1. Preliminaries, background and notation

In several branches of analysis, for instance, the structural theory of topological vector spaces, Schauder basis theory, summability theory, and the theory of functions, the study of sequence spaces occupies a very prominent position. There is an ever-increasing interest in the theory of sequence spaces that has made remarkable advances in enveloping summability theory via unified techniques effecting matrix transformations from one sequence space into another.

Thus, we have several important applications of the theory of sequence spaces, and therefore, we attempt to present a survey on recent developments in sequence spaces and their different kinds of duals.

In many branches of science and engineering, we deal with different kinds of sequences and series, and when we deal with these, it is important to check their convergence. The use of infinite matrices is of great importance, we can bring even the bounded or divergent sequences and series in the domain of convergence. So we can say that the theory of sequence spaces and their matrix maps is the bigger scale to measure the convergence property. Summability can be roughly considered as the study of linear transformations on sequence spaces. The theory

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 sequence or series.

ω ¼ x ¼ ð Þ xk f g : xk ∈ R or C :

( )

k

k

j j xk <sup>p</sup> <sup>&</sup>lt; <sup>∞</sup>

∥x∥<sup>∞</sup> ¼ sup k ∣xk∣

� �

� �

k

jxkj < ∞

Nature of Phyllotaxy and Topology of H-matrix http://dx.doi.org/10.5772/intechopen.74676 77

xk ¼ l; l ∈ C

xk ¼ 0

,ð Þ 0 < p < ∞

x ¼ ð Þ xk : sup

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

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

x ¼ ð Þ xk :

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

The space lp is a Banach space with the norm,

space of all Cauchy sequences, that is,

2.4. Cauchy sequence

2.5. FK-space

2.6. BK-space

coordinates [3–6].

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

<sup>∥</sup>x∥<sup>p</sup> <sup>¼</sup> <sup>X</sup>

k

j j xk <sup>p</sup> !<sup>1</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

<sup>C</sup> : <sup>x</sup> <sup>¼</sup> ð Þ xk : <sup>j</sup>x<sup>n</sup> � xm f g j ! 0 as <sup>n</sup>; <sup>m</sup> ! <sup>∞</sup>

A sequence space X is called an FK-space if it is a complete linear metric space with continuous

A BK-space is a normed FK-space, that is, a BK-space is a Banach space with continuous

coordinates pn : X ! C defined by pnð Þ¼ x xn for all x∈ X and every n∈ N [1, 2].

p

, 1 ≤ p < ∞

X k

( )

x ¼ ð Þ xk : lim

x ¼ ð Þ xk : lim

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

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.
