A Survey on some Paranormed Sequence Spaces

Eberhard Malkowsky, Feyzi Başar

This paper presents a survey of most of the known fundamental results involving the sequence spaces $\ell(p)$, $c_0(p)$, $c(p)$ and $\ell_\infty(p)$, $w_0(p)$, $w(p)$ and $w_\infty(p)$, $f_0(p)$ and $f(p)$. These spaces are generalizations of the classical $BK$ spaces $\ell_p$, $c_0$, $c$ and $\ell_\infty$, the spaces $w^p_0$, $w^p$ and $w^p_\infty$ f sequences that are strongly summable to zero, strongly summable and strongly bounded with index $p$ by the Cesàro 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 $\alpha-$, $\beta-$, $\gamma-$, 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 $\ell(p)$, $c_0(p)$, $c(p)$ and $\ell_\infty(p)$. 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.