Fr\'{e}chet and (LB) sequence spaces induced by dual Banach spaces of discrete Ces\`{a}ro spaces

TL;DR

This paper investigates dual Banach sequence spaces related to discrete Cesàro spaces, revealing their structural properties, differences from classical spaces, and their unique basis characteristics.

Contribution

It introduces and analyzes the properties of the dual sequence spaces d(p+) and d(p-), which were not previously studied, highlighting their similarities and differences with Cesàro spaces.

Findings

d(p+) is isomorphic to an infinite order power series Fréchet space.

d(p-) lacks an absolute basis, unlike Cesàro spaces.

Properties of d(p+) and d(p-) differ significantly from classical sequence spaces.

Abstract

The Fr\'{e}chet (resp.\ (LB)) sequence spaces $ces(p+) := \cap_{r > p} ces(r), 1 \leq p < \infty $ (resp.\ $ ces (p-) := \cup_{ 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_{-}}).$ 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 Ces\`{a}ro 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 Fr\'{e}chet 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.