Abstract Ces\`aro Spaces. I. Duality

TL;DR

This paper explores the duality of abstract Cesàro spaces, generalizing known results for sequence and function spaces, and reveals significant differences in duality behavior between finite and infinite intervals.

Contribution

It provides a simple, elementary method to determine the duals of abstract Cesàro spaces, highlighting the general nature of the duality problem beyond specific cases.

Findings

Duals of abstract Cesàro spaces are characterized in a general setting.

Elementary proofs simplify understanding of Cesàro space duality.

Distinct duality phenomena occur between finite and infinite intervals.

Abstract

We study abstract Ces\`aro spaces $CX$, which may be regarded as generalizations of Ces\`aro sequence spaces $ces_p$ and Ces\`aro function spaces $Ces_p(I)$ on $I = [0,1]$ or $I = [0,\infty)$, and also as the description of optimal domain from which Ces\`aro operator acts to $X$. We find the dual of such spaces in a very general situation. What is however even more important, we do it in the simplest possible way. Our proofs are more elementary than the known ones for $ces_p$ and $Ces_p(I)$. This is the point how our paper should be seen, i.e. not as generalization of known results, but rather like grasping and exhibiting the general nature of the problem, which is not so easy visible in the previous publications. Our results show also an interesting phenomenon that there is a big difference between duality in the cases of finite and infinite interval.