Fréchet and (LB) sequence spaces induced by dual Banach spaces of discrete Cesàro spaces

Abstract

The Fréchet (resp.\ (LB)) sequence spaces $ces(p+):={\cap }_{r>p}ces(r),1\le p<\infty$ (resp.\ $ces(p-):={\cup }_{1<r<p}ces(r),1<p\le \infty ),$ are known to be very different to the classical sequence spaces ${\ell }_{p+}$ (resp., ${\ell }_{p_{-}}).$ Both of these classes of non-normable spaces $ces(p+),ces(p-)$ are defined via the family of reflexive Banach sequence spaces $ces(p),1<p<\infty .$ The dual Banach spaces $d(q),1<q<\infty ,$ of the discrete Cesàro spaces $ces(p),1<p<\infty ,$ were studied by G.\ Bennett, A.\ Jagers and others. Our aim is to investigate in detail the corresponding sequence spaces $d(p+)$ and $d(p-),$ which have not been considered before. Some of their properties have similarities with those of $ces(p+),ces(p-)$ but, they also exhibit differences. For instance, $ces(p+)$ is isomorphic to a power series Fréchet space of order 1, whereas $d(p+)$ is isomorphic to such a space of infinite order. Every space $ces(p+),ces(p-)$ admits an absolute basis but, none of the spaces $d(p+),d(p-)$ have any absolute basis.