with the norm,

$$\|\mathfrak{x}\|\_{p} = \left(\sum\_{k} |\mathfrak{x}\_{k}|^{p}\right)^{\frac{1}{p}}, 1 \le p < \infty$$

### 2.4. Cauchy sequence

originated from the attempts of mathematicians to assign limits to divergent sequences. The classical summability theory deals with the generalization of the convergence of sequences or series of real or complex numbers. The idea is to assign a limit of some sort to divergent sequences or series by considering a transform of a sequence or series rather than the original

The earliest idea of summability theory was perhaps contained in a letter written by Leibnitz to C. Wolf (1713) in which he attributed the sum 1/2 to the oscillatory series �1+1�1 + …. Frobenius in (1880) introduced the method of summability by arithmetic means, which was generalized by Cesàro in (1890) as the (C,K) method of summability. Toward the end of the nineteenth century, study of the general theory of sequences and transformations on them attracted mathematicians, who were chiefly motivated by problems such as those in summability theory, Fourier series, power series and system of equations with infinitely many variables. Presenting some basic definitions and notations that are involved in the present work, the author proposes to give a brief resume of the hitherto obtained results against the background

of which the main results studied in the present chapter suggest themselves.

Here, we state a few conventions which will be used throughout the chapter.

<sup>α</sup> f nð Þ, we mean the sum of all values of f nð Þ for which α ≤ n ≤ β. In the case β < α, then we

<sup>k</sup>¼<sup>1</sup> xk and we shall sometimes write as <sup>P</sup>xk

Summations are over 0, 1, 2, …, when there is no indication to the contrary. If ð Þ¼ xk ð Þ x1; x2;…

A sequence space is a set of scalar sequences (real or complex) which is closed under coordinate-wise addition and scalar multiplication. In other words, a sequence space is a linear

<sup>k</sup> xk we mean <sup>P</sup><sup>∞</sup>

sequence or series.

76 Matrix Theory-Applications and Theorems

2. Notations and symbols

2.1. Symbols N, C, R and A

The symbols are denoted as follows:

A: The infinite matrix ð Þ ank , nð Þ ; k ¼ 1; 2;… .

N: Set of non-negative integers.

C: Set of complex numbers.

2.2. Summation convention

is a sequence of terms, then, by P

2.3. The spaces ω, l∞, c, c0, lp

incase where no possible confusion arises.

subspace of the space ω of all complex sequences, that is,

R: Set of real numbers.

take this to be zero.

By P<sup>β</sup>

A sequence <sup>x</sup> <sup>¼</sup> ð Þ xk is called a Cauchy sequence if and only if <sup>∣</sup>xn � xm<sup>∣</sup> ! <sup>0</sup> ð Þ <sup>m</sup>; <sup>n</sup> ! <sup>∞</sup> that is for any <sup>e</sup> <sup>&</sup>gt; 0, there exists <sup>N</sup> <sup>¼</sup> <sup>N</sup>ð Þ<sup>E</sup> such that <sup>∣</sup>x<sup>n</sup> � xm<sup>∣</sup> <sup>&</sup>lt; <sup>E</sup> for all n, m <sup>≥</sup> <sup>N</sup>. By <sup>C</sup>, we denote the space of all Cauchy sequences, that is,

$$\mathfrak{C}: \{ \mathfrak{x} = (\mathfrak{x}\_k) : |\mathfrak{x}^n - \mathfrak{x}^m| \to 0 \text{ as } n, m \to \infty \}$$

### 2.5. FK-space

A sequence space X is called an FK-space if it is a complete linear metric space with continuous coordinates pn : X ! C defined by pnð Þ¼ x xn for all x∈ X and every n∈ N [1, 2].

### 2.6. BK-space

A BK-space is a normed FK-space, that is, a BK-space is a Banach space with continuous coordinates [3–6].
