A Survey On Some Paranormed Sequence Spaces

Abstract

This paper presents a survey of most of the known fundamental
results involving the sequence spaces ℓ(p), c0(p), c(p) and ℓ1(p), w0(p), w(p)
and w1(p), f0(p) and f(p). These spaces are generalizations of the classical
BK spaces ℓp, c0, c and ℓ1, the spaces wp
0, wp and wp
1 of sequences that
are strongly summable to zero, strongly summable and strongly bounded with
index p by the Ces`aro method of order 1, and of almost null and almost
convergent sequences, respectively. The results inlude the basic topological
properties of the generalized spaces, the complete lists of their known α–,
β–, γ–, functional and continuous duals, and the characterizations of many
classes of matrix transformations between them, in particular, the complete
list of characterizations of matrix transformations between the spaces ℓ(p),
c0(p), c(p) and ℓ1(p). Furthermore, a great number of interesting special
cases are included. The presented results cover a period of four decades. They
are intended to inspire the inreasing number of researchers working in related
topics, and to provide them with a comprehensive collection of results they
may find useful for their work.