On the dual of Ces\`aro function space
Anna Kami\'nska, Damian Kubiak

TL;DR
This paper characterizes the dual space of Cesàro function spaces using an isometric representation, introduces the concept of essential -concave majorant, and explores geometric properties of these spaces.
Contribution
It provides a new isometric representation of the dual space of Cesàro function spaces and introduces the notion of essential -concave majorant.
Findings
Every slice of the unit ball has diameter 2
Cesro spaces lack the Radon-Nikodym property
Cesro spaces are not locally uniformly convex or dual spaces
Abstract
The goal of this paper is to present an isometric representation of the dual space to Ces\`aro function space , , induced by arbitrary positive weight function on interval where . For this purpose given a strictly decreasing nonnegative function on , the notion of essential -concave majorant of a measurable function is introduced and investigated. As applications it is shown that every slice of the unit ball of the Ces\`aro function space has diameter 2. Consequently Ces\`aro function spaces do not have the Radon-Nikodym property, are not locally uniformly convex and they are not dual spaces.
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Taxonomy
TopicsAdvanced Banach Space Theory · Advanced Harmonic Analysis Research · Optimization and Variational Analysis
