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

Abstract

The Fréchet $(\mbox{resp.}, (\mbox{LB})\mbox{-})$ sequence spaces $ces(p+) := \bigcap_{r > p} ces(r), 1 \leq p < \infty $ (resp. $ ces (p\mbox{-}) := \bigcup_{ 1 < r < p} ces (r), 1 < p \leq \infty),$ are known to be very different to the classical sequence spaces $ \ell_ {p+} $ (resp., $ \ell_{p_{\mbox{-}}}).$ Both of these classes of non-normable spaces $ ces (p+), ces (p\mbox{-})$ are defined via the family of reflexive Banach sequence spaces $ ces (p), 1 < p < \infty .$ The \textit{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\mbox{-}),$ which have not been considered before. Some of their properties have similarities with those of $ ces (p+), ces (p\mbox{-})$ but, they also exhibit differences. For instance, $ ces (p+)$ is isomorphic to a power series Fréhet space of order 1 whereas $ d (p+) $ is isomorphic to such a space of infinite order. Every space $ ces (p+), ces (p\mbox{-}) $ admits an absolute basis but, none of the spaces $ d (p+), d (p\mbox{-})$ have any absolute basis.